1  Introduction

1.1  How Many Digits Should Be Used?

Here are some simple rules that apply whenever you are writing down a number:

1.

Use many enough digits to avoid unintended loss of information.

2.

Use few enough digits to be reasonably convenient.

Important note: The previous two sentences tell you everything you need to know for most purposes, including real-life situations as well as academic situations at every level from primary school up to and including introductory college level. You can probably skip the rest of this document.

3.

When using a calculator, it is good practice to leave intermediate results in the machine. This is simultaneously more accurate and more convenient than writing them down and then keying them in again. For details on this, see section 7.11.

Seriously: The primary rule is to use plenty of digits. You hardly even need to think about it. Too many is vastly better than too few. To say the same thing the other way: If you ever have more digits than you need and they are causing major inconvenience, then you can think about reducing the number of digits.

If you want more-detailed guidance, some ultra-simple procedures are outlined in section 2. If you want even more guidance, the details on how to do things right are discussed in section 8.2. For a discussion of the effect of roundoff, see section 8.6. For a discussion of why using “sig figs” is insane, see section 1.3. There is also a complete table of contents.

1.2  What About Uncertainty?

4.

In many cases, when you write down a number, you need not and should not associate it with any notion of uncertainty.

• One way this can happen is if you have a number with zero uncertainty. If you roll a pair of dice and observe five spots, the number of spots is 5. This is a raw data point, with no uncertainty whatsoever. So just write down the number. Similarly, the number of centimeters per inch is 2.54, by definition, with no uncertainty whatsoever. Again: just write down the number.
• Another possibility is that there is a cooked data blob, which in principle must have “some” uncertainty, but the uncertainty is too small to be interesting. It is insignificant. It is unimportant. It is immaterial. There are plenty of situations a moderately rough approximation is sufficient. There are even some situations where an extremely rough approximation is called for, as in so-called “Fermi” problems. This point is discussed in reference 1.

  Along the same lines, here is a less-extreme example that arises in the introductory chemistry class. Suppose the assignment is to balance the equation for the combustion of gasoline, namely

  a C8H18 + b O2   →   x CO2 + y H2O

  (1)

  by finding numerical values for the coefficients a, b, x, and y. The conventional answer is (a, b, x, y) = (2, 25, 16, 18). The outcome of the real reaction must have “some” uncertainty, because there will generally be some nonidealities, including the presence of other molecules such as CO or C₆₀, not to mention NO₂ or whatever. However, my point is that we don’t necessarily care about these nonidealities. We can perfectly well find the idealized solution to the idealized equation and postpone worrying about the nonidealities and uncertainties until much, much later.

  As another example, suppose you use a digital stopwatch to measure some event, and the reading is 1.234 seconds. We call this number the indicated time, and we distinguish it from the true time of the event, as discussed in section 5.5. In principle, there is no chance that the indicated time will be exactly equal to the true time (since true time is a continuous variable, whereas the indicated time is quantized). However, in many cases you may decide that it is close enough, in which case you should just write down the indicated reading and not worry about the quantization error.

5.

The best way to understand uncertainty is in terms of probability distributions. Such things are discussed in section 3.1 and in more detail in reference 2.

6.

Suppose we wish to describe a probability distribution, and further suppose it is a simple one-dimensional distribution, such as the one shown in figure 1. (There’s a lot going on in this figure; for details, see reference 2.) Any Gaussian distribution (also called a normal distribution, or simply a Gaussian) can be described in terms of two numbers, namely the nominal value and the uncertainty. One good notation for this uses an expression of the form 1.234 ± 0.055, where the first numeral (in this case 1.234) represents the nominal value, and the second numeral (in this case 0.055) represents the width of the distribution, i.e. the absolute uncertainty. (Typically, but not necessarily, the standard deviation is used to quantify the width. The region spanned by ± one standard deviation, i.e. ± 0.055, is indicated by yellow shading in the diagram.) This notation works for algebraic symbols, too: A±B.

Figure 1: Gaussian Distribution, 1.234±0.55

For numerical (non-algebraic) values, you can write something of the form 1.234(55), where the number in parentheses indicates the uncertainty. The place-value is such that the last digit of the uncertainty lines up with the last digit of the nominal value. Therefore 1.234(55) is just a more-compact way of writing 1.234 ± 0.055.

When a number has been subjected to rounding, the roundoff error is at most a half-count in the last decimal place. If this is the dominant contribution to the uncertainty, we can denote it by 543.2[½]. Beware that the distribution of roundoff errors is nowhere near Gaussian, as discussed in section 8.3.

In cases where you are uncertain about the uncertainty, as sometimes happens, you can write 543.2(x) which represents a “few” counts of uncertainty in the last place. This stands in contrast to 543.2(?) which usually means that the entire value is dubious, i.e. some chance of a gross error (such as measuring the length instead of the width).

If you wish to describe the uncertainty in relative terms (as opposed to absolute terms), it can be expressed using percentages, parts per thousand, parts per million, or something like that, e.g. 2900 ± 0.13% or equivalently 2900 ± 1300ppm.

(Note that in the expression 1.234 ± 0.055 we have two separate numbers represented by two separate numerals, which makes sense. This stands in contrast to the “sig figs” notation, which tries to represent two numbers using a single numeral, which is a very bad idea.)

7.

Suppose we have a distribution over x and a distribution over y. If the two distributions are known to be uncorrelated, you can get away with describing each one separately, with one nominal value and one width apiece. However, that only works in certain special situations. More generally, you need to talk about the distribution over x and y jointly. Knowing the width in the x-direction and the width in the y-direction is nowhere near sufficient to give a complete description of the joint distribution, because correlations can produce an elongated, cockeyed distribution, as in figure 2. For details, see section 7.7, section 7.23, and section 9.3.

Figure 2: Correlated Uncertainties

If you have N variables that are statistically independent and Gaussian distributed, you can describe the uncertainty in terms of N variances. (The standard deviation is the square root of the variance.)

If you have N variables that are correlated, to describe an N-dimensional Gaussian distribution requires a covariance matrix which has N2 entries. The plain old variances are the diagonal elements of the covariance matrix, and they don’t tell the whole story, especially when N is large.

In the real world, there are commonly nontrivial correlations involving several variables – or several thousand variables. In other words, there are lots of nontrivial off-diagonal matrix elements in the covariance matrix.

As a corollary, you should not become too enamored of the notation 1.234 ± 0.055 or 1.234(55), because that only allows you to keep track of the N variances, not the N² covariances.

8.

Let us continue with the stopwatch example that was introduced in item 4. Suppose we make two observations. The first reading is 1.234 seconds, and the second reading is just the same, namely 1.234 seconds. Meanwhile, however, you may believe that if you repeated the experiment many times, the resulting set of readings would have some amount of scatter, namely ± 0.01 seconds. The two observations that we actually have don’t show any scatter at all, so your estimate of the uncertainty remains hypothetical and theoretical. Theoretical information is still information, and should be written down in the lab book, plain and simple. For example, you might write a sentence that says “Intuition suggests the timing data is reproducible ± 0.01 seconds.” It would be even better to include some explanation of why you think so. The principle is simple: Write down what you know. Say what you mean, and mean what you say.

9.

The same principle applies to the indicated values. The recommend practice is to write down each indicated value, as-is, plain and simple.

You are not trying write down the true values. You don’t know the true values (except insofar as the indicated values represent them, indirectly), as discussed in section 5.5. You don’t need to know the true values, so don’t worry about it. The rule is: Write down what you know. So write down the indicated value.

Also: You are not obliged to attribute any uncertainty to the numbers you write down. Normal lab-book entries do not express an uncertainty using A±B notation or otherwise, and they do not “imply” an uncertainty using sig figs or otherwise. We are always uncertain about the true value, but we aren’t writing down the true value, so that’s not a concern. For an example of how this works, see table 5 in section 6.4.

Some people say there must be some uncertainty “associated” with the number you write down, and of course there is, indirectly, in the sense that the indicated value is “associated” with some range of true values. We are always uncertain about the true value, but that does not mean we are uncertain about the indicated value. These things are “associated” ... but they are not the same thing.

In a well-designed experiment, things like readability and quantization error usually do not make a large contribution to the overall uncertainty anyway, as discussed in section 5.8. Please do not confuse such things with “the” uncertainty.

10.

There must be some “calibration rule” that connects each indicated value to the corresponding range of true values. Be sure to write this rule in the lab book somewhere, unless it is super-obvious.

It suffices to write down the rule just once; you do not need to restate the rule every time you take a reading. Later, when you are analyzing the data, you can apply the rule to each of the readings.¹ As a familiar example of such a rule, you might say “all readings are uncertain due to Poisson statistics”. For another familiar example, see section 6.1.

11.

Before you report “the” uncertainty in your results, make sure you have identified all significant contributions to the uncertainty. It does no good to carefully calculate one contribution while overlooking other contributions. See section 5.8. Also, watch out for correlated uncertainties. See section 7.18.

12.

When describing a distribution, state what family of distributions you are talking about, unless this is obvious from context. Examples include Gaussian, square, triangular, Bernoulli, Poisson, et cetera. See section 8.5 and section 13.8 for why this is important. See reference 2 for a discussion of some common distributions, and the relationships between them.

13.

It is usually a good practice to keep all the original data. When reading an instrument, read it as precisely as the instrument permits, and write down the reading “as is” ... without any conversions, any roundoff, or anything else. See section 8.4 for details (including the rare and tricky possible exceptions).

1.3  What About Significant Figures?

14.

No matter what you are trying to do, significant figures are the wrong way to do it.

When writing, do not use the number of digits to imply anything about the uncertainty. If you want to describe a distribution, describe it explicitly, perhaps using expressions such as 1.234±0.055, as discussed in section 1.2.

When reading, do not assume the number of digits tells you anything about the overall uncertainty, accuracy, precision, tolerance, or anything else, unless you are absolutely sure that’s what the writer intended ... and even then, beware that the meaning is very unclear.

Significant-digit dogma destroys your data and messes up your thinking in many ways, including:

1. Given a distribution that can be described by an expression such as A±B, such as 1.234±0.055, converting it to sig figs gives you an excessively crude and erratic representation of the uncertainty, B. See section 8.6.3 and especially section 17.5.
2. Converting to sig figs also causes excessive roundoff error in the nominal value, A. This is a big problem. See section 7.12 for a concrete example.
3. Sig figs cause people to misunderstand the distinction between roundoff error and uncertainty. See section 7.12 and section 6.4.
4. Sig figs cause people to misunderstand the distinction between uncertainty and significance. See section 14, especially section 14.3.
5. Sig figs cause people to misunderstand the distinction between the indicated value and the corresponding range of true values. See section 5.5.
6. Sig figs cause people to misunderstand the distinction between distributions and numbers. Distributions have width, whereas numbers don’t. Uncertainty is necessarily associated with some distribution, not with any particular point that might have been drawn from the distribution. See section 3.1, section 6.4, and reference 2.
7. As a consequence, sig figs make people hesitate to write down numbers. They think they need to know the amount of supposedly “associated” uncertainty before they can write the number, when in fact they don’t. Very commonly, there simply isn’t any “associated” uncertainty anyway, as discussed in item 4.
8. Sig figs weaken people’s understanding of the axioms of the decimal numeral system. See section 17.5.7.
9. Sig figs provide no guidance as to the appropriate decimal representation for repeating decimals such as 80 ÷ 81, or irrational numbers such as √2 or π.
10. Sig figs give people the idea that N nominal values should be associated with N uncertainties, which is just crazy. In fact the number of uncertainties scales like (N² + N)/2, as discussed in section 9.3.
11. As a related point, sig figs is predicated on «propagation of error» which in turn assumes that things are uncorrelated, when in fact there are often lots of correlations. This causes the error-estimates to fail without warning.
12. The sig figs approach cannot possibly apply to algebraic variables such as A±B, so you are going to have to learn the A±B representation anyway. Having learned it, you might as well use it for decimal numerals such as 1.234±0.055. See section 17.5.5.
13. Sig figs cause people to misunderstand the distinction between representation of uncertainty and propagation of uncertainty. See section 7.1.
14. Et cetera

For a more detailed discussion of why sig figs are a bad idea, see section 17 and reference 3

2  Pedagogical Digression – Extreme Simplifications

2.1  Postponing Uncertainty

In an introductory chemistry class, you should start with some useful chemistry ideas, such as atoms, molecules, bonds, energy, atomic number, nucleon number, etc. — without worrying about uncertainty in any form, and double-especially without introducing ideas (such as sig figs) that are mostly wrong and worse than useless.

Roundoff procedures are necessary, so learn that. Scientific notation is worthwhile, so learn that. The “sig figs” rules that you find in chemistry books are not necessary and are not worthwhile, so the less said about them, the better.

In place of the “sig figs” rules, you can use the following guidelines:

• Keep all the original data. Do not round off the original data. See item 13 above.
• In the introductory class, the following “house rules” apply:

  Basic 3-digit rule: For a number in scientific notation, the rule is simple: For present purposes, you are allowed to round it off to three digits (i.e. two decimal places).

  Example: 1.23456×10⁸ may be rounded to 1.23×10⁸

  For a number not in scientific notation, the rule is almost as simple: convert to scientific notation, then apply the aforementioned 3-digit rule. (Afterwards, you can convert back, or not, as you wish.)

  The point of these rules is to limit the amount of roundoff error. As a corollary, you are allowed to keep more than three digits if you wish, for any reason, or for no reason at all. This is makes sense because it introduces even less roundoff error. As another corollary, trailing zeros may always be rounded off, since that introduces no roundoff error at all.

  Example: 1.80 may be rounded to 1.8, since that means the same thing. Conversely 1.8 can be represented as 1.80, 1.800, 1.8000000, et cetera.

  These rules apply to intermediate steps as well as to final results.

  These “house rules” apply unless/until you hear otherwise. They tell you what is considered significant at the moment. As such, they have zero portability outside the introductory class, and even within this class we will encounter some exceptions (as in section 7.8 for example). Still, for now three digits is enough. There is method to this madness, but now is not the time to worry about it. We have more important things to worry about.

These rules differ in several ways from the “sig figs” rules that you often see in introductory chemistry textbooks.

• First of all, these rules are much simpler.
• Secondly, the conceptual basis is different. The “sig figs” rules in the textbooks are a crude attempt to keep track of uncertainty. Despite the name, those “sig figs” rules do not even attempt to express significance. (See section 14.3 for details on this.) The roundoff rules given here are actually based on significance, i.e. on the importance of the numbers and how they will be used downstream. They say nothing about the accuracy, precision, or uncertainty of the numbers. See section 17.5.2.

  This is important because of the following contrast:

  Every time you write down a number, you have to write down a definite number of digits, and this almost always involves rounding off. Therefore you must have a roundoff rule or some similar guidance as to how many digits are needed.

  There are many cases when you want to write down a number without any indication of uncertainty.

  A roundoff rule is necessary and harmless (unless abused).

  A “sig figs” rule that forces a connection between the number of digits and the uncertainty is unnecessary and harmful.

• Thirdly, these rules (unlike the textbook “sig figs” rules) permit you to get rid of trailing zeros. This is important because it means these rules are consistent with the axioms of the decimal number system that we all learned in 3rd grade and reviewed every year since then: 1.80 is a rational number. It is by definition equal to 180/100, which when written in lowest terms is 9/5. Similarly 1.800 is by definition equal to 1800/1000, which is also exactly equal to 9/5.

Remember, these are roundoff rules. Do not confuse roundoff with uncertainty. Roundoff error is just one contribution to the overall uncertainty. Knowing how much roundoff has occurred gives you a lower bound on the overall uncertainty, but this lower bound is rarely the whole story. Looking at the number of digits in a numeral gives you an upper bound on how much roundoff has occurred. (This is not a tight upper bound, since the number might be exact, i.e. no roundoff at all.) At the end of the day, the number of digits tells you nothing about the overall uncertainty.

Roundoff error is in the category of things that we generally do not need to know very precisely, so long as it is small enough. Uncertainty is not in this category, for reasons discussed in section 4.4.

2.2  Range of Numbers (as a Simplified Distribution)

As discussed in section 3.1, an expression such as 1.234±0.055 does not represent a number, but rather a distribution over numbers, i.e. a probability distribution. Unfortunately, people sometimes use sloppy shorthand shorthand expressions, perhaps referring to the «random variable» x or the «uncertain quantity» x, such that x = 1.234±0.055. Beware that this shorthand causes endless confusion. When in doubt, it is best to think of 1.234±0.055 as describing a distribution.

As a compromise, in the all-too-common situation where somebody wants to learn about uncertainty but doesn’t have a very strong background in probability, we can simplify things by talking about an interval or equivalently a range of numbers.

Note: “interval” is an official mathematical term, while “range of numbers” is more likely to be understood by non-experts.

Working with intervals is easier than working with distributions. You can draw a range of numbers on the number line much more easily than you can draw a probability distribution. It is not an ideal solution, but it is a way to get started. (In contrast, the idea of so-called «random variables» is not good, not as a starting point or anything else.)

In order of decreasing power, sophistication, and reliability:

In order of decreasing simplicity:

In any case, the fundamental point is that some situations cannot be described by a single “number”. Instead, they are better described by a whole range of numbers that are consistent with our knowledge of the situation. The extent of the range expresses the uncertainty. One way to explain this is in terms of hedging a bet. If you roll a pair of dice, the most likely outcome is 7 ... but that outcome occurs less than 17% of the time. If you want to be right more than half of the time, you can’t do it by betting on any single number, but you can do it by betting on a range of numbers.

So, if you want, you can simplify the following discussion (with only a modest reduction in correctness) by crossing out every mention of “probability distribution” and replacing it with “range of numbers”.

*   Contents

• 1  Introduction
  • 1.1  How Many Digits Should Be Used?
  • 1.2  What About Uncertainty?
  • 1.3  What About Significant Figures?
• 2  Pedagogical Digression – Extreme Simplifications
  • 2.1  Postponing Uncertainty
  • 2.2  Range of Numbers (as a Simplified Distribution)
• 3  Foundations and Prerequisites
  • 3.1  What Is Uncertainty?
• 4  The Importance of Uncertainty
  • 4.1  Sometimes It’s Unimportant or Nonexistent
  • 4.2  Separate versus Bundled Parameters
  • 4.3  Raw Data Points versus Cooked Data Blobs
  • 4.4  Weighing the Evidence
  • 4.5  Significant Figures, or Not
• 5  Fundamental Notions of Uncertainty
  • 5.1  Some Things are Certain, and Some are Uncertain
  • 5.2  Uncertainty ≡ Probability Distribution
  • 5.3  Analog Measurements
  • 5.4  Digital Measurements
  • 5.5  Indicated Value versus True Values
  • 5.6  Uncertainty ± Error ≠ Mistake
  • 5.7  Probably Almost Correct
  • 5.8  Identify All Contributions to the Uncertainty
  • 5.9  Empirical Distributions versus Theoretical Distributions
  • 5.10  Terminology and Notation
  • 5.11  How Not to Represent Uncertainty
  • 5.12  Uncertainty, Standard Deviation, Confidence Limits, etc.
• 6  Reading an Instrument
  • 6.1  Example: Reading a Meter
  • 6.2  Example: Reading a Burette using Simple Interpolation
  • 6.3  Example: Reading a Burette using Fancy Interpolation
  • 6.4  Analyzing an Ensemble of Readings
  • 6.5  Standard Deviation versus Uncertainty versus Error
  • 6.6  Example: Decimal Interpolation Between Graduations
  • 6.7  Readability is Only Part of the Uncertainty
• 7  Propagation of Uncertainty
  • 7.1  Overview
  • 7.2  Simple Example: Multi-Step Arithmetic
  • 7.3  Guard Digits (Preview)
  • 7.4  Example: Beyond First Order: 1 to the 40th Power
  • 7.5  Example: Beyond First Order: Momentum and Energy
  • 7.6  Example: Non-Differentiable: Time = Distance / Rate
  • 7.7  Example: Correlated Data: Charge-to-Mass Ratio
  • 7.8  Example: Solving a Quadratic Polynomial for the pH
  • 7.9  Example: Solving a Cubic Polynomial for the pH
  • 7.10  Another Example: Multi-Step Relativity
    • 7.10.1  Correct Direct Calculation
    • 7.10.2  Unsuccessful Double-Precision Direct Calculation
    • 7.10.3  Gross Failure: Sig Figs
    • 7.10.4  Algebraic Simplification
  • 7.11  Discussion: Loss of Significance
  • 7.12  Example: Signal Averaging: Extracting a Signal from Noise
  • 7.13  Example: The Effect of Roundoff Error
  • 7.14  Crank Three Times™
  • 7.15  Another Example: Magnesium Mass, Preliminary Attempt
  • 7.16  Magnesium Mass, Monte Carlo Solution
  • 7.17  Exercise
    • 7.17.1  Discussion: Mg Mass
  • 7.18  Reporting Correlated Uncertainties
  • 7.19  Another Example: Solving a Quadratic via Monte Carlo
  • 7.20  Step-by-Step First-Order Propagation of Uncertainty
    • 7.20.1  Disclaimers
    • 7.20.2  Step-by-Step Propagation Rules
    • 7.20.3  More Disclaimers
  • 7.21  OK Example: Step-by-Step Propagation
  • 7.22  Amplification of Uncertainty
  • 7.23  Counterexample: Step-by-Step Propagation
  • 7.24  Curve Fitting – Least Squares and Otherwise
  • 7.25  Choosing a Method of Propagation
• 8  How Much Accuracy Is Enough? How Much Data Is Enough?
  • 8.1  Why is this hard?
  • 8.2  How To Do It Right – Basic Recommendations
  • 8.3  Indicating Roundoff and Truncation
  • 8.4  Keep All the Original Data
  • 8.5  Report the Form of the Distribution
  • 8.6  The Effect of Rounding
    • 8.6.1  Rounding Off a Gaussian
    • 8.6.2  Rounding Off a Histogram
    • 8.6.3  Nonuniform Relative Error
    • 8.6.4  Roundoff Error is Not Necessarily Random
    • 8.6.5  Correlations
  • 8.7  Comparisons, with or without Rounding
  • 8.8  Guard Digits
  • 8.9  «Final» Results : Guard Digits; Actual Significance
  • 8.10  Too Many Digits, Oh My!
  • 8.11  How To Avoid Introducing Sig Figs
  • 8.12  Psychological Issues
  • 8.13  How To Survive an Attack of Sig Figs
  • 8.14  Sensitivity Analysis, On-Line Analysis, and Cross-Checking
• 9  Correlation and Covariance
  • 9.1  Example: Electron Charge-to-Mass Ratio
  • 9.2  Example: Common Mode versus Differential Mode
  • 9.3  Example: Mass and Gravitation (I)
  • 9.4  Dealing with Correlations
  • 9.5  Example: Mass and Gravitation (II)
• 10  “Correctness” versus Agreement
  • 10.1  Your Data is Your Data
  • 10.2  Measurement Errors versus Modeling Errors
• 11  Samples, Sets, Groups, or Clusters of Observations
  • 11.1  Particles and Clusters
  • 11.2  Estimators
  • 11.3  Terminology
  • 11.4  Mean, Variance and Standard Deviation
• 12  Contributions to the Uncertainty
  • 12.1  Statistical Fluctuations
  • 12.2  Roundoff Error
  • 12.3  Series Truncation Error
  • 12.4  Ignorance versus Randomness
• 13  Categories of Uncertainty – and Related Notions
  • 13.1  Tolerance
  • 13.2  Precision
  • 13.3  Accuracy
  • 13.4  Readability and Reproducibility
  • 13.5  Systematic versus Non-Systematic Uncertainty
  • 13.6  Scatter
  • 13.7  “Experimental Error”
  • 13.8  Other Limits to the Notion of Uncertainty
• 14  Significance
  • 14.1  Significant ≡ Worth Knowing
  • 14.2  Users Decide
  • 14.3  Significance versus Uncertainty
• 15  Analysis Plus Synthesis – Closing the Loop
• 16  The Definition of Probability
• 17  More than You Ever Wanted to Know about Sig Figs
  • 17.1  Supposed Goals
  • 17.2  OK: Sig figs ↔ Place Value
  • 17.3  Mostly OK: Sig figs ↔ Roundoff
  • 17.4  Abomination: Sig Figs ↔ Uncertainty
    • 17.4.1  If You Mean Place Value, Say Place Value
    • 17.4.2  Observations versus Distributions
    • 17.4.3  Example of Sig-Figs and Roundoff: Calculating Density
    • 17.4.4  Uncertainty, Insignificance, and Guard Digits
    • 17.4.5  Bogus Apology: “Rough Uncertainty”
  • 17.5  Excessively Rough Uncertainty
    • 17.5.1  Sectarian Differences
    • 17.5.2  Exact Numbers Are Not Describable Using Sig Figs
    • 17.5.3  Null Experiments Are Not Describable Using Sig Figs
    • 17.5.4  Some Other Inexact Numbers Are Not Describable Using Sig Figs
    • 17.5.5  Algebra
    • 17.5.6  Units Won’t Solve the Problem
    • 17.5.7  Decimals Are Exact By Definition
    • 17.5.8  Ambiguity Is Not an Advantage
• 18  Appendix: Place Value and Mantissa Digits
• 19  Appendix: Resistor Values
• 20  References

3  Foundations and Prerequisites

3.1  What Is Uncertainty?

The best way to understand uncertainty is in terms of probability distributions. The idea of probability is intimately connected with the idea of randomness.

The make use of this idea, you have to identify the relevant ensemble, i.e. the relevant probability distribution, i.e. the relevant probability measure. Consider for example the star cluster shown in figure 3. There are two ways to proceed:

Distribution (A)

You could pick one particular star and re-measure its celestial coordinates again and again. This would produce a sample with some tiny “error bars”.

Distribution (B)

You could randomly pick various stars from the cluster and measure the coordinates of each one.

These are both perfectly good distributions; they’re just not the same distribution. There are innumerable other distributions you could define. It is often nontrivial to decide which distribution is most informative in any given situation. There is no such thing as «the» all-purpose probability distribution.

To calculate the width of the cluster in figure 3, the conventional and reasonable approach is to measure a great many individual stars and then let the data speak for itself. Among other things, you could calculate the mean and standard deviation of the ensemble of star-positions.

In contrast, you cannot use the width of distribution (A) to infer anything about the width of distribution (B). You could measure each individual star ten times more accurately or ten times less accurately and it would have no effect on your value for the width of the cluster. Therefore the whole idea of “propagation of uncertainty” is pointless in this situation.

Figure 3: M13 Globular Cluster in Hercules

The contrast between figure 4 and figure 5 offers another good way of looking at the same fundamental issue. In both figures, the red dashed curve represents the distribution of x in the underlying population, i.e. in the star cluster as a whole. In figure 4, the orange-shaded region represents the joint probability that that x occurs in the population and rounds off to 5 (rounding to the nearest integer). Similarly, the blue-shaded region represents the joint probability that that x occurs in the population and rounds off to 2. This is a small, not-very-probable region.

Figure 4: The Population		Figure 5: Two of the Subpopulations

Meanwhile, in figure 5, the orange-shaded region represents the conditional probability of finding x in the population, conditioned on x rounding off to 5. Roughly speaking, this corresponds to the uncertainty on the position of a single star, after it has been picked and measured. In a well-designed experiment, this has almost nothing to do with the width of the distribution as a whole (i.e. the population as a whole). Similarly, the blue-shaded region represents the conditional probability of finding x in the population, conditioned on x rounding off to 2. In this figure, the area under the blue curve and orange curve are normalized to unity, as is appropriate for conditional probabilities. The area under the red curve is also normalized to unity. The sum of the joint probabilities, summed over all colors, is normalized.

These are all perfectly good distributions, just not the same distribution. This often leads to confusion at the most basic conceptual level, because the language is ambiguous: When somebody says “the error bars on x are such-and-such” it is not the least bit obvious whether they are talking about the unconditional distribution (i.e. the underlying population, i.e. the star cluster as a whole), or about the conditional distribution (i.e. the precision of a single measurement, after a particular star has been picked and measured).

To summarize, when you write “5” in the lab notebook there are at least three concepts to consider.

• The indicated value is x_i = 5, with no uncertainty whatsoever.
• This corresponds to some range of true values, {x_t | x_i=5} as represented by the orange-shaded region in figure 5. This is often a very peculiar distribution, not a normal Gaussian. However, in a well-designed experiment the details don’t matter much, provided the distribution has a “small enough” width.
• There is also the unconditional range of true values {x}, as represented by the red dashed curve in the figures. This is often an important thing to measure. It usually requires obtaining a great many x_i values and then applying statistical formulas.

There is yet more ambiguity because you don’t know how much the error bars contribute to the bias as opposed to the variance. For example, if you round π to 3.14, it contributes precisely nothing to the variance, because every time you do that the roundoff error is the same. It does however introduce a bias into the calculation.

Beware: The fact that the conditional probability has some nonzero width is often used as a pretext for teaching about «sig figs», even though in a well-designed experiment it is irrelevant.

In any case, it is not recommended to describe uncertainty in terms of “random numbers” or “uncertain quantities”. As John von Neumann and others have pointed out, there is no such thing.

People do commonly speak in terms of “random numbers” or “uncertain quantities”, but that doesn’t make it right. These must be considered idiomatic expressions and misnomers. See section 4.3 and section 5.2 for more on this.

An ultra-simple notion of distribution is presented in section 2.2. A more robust but still intuitive and informal introduction to the idea of probability distributions and probability measures can be found in section 4.3 and section 5.2. If you want a cheap and easy experiment that generates data with a nontrivial distribution, partly random and partly not, consider tack-tossing, as discussed in reference 4. Some tack-tossing data is presented in figure 6 and figure 7. For a more formal, systematic discussion of how to think about probability, see reference 2.

You need to understand the distinction between a number and a distribution before you do anything with uncertainty. Otherwise you’re just pushing around symbols without understanding what they mean.

Figure 6: Binomial Distribution : Diaspogram		Figure 7: Binomial Distribution : XY Scatter Plot

4  The Importance of Uncertainty

4.1  Sometimes It’s Unimportant or Nonexistent

Sometimes there is uncertainty, but it is unimportant, as mentioned in section 2.1 and especially section 5.1.

Moreover, sometimes there is no uncertainty, and it would be quite wrong to pretend there is, especially when dealing with raw data or when dealing with a particular data point drawn from a distribution, as discussed in section 5.2.

4.2  Separate versus Bundled Parameters

Suppose we have a distribution over x – perhaps the distribution shown in figure 1 – and the distribution is described by a couple of parameters, the mean A and and the standard deviation B. Consider the contrast:

4.3  Raw Data Points versus Cooked Data Blobs

We must distinguish between raw data points and cooked data blobs. These are different, as surely as a scalar is different from a high-dimensional vector. As an example of what I’m talking about, consider the following contrast:

Figure 8: Samples Drawn from a Gaussian		Figure 9: Samples with Misbegotten Error Bars

See also section 5.2. These two figures, and the associated ideas, are discussed in more detail in reference 2.

Suppose on Monday we roll a pair of slightly-lopsided dice 400 times, and observe the number of spots each time. Let x_i represent the number of spots on the ith observation. This is the raw data: 400 raw data points. It must be emphasized that each of these raw data points has no error bars and no uncertainty. The number of spots is what it is, period. The points are zero-sized pointlike points.

On Tuesday we have the option of histogramming the data as a function of x and calculating the mean (A) and standard deviation (B) of the distribution.

Using the raw data to find values for A and B can be considered an example of curve fitting. (See section 7.24 for more about curve fitting.) It is also an example of modeling. We are fitting the data to a model and determining the parameters of the model. (For ideal dice, the model would be a triangular distribution, but for lopsided dice it could be much messier. Beware that using the measured standard deviation of the set of raw data points is not the best way to determine the shape or even the width of the model distribution. This is obvious when there is only a small number of raw data points. See section 11.4 and reference 2 for details on this.)

If we bundle A and B together (as defined in section 4.2), we can consider A±B as a single object, called a blob, i.e. a cooked data blob. We have the option of trading in 400 raw data points for one cooked data blob. This cooked data blob represents a model distribution, which is in turn represented by two numbers, namely the mean and the standard deviation.

So, this is one answer to the question of why uncertainty is important: It is sometimes more convenient to carry around one cooked data blob, rather than hundreds, thousands, or millions of raw data points. Cooking the data causes a considerable loss of information, but there is sometimes a valuable gain in convenience.

Note that if somebody gives you a cooked data blob, you can – approximately – uncook it using Monte Carlo, thereby returning to a representation where the distribution is represented by a cloud of zero-sized points. That is, you can create a set of artificial raw data points, randomly distributed according to the distribution described by the cooked data blob.

Here’s a story that illustrates an important conceptual point:

Suppose we are using a voltmeter. The manufacturer (or the calibration lab) has provided a calibration certificate that says anything we measure using this voltmeter will be uncertain plus-or-minus blah-blah percent. In effect, they are telling us that there is an ensemble of voltmeters, and there is some spread to the distribution of calibration coefficients.

Note that any uncertainty associated with the ensemble of voltmeters is not associated with any of the raw data points. This should be obvious from the fact that the ensemble of voltmeters existed before we made any observations. This ensemble is owned by the manufacturer or the calibration lab, and we don’t get to see more than one or two elements of the ensemble. So we rely on the calibration certificate, which contains a cooked data blob describing the whole ensemble of voltmeters.

Now suppose we make a few measurements. This is the raw data. It must be emphasized that each of these raw data points has no error bars and no uncertainty. The data is what it is, period.

At the next step, we can use the raw data plus other information including the calibration certificate to construct a model distribution. The ensemble of voltmeters has a certain width. It would be a tremendous mistake to attribute this width to each of the raw data points, especially considering that the calibration coefficient is likely to be very strongly correlated across all of our raw data.

See section 13.6 for more on this.

4.4  Weighing the Evidence

When dealing with a cooked data blob, it is sometimes very important to keep track of the width of the blob, i.e. the uncertainty. Far and away the most common reason for this has to do with weighing the evidence. If you are called upon to make a judgment based on a collection of evidence, the task is straightforward if all of the evidence is equally reliable. On the other hand, if some of the evidence is more uncertain than the rest, you really need to know how uncertain it is.

Here’s a non-numerical example: Suppose you are on a jury. there are ten witnesses who didn’t see what happened, and one who did. It should go without saying that you really, really ought to give less weight to the uncertain witnesses.

Now let’s do a detailed numerical example. Suppose we are trying to diagnose and treat a patient who has some weird symptoms. We have run 11 lab tests, 10 of which are consistent and suggest we should try treatment “A” while the 11th test suggests we should try treatment “B”.

In the first scenario, all 11 observations have the same uncertainty. This situation is depicted in figure 10. Each of the observations is shown as a Gaussian (bell-shaped curve) such that the width of the curve represents the uncertainty.

Figure 10: Equally-Weighted Evidence

In a situation like this, where the observations are equally weighted, it makes sense to average them. The average x-value is shown by the black dot, and the uncertainty associated with the average value is shown by the error bars sticking out from the sides of the dot. We could have represented this by another Gaussian curve, but for clarity we represented it as a dot with error bars, which is another way of representing a probabilistic distribution of observations.

We see that the average is about x=0.1, which is slightly to the right of x=0. The outlier (the 11th observation) has pulled the average to the right somewhat, but only somewhat. The outlier is largely outvoted by the other 10 observations.

Scenario #2 is the same as scenario #1 except for one detail: The 11th observation was obtained using a technique that has much less uncertainty. This situation is shown in figure 11. (We know the 11th curve must be taller because it is narrower, and we want the area under each of the curves to be the same. For all these curves, the area corresponds to the total probability of the measurement producing some value, which must be 100%.)

Figure 11: Unequally-Weighted Evidence

When we consider the evidence, we must give each observation the appropriate weight. The observation with the small uncertainty is given greater weight. When we take the appropriately-weighted average, it gives us x=0.91. This is represented by the black dot in figure 11. Once again the uncertainty in the average is represented by error bars sticking out from the black dot.

It should be obvious that the weighted average (figure 11) is very, very different from from the unweighted average (figure 10).

In particular, suppose the yellow bar in the diagram represents the decision threshold. With unweighted data, the weight of the evidence is to the left of the threshold, and we should try treatment “A”. With weighted data, the weight of the evidence is to the right of the threshold, and we should try treatment “B”.

On the third hand, when considering these 11 observations collectively, it could be argued that the chi-square is so bad that we ought to consider the possibility that all 11 are wrong, but let’s not get into that right now. Properly weighing the evidence would be just as important, just slightly harder to visualize, if the chi-square were lower.

This could be a life-or-death decision, so it is important to know the uncertainty, so that we can properly weigh the evidence.

4.5  Significant Figures, or Not

The “significant figures” approach is intrinsically and incurably unable to represent uncertainty to better than the nearest order of magnitude; see section 8.6 for more on this. What’s worse, the way that sig figs are used in practice is even more out-of-control than that; see section 17.5.1 for details.

Everyone who reports results with uncertainties needs to walk a little ways in the other guy’s moccasins, namely the guy downstream, the guy who will receive those results and do something with them. If the uncertainty is only reported to the nearest order of magnitude, it makes it impossible for the downstream guy to collect data from disparate sources and weigh the evidence.

To say the same thing the other way, it is OK to use sig figs if you are sure that nobody downstream from you will ever use your data in an intelligent way, i.e. will never want to weigh the evidence.

Tangential remark: Just to rub salt into the wound: In addition to doing a lousy job of representing the uncertainty ΔX, the sig-figs rules also do a lousy job of representing the nominal value ⟨X⟩ because they introduce excessive roundoff error. However that is not the topic of this section.

5  Fundamental Notions of Uncertainty

5.1  Some Things are Certain, and Some are Uncertain

Some things are, for all practical purposes, completely certain. For example:

• Recently I bought a carton of eggs, and counted how many eggs it contained. The answer was 12. That means 12, exactly, with no uncertainty. I am quite certain that there were 12±0 eggs in that carton. That’s my story, and I’m sticking with it.
• Similarly, I don’t know everything there is to know about the moon, and I don’t know everything there is to know about cheese, but I am certain for all practical purposes that the moon is not made of green cheese.

On the other hand, there is a very wide class of processes that lead to a distribution of possible outcomes, and these are the main focus of today’s discussion. Some introductory examples are discussed in section 5.2.

5.2  Uncertainty ≡ Probability Distribution

The only way to really understand uncertainty is in terms of probability distributions. You learned in grade-school how to add, subtract, multiply, and divide numbers ... but in order to deal with uncertainties you will have to add, subtract, multiply and divide probability distributions. This requires a tremendously higher level of sophistication.

If you want a definition of probability, in fundamental and formal terms, please see reference 2. For the present purposes we can get along without that, using instead some simple intuitive notions of probability, as set forth in the following examples.

As a first example, suppose we roll an ordinary six-sided die and observe the outcome. The first time we do the experiment, we observe six spots, which we denote by x₁=6. The second time, we observe three spots, which we denote by x₂=3. It must be emphasized that each of these observations has no uncertainty whatsoever. The observation x₁ is equal to 6, and that’s all there is to it.

If we repeat the experiment many times, ideally we get the probability distribution X shown in figure 12. To describe the distribution X, we need to say three things: the outline of the distribution is rectangular, the distribution is centered at x=3.5, and the distribution has a half-width at half-maximum (HWHM) of 2.5 units (as shown by the red bar).

Figure 12: Probability Distribution for a Six-Sided Die

The conventional but abusive notation for describing such a situation is to write x=3.5±2.5, where x is called a «random variable» or an «uncertain quantity». I do not recommend this notation or this way of thinking about things. However, it is sometimes encountered, so we need a way of translating it into something that makes more sense.

An expression of the form 3.5±2.5 is a fine way to describe the distribution X. So far so good. There are however problems with the x that we encounter in expressions such as x = 3.5±2.5. In this narrow context evidently x is being used to represent the distribution X, while in other contexts the same symbol x is used to represent an outcome drawn from X, or perhaps some sort of abstract “average” outcome, or who-knows-what. This is an example of form not following function. Remember, there is a profound distinction between a number and some distribution from which that number might have been randomly drawn. See section 6.4 for more on this.

When you see the symbol x, it is important to appreciate the distinction between x=3.5±2.5 (which is abusive shorthand for the distribution X) and particular outcomes such as x₁=6 and x₂=3 (which are plain old numbers, not distributions):

Now suppose we roll two dice, not just one. The first time we do the experiment, we observe 8 spots total, which we denote by x₁=8. The second time, we observe 11 spots, which we denote by x₂=11. If we repeat the experiment many times, ideally we get the probability distribution X shown in figure 13. To describe the distribution X, we need to say that the outline of the distribution is symmetrical and triangular, the distribution peaks at x=7, and the distribution has a half-width at half-maximum (HWHM) of 3 units (as shown by the red bar).

Figure 13: Probability Distribution for a Pair of Dice

Next suppose the outcomes are not restricted to being integers. Let one of the outcomes be x₃=25.37. Once again, these outcomes are drawn from some distribution X.

We can round off each of the original data points x_i and thereby create some rounded data, y_i. For example, x₃=25.37 and y₃=25.4. We can also calculate the roundoff error q_i := y_i − x_i. In our example, we have q₃=0.03. Given a large number of such data points, we can calculate statistical properties such as the RMS roundoff error. Each x_i is drawn from the distribution X, while each y_i is drawn from some different distribution Y, and each q_i is drawn from some even-more-different distribution Q.

Consider the probability distribution represented by the colored bands in figure 14. There is a distribution over y-values, centered at y=2. Green represents ±1σ from the centerline, yellow represents ±2σ, and magenta represents ±3σ. The distribution exists as an abstraction, as a thing unto itself. The distribution exists whether or not we draw any points from it.

Figure 14: The Distribution Exists Unto Itself

Meanwhile in figure 15, the small circles represent data points drawn from the specified distribution. The distribution is independent of x, and the x-coordinate has no meaning. The points are spread out in the x-direction just to make them easier to see. The point here is that randomness is a property of the distribution, not of any particular point drawn from the distribution.

According to the frequentist definition of probability, if we had an infinite number of points, we could use the points to define what we mean by probability ... but we have neither the need nor the desire to do that. We already know the distribution. Figure 14 serves quite nicely to to define the distribution of interest.

Figure 15: Randomness = Property of the Distribution

By way of contrast, it is very common practice – but not recommended – to focus attention on the midline of the distribution, and then pretend that all the uncertainty is attached to the data points, as suggested by the error bars in figure 16.

Figure 16: Random Points = Misconception

In particular, consider the red point in these figures, and consider the contrasting interpretations suggested by figure 15 and figure 16.

Things get more interesting when the model says the uncertainty varies from place to place, as in figure 17. The mid-line of the band is a power law, y = x^3.5. The uncertainty has two components: an absolute uncertainty of 0.075, “plus” a relative uncertainty of 0.3 times the y-value. The total uncertainty is found by adding these two components in quadrature.

This sort of thing is fairly common. For instance, a the calibration certificate for a voltmeter might say the uncertainty is such-and-such percent of the reading plus this-or-that percent of full scale.

Figure 17: Band Plot: Absolute plus Relative Error

Note that on the left side of the diagram, the total uncertainty – the width of the band – is dominated by the absolute uncertainty, whereas on the right side of the diagram, the total uncertainty is dominated by the relative uncertainty.

Figure 18 shows the same data, plotted on log/log axes. Note that log/log axes are very helpful for visualizing some aspects of the data, such as the fact that the power law is a straight line in this space. However, log/log axes can also get you into a lot of trouble. One source of trouble is the fact that the error bands in figure 17 extend into negative-y territory. If you take the log of negative number, bad things are going to happen.

Figure 18: Band Plot: Absolute plus Relative Error; Log/Log Axes

In figure 18, the red downward-pointing triangles hugging the bottom edge of the triangle correspond to off-scale points. The abscissa is correct, but the ordinate of such points is unplottable.

The spreadsheet used to create this figures is given in reference 5.

Band plots (as in figure 15 or figure 17) are extremely useful. The technique is not nearly as well known as it should be. As a related point, it is extremely unfortunate that the commonly-available plotting tools do not support this technique in any reasonable way.

Tangential remark: This can be seen as reason #437 why sig figs are a bad idea. In this case, sig figs force you to attribute error bars to every data point you write down, even though that’s conceptually wrong.

Please see reference 2 for a discussion of fundamental notions of probability, including

• The idea that a distribution has width but a point does not.
• Probability density distributions versus cumulative probability distributions.
• Distributions over a continuous variable verus distributions over a discrete variable.

5.3  Analog Measurements

There are lots of analog measurements in the world. For example:

• Every time you draw a scale diagram of the apparatus or draw a graph you are in effect recording some measurements in analog form.
• If you are in the field, and you brought your notebook but not your ruler, it might be perfectly sensible to write down that one beetle was |―――| long while another was |――――| long.
• Et cetera.

Analog measurements are perfectly reasonable. There are ways of indicating the uncertainty of an analog measurement. However, these topics are beyond the scope of the present discussion, and we shall have nothing more to say about them.

5.4  Digital Measurements

Here are the main cases and sub-cases of interest:

1. Sometimes we make a digital measurement of something that is intrinsically digital and discrete, such as the number of beans in a jar, or the number of photons received by a photon counter.
2. Sometimes we make a digital measurement of something that started out as a continuous, analog signal (such as time, distance, voltage, etc.) but was subsequently digitized.
   • Sometimes we use an instrument that accepts an analog input and digitizes it automatically, providing an explicit digital readout. Examples include electronic digital scales, A-to-D converters that measure voltage, time-stamp recorders, et cetera.
   • Sometimes we use an analog instrument and then digitize the reading by hand. Examples include analog voltmeters (where the position of the needle is a continuous variable), burettes (where the position of the meniscus is a continuous variable), et cetera. The point is that when you read such an instrument, you mentally digitize the result. By the time you write the value into the lab book, in the form of a decimal numeral, it is entirely digital.

     Let’s be clear: The incoming signal is analog, and the needle position is analog, but the digits you write into the lab book are digital.

5.5  Indicated Value versus True Values

It helps to distinguish the indicated value from the true values. Let’s consider a couple of scenarios:

Scenario A: We hook a digital voltmeter to a nice steady voltage.

In general, each indicated value corresponds to a range of true values, or some similar distribution over true values. For example, in the case of an ideal voltmeter, the relationship might follow the pattern shown in table 1.

Scenario B: Using a couple of comparators, we arrange to show a green light whenever the voltage is greater than −12 volts and less than +12 volts, and a red light otherwise. That is to say, a “green light” indication corresponds to a true value in the interval 0±12 volts.

Instruments with non-numerical outputs are quite common in industry, used for example in connection with “pass/fail” inspections of incoming or outgoing merchandise. There are many indicators of this kind on the dashboard of your car, indicating voltage, oil pressure, et cetera.

The indicated value will never be exactly equal to the true value. This is particularly obvious in scenario B, where the indicated value is not even numerical, but is instead an abstract symbol.

Still, the indicated value does tell us “something” about the true value. It corresponds to a range of true values, even though it cannot possibly equal the true value.

You should not imagine that things will always be as simple as the examples we have just seen.

• For one thing, in table 1 the true-value intervals are uniformly spaced and non-overlapping, but this is not the general case. A counterexample is presented in section 6.1, where we see nonuniformity and lots of overlap. Also in section 19 the intervals are nonuniform.
• In table 1 the indicated values correspond to rounding off the true values, so the true-value intervals can be “explained” in terms of roundoff error.
  • This is clearly not the case in in table 2, where the interval 0±12 could not possibly have resulted from rounding off decimal digits (since that always results in ± half a count in the last decimal place). We also note that this interval could not possibly be represented by sig figs. Not even close.
  • It is also not the case in section 6.1, where the uncertainty is dominated by calibration issues, not by readability or roundoff. (You can always make roundoff the dominant issue, but only by rounding off too much, to the point where your data is seriously degraded.)
  • It is also not the case with the number of centimeters per inch. Sometimes a decimal such as 2.54 is obtained by rounding off, and sometimes it isn’t.
• Very commonly, computer audio systems use 256 binary levels internally, but the user interface expresses things on a scale of 0 to 100%, in steps of 1%, which means that some indicated values correspond to an interval of two real values, while other readings correspond to an interval of three real values. So this is another example of non-uniform intervals.
• Ditto for the RGB color codes in many computer graphic systems.
• You can’t use “sig figs” ideas to determine the size of the true-value ranges. I’ve seen instruments that count by twos in the last decimal place, and others that count by fives in the last decimal place. In such cases the quantization intervals are much larger than you might guess just by counting digits in the display.

Terminology: The true-value intervals (such as we see in table 1) go by various names. In the context of digital instruments people speak of resolution, quantization error, and/or roundoff error. In the context of analog instruments they speak of resolution and/or readability.

In a well-designed experiment, these issues are almost never the dominant contribution to the overall uncertainty. This leads to an odd contrast:

5.6  Uncertainty ± Error ≠ Mistake

Keep in mind that we are using the word uncertainty to refer to the width of a probability distribution ... nothing more, nothing less.

Sometimes this topic is called “error analysis”, but beware that the word “error” is very widely misunderstood.

In this context, error means the same thing as uncertainty. It refers to the width of the distribution, not to a mistake or blunder. Indeed, we use the concept of uncertainty in order to avoid making mistakes. It would always be a mistake to say the voltage was exactly equal to 1.23 volts, but we might be confident that the voltage was in the interval 1.23±0.05 volts.

The proper meaning of uncertainty (aka “error”) is well illustrated by Scenario B in section 5.5. The comparator has a wide distribution of true voltages that correspond to the “green light” indication. This means we are uncertain about the true voltage. This uncertainty is, however, not a blunder. Absolutely not. The width of the distribution is completely intentional. The width was carefully designed, and serves a useful purpose.

This point is very widely misunderstood. For example, the cover of Taylor’s book on Error Analysis (reference 6) features a crashed train at the Gare Montparnasse, 22 October 1895. A train crash is clearly an example of a shameful mistake, rather than a careful and sophisticated analysis of the width of a distribution. It’s a beautiful photograph, but it conveys entirely the wrong idea.

See also section 8.12.

5.7  Probably Almost Correct

Consider the following contrast:

5.8  Identify All Contributions to the Uncertainty

Accounting for uncertainty is not merely an exercise in mathematics. Before you can calculate the uncertainty in your results, you need to identify all the significant sources of uncertainty. This is a major undertaking, and requires skill and judgment.

For example: The voltmeter could be miscalibrated. There could be parallax error when reading the ruler. There could be bubbles in the burette. The burette cannot possibly be a perfectly uniform cylinder. There could be moisture in the powder you are weighing. And so on and so on.

Four categories of contributions that are almost always present to some degree are fluctuations, biases, calibration errors, and resolution problems aka roundoff errors, as we now discuss.

• Suppose you are looking at a meter and the needle is wiggling all over the place. It doesn’t matter how well calibrated the meter is or how finely graduated the scale is. The inherent scatter in the readings overwhelms other contributions to the uncertainty.
• You can reduce the noise by averaging the signal, but this introduces a bias, insofar as part of the signal you are supposed to be measuring will be outside the passband of the filter. In general there will be noise/bandwidth tradeoffs. Even more generally, there will be variance/bias tradeoffs.
• Now suppose the signal is not fluctuating, and you can read the scale very accurately. In such a case, it may be that calibration errors are the dominant contribution to the uncertainty.
• Suppose you know that a certain landmark is between 32^∘ and 33^∘ north latitude, at some well-known longitude. There is no question of fluctuations, since the landmark is not moving. Also there is no problem with the calibration, since we know where the lines of latitude are, with an uncertainty of well less than 1 meter. Yet still we have many kilometers of uncertainty about the location of the landmark, because our information is too coarsely quantized. We don’t have enough resolution. In other words, we are getting clobbered by roundoff errors.

Remark #1: Remember: Roundoff error is only one contribution to the overall uncertainty. In a well-designed experiment, it is almost never the dominant contribution. See section 8.6 for a discussion of how distributions are affected by roundoff errors.

Remark #2: It is not safe to assume that roundoff errors are uncorrelated. It is not safe to assume that calibration errors are uncorrelated. Beware that many textbooks feature techniques that might work for uncorrelated errors, but fail miserably in practical situations where the errors are correlated.

Remark #3: If one of these three contributions is dominant, it is fairly straightforward to account for it while ignoring the others. On the other hand, if more than one of these contributions are non-negligible, the workload goes up significantly. You may want to redesign the experiment.

If you can’t redesign the experiment, you might still be able to save the day by finding some fancy way to account for the various contributions to the uncertainty. This, however, is going far beyond the scope of this document

Remark #4: More specifically: You usually want to design the experiment so that the dominant contribution to the uncertainty comes from the inherent fluctuations and scatter in the variable(s) of interest. Let’s call this the Good Situation.

It’s hard to explain how to think about this. In the Good Situation, many idealizations and simplifications are possible. For example: since calibration errors are negligible and roundoff errors are negligible, you can more-or-less ignore everything we said in section 5.5 about the distinction between the indicated value and the range of true values. If you always live in the Good Situation, you might be tempted to reduce the number of concepts that you need to learn. If you do that, though, and then encounter a Not-So-Good Situation, you are going to be very confused, and you will suddenly wish you had a better grasp of the fundamentals.

Possibly helpful suggestion: A null experiment – or at least a differential experiment – often improves the situation twice over, because (a) it reduces your sensitivity to calibration errors, and (b) after you have subtracted off the baseline and other common-mode contributions, you can turn up the gain on the remaining differential-mode signal, thereby improving the resolution and readability.

5.9  Empirical Distributions versus Theoretical Distributions

There are many probability distributions in the world, including experimentally-observed distributions as well as theoretically-constructed distributions.

Any set of experimental observations {x_i} can be considered a probability distribution unto itself. In simple cases, we assign equal weight (i.e. equal measure, to use the technical term) to each of the observations. To visualize such a distribution, often the first thing to do is look a scatter plot. For example, figure 34 shows a two-dimensional scatter plot, and figure 37 shows a one-dimensional scatter plot. We can also make a graph that shows how often x_i falls within a given interval. Such a graph is called a histogram. Examples include figure 12, figure 13, and figure 22.

Under favorable conditions, given enough observations, the histogram may converge to some well-known theoretical probability distribution. (Or, more likely, the cumulative distribution will converge, as discussed in reference 2.) For example, it is very common to encounter a piecewise-flat distribution as shown by the red curve in figure 19. This is also known as a square distribution, a rectangular distribution, or the uniform distribution over a certain interval. Distributions of this form are common in nature: For instance, if you take a snapshot of an ideal rotating wheel at some random time, all angles between 0 and 360 degrees will be equally probable. Similarly, in a well-shuffled deck of cards, all of the 52-factorial permutations are equally probable. As another example, ordinary decimal roundoff errors are confined to the interval [-0.5, 0.5] in the last decimal place. Sometimes they are uniformly distributed over this interval and sometimes not. See section 8.3 and section 7.12 for more on this. Other quantization errors (such as discrete drops coming from a burette) contribute an uncertainty that might be more-or-less uniform over some interval (such as ± half a drop).

It is also very common to encounter a Gaussian distribution (also sometimes called a “normal” distribution). In figure 19, the blue curve is a Gaussian distribution. The standard deviation is 1.0, and is depicted by a horizontal green bar. The standard deviation of the rectangle is also 1.0, and is depicted by the same green bar.

Meanwhile, the HWHM of the Gaussian is depicted by a blue bar, while the HWHM of the rectangle is depicted by a red bar.

Figure 19: Gaussian vs. Rectangular; StDev vs. HWHM

Table 3 lists a few well-known families of distributions. See section 13.8 for more on this.

Each of these distributions is discussed in more detail in reference 2.

Each name in table 3 applies to a family of distributions. Within each such family, to describe a particular member of the family (i.e. a particular distribution), it suffices to specify a few parameters. For a symmetrical two-parameter family, typically one parameter specifies the center-position and the second parameter has something to do with the halfwidth of the distribution. The height of the curve is implicitly determined by the width, via the requirement² that the area under the curve is always 1.0.

In particular, when we write A±B, that means A tells us the nominal value of the distribution and B tells us the uncertainty or equivalently the error bar. See section 5.12 for details on the various things we might mean by nominal value and uncertainty.

5.10  Terminology and Notation

Best current practice is to speak in terms of the uncertainty. We use uncertainty in a broad sense. Other terms such as accuracy, precision, experimental error, readability, tolerance, etc. are often used as nontechnical terms ... but sometimes connote various sub-types of uncertainty, i.e. various contributions to the overall uncertainty, as discussed in section 12. In most of this document, the terms “precise” and “precision” will be used as generic, not-very-technical antonyms for “uncertain” and “uncertainty”.

As a related point, see section 13.7 for details on why we avoid the term “experimental error”.

Some guidelines for describing a distribution are given in section 1.2. When writing the nominal value and the standard deviation, be sure to write them separately, using two separte numerals. For example, NIST (reference 7) reports the charge of the electron as

which is by definition equivalent to

 
Note that this value departs from the usual “sig-digs rules” by a wide margin. The reported nominal value ends in not one but two fairly uncertain digits.

For specific recommendations on what you should do, see section 8.2. Also, NIST offers some prescriptions on how to analyze and report uncertainties; see reference 8, reference 9, and reference 10.

Additional discussions of how to do things can be found in reference 11 and reference 12.

5.11  How Not to Represent Uncertainty

The “significant figures” method attempts to use a single decimal numeral to express both the center and the halfwidth of a distribution: the ordinary value of the numeral encodes the center, while the length of the string of digits roughly encodes the halfwidth. This is a horribly clumsy way of doing things.

See section 1.3 and section 17.

5.12  Uncertainty, Standard Deviation, Confidence Limits, etc.

In the expression A±B, we call A the nominal value and B the uncertainty (or, equivalently, the error bar).

We will explicitly avoid giving any quantitative definition for the terms nominal value and uncertainty. This is because there is not complete consensus as how to quantify the expression A±B. When you write such an expression, it is up to you to specify exactly what you mean by it. When you read such an expression, you will have to look at the context to figure out what it means.

• For a mathematically ideal Gaussian distribution, there is a reasonably strong consensus that in the expression A±B, the nominal value (A) is the mean. In this case the mean is also the median and the mode and the center of symmetry, so there is not really much to debate.

  Meanwhile, as for B:

  • Conventionally the uncertainty (B) is the standard deviation, B=σ. The interval from A−σ to A+σ is the 68% confidence interval.
  • In specialized situations, people might be interested in the interval from from A−2σ to A+2σ, which is the 95% confidence interval, or even the interval from from A−3σ to A+3σ, which is the 99.73% confidence interval.

    However, if you are going to use two-sigma or three-sigma error bars, you need to warn people, because this is not what they are expecting. Normally, for a Gaussian, the expression A±B communicates the mean plus-or-minus one sigma.

• For a mathematically ideal symmetrical triangular distribution, again the nominal value A is the mean and the median and the mode and the center of symmetry.

  As for the uncertainty, there are at least two reasonable choices. B could represent the standard deviation, or it could represent the HWHM.

  • The interval from A−σ to A+σ is the 65% confidence interval. This has the advantage of using the A±B symbol the same way for both Gaussian and triangular distributions: it communicates the mean and standard deviation.
  • The interval from A−HWHM to A+HWHM is the 75% confidence interval. This has the advantage of being a natural geometrical feature of the triangle.
• For a mathematically ideal rectangular distribution, once again A conventionally represents the mean and the median and the center of symmetry. The distribution does not have a peak, properly speaking, so there is no mode.

  Again there are reasonable arguments for using the standard deviation to quantify the uncertainty, and also reasonable arguments for using the HWHM. Both are commonly used:

  • The interval from A−σ to A+σ is the 58% confidence interval. This has the advantage of using the A±B symbol the same way for both Gaussian and rectangular distributions: it communicates the mean and standard deviation.
  • The interval from A−HWHM to A+HWHM is the 100% confidence interval. This has the advantage of being a natural geometrical feature of the rectangle. Indeed this is the raison d’être for the rectangular distribution: all of the probability lies within ±1 HWHM of the middle.

In all cases the uncertainty B is more closely related to the halfwidth than to the full width, since the expression A±B is pronounced A plus-or-minus B, not plus-and-minus. That is to say, B represents the plus error bar or the minus error bar separately, not both error bars together.

For a distribution defined by a collection of data, we need to proceed even more carefully. The data itself has a perfectly well defined mean and standard deviation, and you could certainly compute the mean and standard deviation, using the definition directly. These are called the sample-mean and the sample-standard-deviation. These quantities are well defined, but not necessarily very useful. Usually it is smarter to assume that the data is a sample drawn from some underlying mathematically-defined distribution, – called the population – and to use the data to estimate the parameters of the population. The mean of the data might not be the best estimator of the mean of the population. (When the number of data points is not very large, the standard deviation of the sample is a rather badly biased estimator of the standard deviation of the population)

Also, remember: An expression of the form A±B only makes sense provided everybody knows what family of distributions you are talking about, provided it is a well-behaved two-parameter family, and provided everybody knows what convention you are using to quantify the nominal value and the uncertainty. To say the same thing the other way: it is horrifically common for people to violate these provisos, in which case it A±B doesn’t suffice to tell you what you need to know. For example: in figure 19, both curves have the same mean and the same standard deviation, but they are certainly not the same curve. Data that is well described by the blue curve would not be well described by the red curve, nor vice versa.

6  Reading an Instrument

6.1  Example: Reading a Meter

It is very common to have an analog meter where the calibration certificate says the uncertainty is 2% of the reading plus 2% of full scale. The latter number means there is some uncertainty as to the “zero offset” of the meter.

When dealing with uncertainty, it helps to keep in mind the distinction between the indicated value and the true value. As discussed in section 5.5, even when the indicated value is known with zero uncertainty, it usually represents a range of true values with some conspicuously non-zero uncertainty.

This tells us that when the indicated value is at the top of the scale, the distribution of of true values has a relative uncertainty of 3 or 4 percent (depending on whether you think the various contributions are independent). More generally, the situation is shown in table 4.

As you can see in the table, as the readings get closer to the bottom of the scale, the absolute uncertainty goes down, but the relative uncertainty goes up dramatically. Indeed, if the reading is in the bottom part of the scale, you should switch ranges if you can ... but for the moment, let’s suppose you can’t.

Keep in mind that calibration errors are only one of many contributions to the overall uncertainty.

Let’s turn now to another contribution, namely readability. Imagine that the meter is readable to ±2% of full scale. That means it is convenient to express each reading as a two-digit number. You should record both digits, even in the bottom quarter of the range, where the associated uncertainty is so large that the sig figs rules would require you to round off. You should record both digits because:

• Recording both digits is easier than worrying about whether both digits are necessary.
• Rounding off is error-prone and should not be done on the fly.
• Rounding off introduces roundoff error and you don’t want to do that if there’s no need to.
• Even in the best of circumstances, the sig figs method gives only a very crude estimate of the uncertainty. If at some point you switch from two digits to one digit, it would imply that the uncertainty suddenly went up by a factor of ten. It’s silly to do that, given that we have much better information about the uncertainty. We know that it is 2% of the reading plus 2% of full scale.

You should write a note in the lab book saying what you know about the situation:

Calibration good to 2% of reading plus 2% of full scale.
Scale readable to 2%.

Then just record each indicated value, as is. Two decimal places suffice to guarantee that the roundoff error is not larger than the readability interval. Remember that the indicated value is known with zero uncertainty, but represents a distribution of true values.

Writing such a note in the lab book, and then writing the indicated values as plain numbers, is incomparably easier and better than trying to describe the range of true values for every observation on a line-by-line basis.

This upholds the important rule: say what you mean, and mean what you say. Describing the calibration and readability situation and then writing down the indicated values makes sense, because you are writing down what you know, nothing more and nothing less.

Also note that this upholds the rule of specifying the uncertainty separately, rather than trying to encode it using sig figs. You should never try to use one numeral to represent two numbers.

6.2  Example: Reading a Burette using Simple Interpolation

Figure 20 is a photograph³ of some liquid in a burette. For present purposes, this photograph is our raw data. Our task is to read the data, so as to arrive at a numerical reading.

Figure 20: Liquid in a Burette

Let’s start by taking the simple approach. (See section 6.3 for a fancier approach.

To get a decent accuracy, we divide the smallest graduation in half. Therefore readings will be quantized in steps of 0.05 mL. More to the point, that gives us a readability of ±0.025 mL, since the indicated value will differ from the true value by at most half a step in either direction.

Using this approach, I observe that the meniscus is pretty close to the 39.7 graduation. It is not halfway to 39.8, or even halfway to halfway, so it is clearly closer to 39.7 than to 39.75. Therefore I would record the indicated value as 39.7 mL (with a readability of ±0.0125 mL.

6.3  Example: Reading a Burette using Fancy Interpolation

We now start over and re-do the interpolation. We work a lot harder this time, so as to obtain a more accurate result.

It is not always worthwhile to go to this much trouble, but sometimes it is.

• Take a picture using a digital camera. Note that in figure 20, the camera has been carefully lined up so as to minimize parallax errors. Also the lighting has been arranged so that the meniscus shows up clearly.
• Read the picture into a graphics program such as inkscape.
• Magnify it 500% so that squinting is not necessary, as shown in figure 21.
• Look closely at the picture. In this example, you see the following, in order from top to bottom: Background; liquid (darker than the background); lower boundary of the liquid (very dark); bright halo (brighter than the background); background again.

  I choose to define “the” position of the meniscus as the boundary between the dark boundary and the bright halo. Others may choose differently. The choice doesn’t matter much for typical chem-lab purposes (so long as the choice is applied consistently), because when using a burette we are almost always interested in the difference between two readings.

• Fit the boundary of the meniscus to the boundary of a drawn object, such as the red object shown in figure 21. Note that the red object has a partially-transparent interior. This allows you to slide it around and still see what’s behind it. Also note that it has no boundary line. This makes use of a profound mathematical fact: the boundary of a boundary is zero. That is to say, the boundary of the red object has zero width, whereas if you drew a line, you would need to worry about the width of the line.

  It is not hard to position the boundary of the red object against the boundary of the liquid with sub-pixel accuracy. It may help to reduce the opacity of the red object during this step.

• Add additional graduations, as shown by the thin cyan lines in figure 21. Up to a point, counting closely-spaced lines is easier than interpolating between widely-spaced lines ... and interpolation over short distances is easier than interpolation over long distances.
• Read out the position of the boundary of the red object. Forget about the raw-data pixels at this point, because they are blurry while the red object is not. It may help to increase the opacity of the red object during this step.

Figure 21: Liquid in a Burette : Zoomed In

Following this procedure, I decide the indicated value is 39.71, readable to the nearest .01 mL. That is to say, the readability is ±0.005 ml. Note that this approach gives us five times better accuracy, compared to the simple approach in section 6.2.

It is not be necessary to computer-analyze every burette reading. For one thing, in many cases you don’t need to know the reading to this degree of accuracy. Secondly, with a little bit of practice you can read this burette by eye to the nearest 0.01 mL, without the aid of the computer. A detailed analysis is worth the trouble every once in a while, if only to increase your eyeball skills, and to give you confidence in those skills. Interpolating by eye to one tenth of a division is doable, but it is not easy. Nobody was born knowing how to do this.

At some point readability gets mixed up with quantization error aka roundoff error associated with the numbers you write down. In this example, I have chosen to quantize the reading in steps of 0.01 ml. This introduces a roundoff error of ± 0.005 ml ... with a very non-Gaussian distribution.

Remember: In a well-designed experiment, roundoff error is almost never the dominant contribution to the overall uncertainty. In this case, the roundoff error is less than the uncertainty due to my limited ability to see where the meniscus actually is, so I’m not going to worry too much about it.

It is hard to know the readability for sure without repeating the measurement N times and doing some sort of statistical analysis.

For reasons discussed in section 6.1 and section 6.4, you probably do not want to record this in the form 39.71 ± 0.005, because people will interpret that as a statement of “the” uncertainty, whereas readability is only one contribution to the overall uncertainty. It is better to simply make a note in the lab book, saying that you read the burette to the nearest 0.01 mL, or words to that effect.

On top of all that, the meaning of a burette reading may be subject to uncertainty due to the fact that the liquid comes out in discrete drops. There are steps you can take to migitate this. If there are droplets inside the column, or a thin film wetting the surface, this is an additional source of uncertainty, including both scatter and systematic bias.

Last but not least, there will be some uncertainty due to the fact that the burette may not be a perfect cylinder, and the graduations may not be in exactly the right places. Industry-standard tolerances are:

The tolerances apply to the full capacity of the burette. It is likely (but not guaranteed) that the errors will be less if a lesser amount is delivered from the burette.

At the time you make a reading, it is quite likely that you don’t know the overall uncertainty, in which case you should just write down the number with plenty of guard digits.⁴ Make a note of whatever calibration information you have, and make a note about the readability, but don’t say anything about the uncertainty. Weeks or months later, when you have figured out the overall uncertainty, you should report it ... and in most cases you should also report the various things that contributed to it, including things like readability, quantization errors, systematic biases, et cetera.

6.4  Analyzing an Ensemble of Readings

Suppose we perform an ensemble of measurements, namely 100 repetitions of the experiment described in section 6.3. The black vertical bars in Figure 22 are a histogram, showing the results of a numerical simulation.

One thing to notice is that the measurements, as they appear in my lab book, have evidently been rounded off. This is of course unavoidable, since the true value is a continuous, analog variable, while the indicated value that gets written down must be discrete, and must be represented by some finite number of digits. See section 8.6 for more about the effect of rounding. We can see that in the figure, by noticing that only the bins corresponding to round multiples of 0.001 are occupied. The histogram shows data for bins at all multiples of 0.0002, but only every fifth such bin has any chance of being occupied.

Figure 22: Histogram of Burette Readings

In figure 22, the magenta line is a Gaussian with the same mean and standard deviation as the ensemble of measurements. No deep theory is needed here; we just calculate the mean and standard deviation of the data and plot the Gaussian. You can see that the Gaussian is not a very good fit to the data, but it is not too horribly bad, either. It is a concise but imperfect way of summarizing the data.

There is a conceptual point to be made here: Suppose we ignore the black bars in the histogram, and consider only the 100 raw data points plus the cooked data blob. The question arises, how many numbers are we talking about?

The answer is 102, namely the 100 raw data points plus the mean and standard deviation that constitute the raw data blob, i.e. the Gaussian model distribution, as indicated in the following table:

We emphasize that there is only one ± symbol in this entire table, namely the one on the bottom line, where we describe the model distribution. In contrast, at the time measurement #1 is made, we could not possibly know the standard deviation – much less the uncertainty⁵ – of this set of measurements, so it would be impossible to write down 39.37 plus-or-minus anything meaningful. Therefore we just write down 39.37 and move on to the next measurement.

In general, if we have N observations drawn from some Gaussian distribution, we are talking about N+2 numbers. We are emphatically not talking about 2N+2 numbers, because it is conceptually not correct to write down any particular measurement in the form A±B. People do it all the time, but that doesn’t make it right. As mentioned in section 5, a distribution is not a number, and a number is not a distribution.

• Anything you write down in the form x₁ = 39.37 represents a plain old number.
• Anything you write down in the form X = 39.3704 ± 0.0015 represents a distribution, not a number.

In the simplest case, namely N=1, it requires three numbers to describe the measurement and the distribution from which it was drawn. If we unwisely follow the common practice of recording “the measurement” in the form A±B, presumably B represents the standard deviation of the distribution, but A is ambiguous. Does it represent the actual observed reading, or some sort of estimate of the mean of the underlying distribution? When we have only a single measurement, the ambiguity seems mostly harmless, because the measurement itself may be our best estimate of the mean of the distribution. Even if it’s not a very good estimate, it’s all we have to go on.

Things get much stickier when there are multiple observations, i.e. N≥2. In that case, we really don’t want to have N separate estimates of the mean of the distribution and N separate estimates of the standard deviation. That is to say, it just doesn’t make sense to write down N expressions of the form A±B. The only thing that makes any sense is to write down the N measurements as plain numbers, and then separately write down the estimated mean and standard deviation of the distribution ... as in the table above.

6.5  Standard Deviation versus Uncertainty versus Error

Before leaving the burette example, there is one more issue we must discuss. It turns out that during my series of simulated experiments, in every experiment I started out with the exact same volume of liquid, namely 39.3312 mL, known to very high accuracy. Subsequently, during the course of each experiment, the volume of liquid will of course fluctuate, due to thermal expansion and other factors, which accounts for some of the scatter we see in the data in figure 22. Imperfect experimental technique and roundoff error account for additional spread.

Now we have a little surprise. The distribution of measurements is 39.3704 ± 0.0015 mL, whereas the actual amount of liquid was only 39.3312 mL, which is far, far outside the measured distribution. So, how do we explain this?

It turns out that every one of the experiments was done with the same burette, which was manufactured in such a way that its cross-sectional area is too small by one part per thousand. Therefore it always reads high by a factor of 1.001, systematically.

This underlines that point that statistical analysis of your observations will not reveal systematic bias. Standard deviation is precisely defined and easy to calculate, but it is not equivalent to uncertaintly, let alone error. For more on this, see section 13, especially section 13.5 and section 13.6.

6.6  Example: Decimal Interpolation Between Graduations

Suppose I’m measuring the sizes of some blocks using a ruler. The ruler is graduated in millimeters. If I look closely, I can measure the blocks more accurately than that, by interpolating between the graduations. As pointed out by Michael Edmiston, sometimes the situation arises where it is convenient to interpolate to the nearest 1/4th of a millimeter. Imagine that the blocks are slightly misshapen so that it is not possible to interpolate more accurately than that.

Let’s suppose you look in my lab notebook and find a column containing the following numbers:

---

40

40.25

40.75

41

Table 6: Length of Blocks, Raw Data

---

and somewhere beside the column is a notation that all the numbers are rounded to the nearest 1/4th of a millimeter. That means that each of these numbers has a roundoff error on the order of ±1/8th of a millimeter. As always, the roundoff errors are not Gaussian-distributed. Roundoff errors are one contribution to the uncertainty. In favorable situations this contribution is flat-distributed over the interval ±1/8 mm, but the actual situation may not be nearly so favorable, as discussed in section 7.12, but let’s not worry about that right now.

If we worshipped at the altar of sig digs, we would say that that the first number (40) had one “sig dig” and therefore had an uncertainty of a few dozen units. However, that would be arrant nonsense. The actual uncertainty is a hundred times smaller than that. The lab book says the uncertainty is 1/8th of a unit, and it means what it says.

At the other end of the spectrum, the fact that I wrote 40.75 with two digits beyond the decimal point does not mean that the uncertainty is a few percent of a millimeter (or less). The actual uncertainty is ten times larger than that. The lab book says that all the numbers are rounded to the nearest 1/4th of a millimeter, and it means what it says.

The numbers in table 6 are perfectly suitable for typing into a computer for further processing. Other ways of recording are also suitable, but it is entirely within my discretion to choose among the various suitable formats that are available.

The usual ridiculous “significant digits rules” would compel me to round off 40.75 to 40.8. That changes the nominal value by 0.05mm. That shifts the distribution by 40% of its half-width. Forty percent seems like a lot. Why did I bother to interpolate to the nearest 1/4th of a unit, if I am immediately forced to introduce a roundoff error that significantly adds to the uncertainty? In contrast, writing 3/4ths as .75 is harmless and costs nothing.

Bottom line: Paying attention to the “sig digs rules” is unnecessary at best. Good practice is to record the nominal value and the uncertainty separately. Keep many enough digits to make sure there is no roundoff error. Keep few enough digits to be reasonably convenient. Keep all the original data. See section 8.2 for more details.

Even more-extreme examples can be found. Many rulers are graduated in 1/8ths of an inch. This is similar to the example just discussed, except that now it is convenient to write things to three decimal places (not just two). Again the sig figs rules mess things up.

More generally: Any time your measurements are quantized with a step-size that doesn’t divide 10 evenly, you can expect the “sig digs rules” to cause trouble.

6.7  Readability is Only Part of the Uncertainty

Consider the contrast:

And another, separate contrast:

I’ve seen alleged rules that say you should read instruments by interpolating to 1/10th of the finest scale division, and/or that the precision of the instrument is 1/10th of the finest scale division. In some situations those rules reflect reality, but sometimes they are wildly wrong.

When choosing or designing an instrument for maximum accuracy, usually you should arrange it so that the dominant contribution to the overall uncertainty is is set by some sort of noise, fluctuations, or fuzz. That makes sense, because if the reading is not fuzzy, you can usually find a way to apply some some magnification and get more accuracy very cheaply.

7  Propagation of Uncertainty

7.1  Overview

Consider the following scenario: Suppose we know how to calculate some result x_i as a function of some inputs a_i, b_i, and c_i:

We assume the functional form of f(...) is known. That’s fine as far as it goes. The next step is to understand the uncertainty. To do that, we need to imagine that the numbers a_i, b_i, and c_i are drawn from known distributions A, B, and C respectively, and we want to construct a distribution X with the following special property: Drawing an element x_i at random from X is the same as drawing elements from A, B, and C and calculating x_i via equation 6.

This topic is called propagation of uncertainty. The idea is that the uncertainty “propagates” from the input of f(...) to the output.

If we are lucky, the distribution X will have a simple form that can be described in terms of some nominal value ⟨X⟩ plus-or-minus some uncertainty [X]. If we are extra lucky, the nominal value of X will be related to the nominal values of A, B, and C by direct application of the same function f(...) that we saw in equation 6, so that

Beware that propagation of uncertainty suffers from three categories of problems, namly Misrepresentation, Malexpansion, and Correlation. That is:

1. Misrepresentation: The sig-figs approach cannot even represent uncertainty to an acceptable accuracy. Representation issues are discussed in section 8.2. You could fix the representation using the ⟨A⟩±[A] notation or some such, but then both of the following problems would remain.
2. Malexpansion: The step-by-step first-order approach fails if the first-order Taylor expansion is not a good approximation, i.e. if there is significant nonlinearity. The step-by-step approach fails even more spectacularly if the Taylor series fails to converge. See e.g. section 7.19, section 7.6, and section 7.5.
3. Correlation: The whole idea of a data blob of the form ⟨A⟩±[A] goes out the window if one blob is correlated with another. See e.g. section 7.7.

Let’s consider how these issue affect the various steps in the calculation:

Step 0: We need a way to represent the uncertainty of three input distributions A, B, and C.

Step 1: We need a way to calculate the properties (including the uncertainty) of the new distribution X.

Step 2: After we know the uncertainty of X, we need a way to represent it.

Steps 0 and 2 are representation issues, while step 1 is a propagation issue. The propagation rules are distinct from the representation issues, and are very much more complicated. The propagation rules might fail if the Taylor expansion isn’t a good approximation ... and might also fail if there are correlations in the data.

Beware that the people who believe in sig figs tend to express both the representation rules and the propagation rules in terms of sig figs, and lump them all together, but this is just two mistakes for the price of one. As a result, when people speak of “the” sig figs rules, you never know whether they are talking about the relatively-simple representation rules, or the more complicated propagation rules.

Sig figs cause people to misunderstand the distinction between representation of uncertainty and propagation of uncertainty.

In reality, when dealing with real raw data points or artificial (Monte Carlo) raw data points, the representation issue does not arise. The raw data speaks for itself.

In practice, the smart way to propagate uncertainties is:

1.

Use the Crank Three Times™ method, as described in section 7.14. This works for a wide range of simple problems. This contructs three elements of the distribution X. If you are lucky, this is a representative sample. The best thing is, in cases where it doesn’t work, you will almost certainly know it. That’s because (unless you are very unlucky) you will get lopsided error bars, or worse, and this tells you a more powerful method is needed.

2.

Use the Monte Carlo method, as described in section 7.16. This is very easy to do using the spreadsheet program on your computer. This constructs a representation of the the distribution X, representing it as a cloud of zero-sized points.

This is tremendously advantageous, because the uncertainty is now represented by the width of the cloud. The individual points have no width, so you can use ordinary algebra to calculate whatever you want, point-by-point, step-by-step. This is very much simpler – and more reliable – than trying to attach uncertainty to each point and then trying to propagate the uncertainty using calculus-based first-order techniques.

In order to really understand the propagation of uncertainty, we must learn a new type of arithmetic: We will be performing computations on probability distributions rather than on simple numbers.

7.2  Simple Example: Multi-Step Arithmetic

This subsection shows the sort of garbage that results if you try to express the propagation rules in terms of sig figs.

Let’s start with an ultra-simple example

where each of the addends has an uncertainty of ±10%, normally and independently distributed.

Common sense suggests that the correct answer is x = 3.6 with some uncertainty. You might guess that the uncertainty is about 10%, but in fact it is less than 6%, as you can verify using the methods of section 7.16 or otherwise.

In contrast, the usual “significant digits rules” give the ludicrous result x=2. Indeed the “rules” set each of the parenthesized sub-expressions is equal to 2.

This is a disaster. Not only do the “sig figs rules” get the answer wrong, they get it wrong by a huge margin. They miss the target by seven times the radius of the target!

To understand what’s going on here, consider the innermost parenthesized sub-expression, namely (2 + 0.4).

Step 1 (propagation): The sum is 2.4, obviously. Let’s assume this is the nominal value of the result-distribution. Let’s also assume the uncertainty is calculated in the usual way, so that the uncertainty on the sum is at least as great as the uncertainty on the addends. Neither of these assumptions is entirely safe, but let’s assume them anyway, so as to construct a best-case scenario.

Step 2 (representation): Since the sum (2.4) has more uncertainty than the first addend (2), it should be represented by at most the same number of sig figs, so we round it off. We replace 2.4 with 2. This is a disaster.

Repeatedly adding 0.4 causes the same disaster to occur repeatedly.

The fundamental issue here is that the sig figs rules require you to keep rounding off until roundoff error becomes the dominant contribution to the uncertainty. This is a representation issue, but it interacts with the propagation issue as follows: The more often you apply the sig figs representation rules, the worse off you are ... and the whole idea of propagation requires you to do this at every step of the calculation.

Rounding off always introduces some error. This is called roundoff error or quantization error. Again: One of the fundamental problems with the sig figs rules is that in all cases, they demand too much roundoff.

This problem is even worse than you might think, because there is no reason to assume that roundoff errors are random. Indeed, in equation 8 the roundoff errors are not random at all; the roundoff error is 0.4 at every step. These errors accumulate linearly. That is, in this multi-step calculation, the overall error grows linearly with the number of steps. The errors do not average out; they just accumulate. Guard digits are a good way to solve part of the problem, as discussed in section 7.3 and section 8.8.

7.3  Guard Digits (Preview)

Let’s take another look at the multi-step calculation in equation 8. Many people have discovered that they can perform multi-step calculations with much greater accuracy by using the following approach: At each intermediate step of the calculation, the use more digits than would be called for by the sig figs rules. These extra digits are called guard digits, as discuseed in See section 8.8. Keeping a few guard digits reduces the roundoff error by a few orders of magnitude. When in doubt, keep plenty of guard digits on all numbers you care about.

Guard digits do not, however, solve all the world’s problems. In particular, suppose you were using the sig figs rules at every step (as in section 7.2) in an attempt to perform “propagation of error”. (Propagation is, after all, the topic of this whole section, section 7). The problem is, step-by-step first-order propagation is almost never reliable, even if you use plenty of guard digits. The first reason why it is unreliable is that the first-order Taylor approximation often breaks down. Furthermore, even if you could fix that problem, the approach fails if there are correlations. There’s a proverb that says imperfect information is better than no information, but that proverb doesn’t apply here, because we have much better ways of getting information about the uncertainty, such as the Crank Three Times™ method.

When there is noise (i.e. uncertainty) in your raw data, guard digits don’t make the raw noise any smaller ... they just make the roundoff errors smaller.

See section 8.8 and section 8.9 for more discussion of guard digits. See section 12 for more discussion of various contributions to the uncertainty.

7.4  Example: Beyond First Order: 1 to the 40th Power

Exponentials show up in a wide variety of real-life situations. For example, the growth of bacteria over time is exponential, under favorable conditions.

As a simple example, let x=1 and consider raising it to the 40th power, so we have y = x⁴⁰. Then y=1. It couldn’t be simpler.

Next, consider x that is only “near” 1. We draw x from the rectangular distribution 1.0±0.05. We compute y = x⁴⁰, and look at the distribution over y-values. Roughly speaking, this is the distribution over the number of bacteria in your milk, when there is a distribution over storage temperatures. The results are diagrammed in figure 23 and figure 24. Note that figure 23 is zoomed in to better portray the red curve, at the cost of clipping the blue spike; the distribution over x actually peaks at dP/dx=10.

Figure 23: y=x40 for x near 1 : Density		Figure 24: y=x40 for x near 1 : Cumulative

As you can see, the y-values are spread over the interval from 0.13 to 7.04. Hint: that’s 1/e² to e².

What’s worse is that the distribution is neither rectangular nor Gaussian, not even close. It is strongly peaked at the low end. The HWHM is very small, while the overall width is enormous. The mode of the distribution is not 1, the mean is not 1, and the median is not 1. So the typical abscissa (x=1) does not map to the typical ordinate.

This is an example where Crank Three Times gives spectacularly asymmetric error bars, which is a warning. There are lots of distributions in this world that cannot be described using the notion of “point plus error bars”.

This is not primarily a «sig figs» problem. However, as usual, no matter what you are doing, you can always make it worse by using «sig figs». The uncertainty on y is larger than y, so «sig figs» cannot even represent this result! If you tried, you would end up with zero significant digits.

Also, the usual propagation rules, as taught in conjunction with «sig figs», say that x multiplied by x has the same number of «sig figs» as x. Do that 40 times and you’ve still got the same number. So the «sig figs» alleged uncertainty on y is just 0.05 ... but reality begs to differ.

7.5  Example: Beyond First Order: Momentum and Energy

Suppose we have a bunch of particles in thermal equilibrium. The x component of momentum is Gaussian distributed, with mean 0 and standard deviation √mkT. The distribution is the same for the y and z components. For simplicity, lets choose units such that m=1, and momentum is equal to velocity. A scatter plot of the x and y components is shown in figure 25.

Figure 25: Thermal Distribution of Velocities

The kinetic energy of any given particle is p²/(2m). The uncertainty in the mass is negligible in this situation. This situation is simple enough that the right answer can be found analytically, as some guy named Maxwell did in the mid-1800s. You can also find the right answer using Monte Carlo techniques. If the situation were even slightly more complicated, Monte Carlo would be the only option.

If you calculate the energy for an ensemble of such particles, the cumulative probability is shown in figure 26. Similarly, the probability density distribution is shown in figure 27. The dashed red line shows the exact analytic result, i.e. the Maxwell-Boltzmann distribution.

Figure 26: Maxwell-Boltzmann Distribution of Energy (3D)		Figure 27: Maxwell-Boltzmann Distribution of Energy (3D)
Cumulative Probability		Probability Density

If you tried to obtain the same result using step-by-step propagation of uncertainty, starting from the thermal distribution of velocities, things would not go well. Using the procedure given in section 7.20.2, you would find that the relative uncertainty was infinite. Forging ahead, applying the formula without regard to the provisos in the rule, this would imply an energy of zero plus-or-minus infinity. This is nowhere close to the right answer.

We can discuss the failure of the step-by-step approach in terms of the unholy trinity of Misrepresentation, Malexpansion, and Correlation.

• Sig figs cannot represent the distribution of velocities. It cannot represent 0±√mkT or anything remotely like that.
• The first-order Taylor expansion fails. There is no first-order term in the expansion for E in terms of p.
• Correlation issues are not a problem in this situation.

This example and the next one were chosen because they are simple, and because they make obvious the failure of the step-by-step approach. Beware that in situations that are even slightly more complex, the step-by-step approach will fail and give you wrong answers with little or no warning.

7.6  Example: Non-Differentiable: Time = Distance / Rate

Suppose we have a long, narrow conference table. We start a particle in the middle of the table. At time t=0 we give it a velocity based on a thermal distribution, zero plus-or-minus √kT/m. Thereafter it moves as a free particle, moving across the table. We want to know how lot it takes before the particle falls off the edge of the table. A scatter plot of the velocity is shown in figure 25. For present purposes, only the x component matters, because the table is narrow in the x direction and very very long in the y direction.

If we take the Monte Carlo approach, this is an ultra-simple “time = distance / rate” problem. For each element of the ensemble, the time to fall off is:

where w is the width of the table, and v is the velocity.

The cumulative probability distribution is shown in figure 28. A histogram of the probability density is shown in figure 29.

Figure 28: Time to Fall Off		Figure 29: Time to Fall Off
Cumulative Probability		Probability Density

Beware that not all the data is visible in these figures. Given an ensemble of 1000 points, it would not be uncommon to find the maximum time to be greater than 1000 units, or indeed greater than 2000 units. The maximum-time point corresponds to the minimum-velocity point, and velocites near zero are not particularly uncommon. That means that the probability density distribution converges only very slowly toward zero at large times. As a consequence, the mean of the distribution is large, vastly larger than the mode. The standard deviation could be in the hundreds, which is vastly larger than the HWHM.

We can contrast the Monte Carlo approach to step-by-step first-order propagation. The latter fails miserably. In the first step, we need to take the absolute value of the velocity. To calculate the uncertainty, we need the derivative of this, evaluated at the origin, but alas absolute value is not a differentiable function at the origin. In the second step, we need to take the reciprocal, which is not even a function at the origin, much less a differentiable function.

This example and the previous one were chosen because they are simple, and because they make obvious the failure of the step-by-step approach. Beware that in situations that are even slightly more complex, the step-by-step approach will fail and give you wrong answers with little or no warning.

Extensions: This simple example is part of a larger family. It can be extended and elaborated in various ways, including:

• Motion in 2 or 3 dimensions, not just one.
• Uncertainty in the initial position, not just velocity. At some level, this is required by the Heisenberg uncertainty principle.
• Motion in the presence of a nontrivial potential. Example: time to fall over, for the 1D motion of a thin flat slab balanced on its edge. Example: time to fall over, for the 2D motion of a pencil balanced on its point.

7.7  Example: Correlated Data: Charge-to-Mass Ratio

Suppose we want to know the charge-to-mass ratio for the electron, i.e. the e/m ratio. This is useful because it shows up in lots of places, for instance in the formula for the cyclotron frequency (per unit field).

We start by looking up the accepted values for e and m, along with the associated uncertainties. Here are the actual numbers, taken from the NIST website:

At this point it is amusing to calculate the e/m ratio by following the propagation-of-error rules that you see in textbooks. Ask yourself, What is the calculated uncertainty for the e/m ratio, when calculated this way? Choose the nearest answer:

a) 22 ppb

b) 33 ppb

c) 44 ppb

d) 50 ppb

e) 66 ppb

Note: Ordinarily I play by the rule that says you are expected to use everything you know in order to get the real-world right answer. Ordinarily I despise questions where knowing the right answer will get you into trouble. However ... at the moment I’m making a point about the method, not trying to get the right answer, so this rule is temporarily suspended. You’ll see why shortly.

If we carry out the calculation in the usual naïve way, we assume the uncertainties are uncorrelated, so we can add the relative uncertainties in quadrature:

so the full result is

We can contrast this with the real-world correct value:

The real uncertainty is vastly less than the naïvely-calculated uncertainty.

We can understand this as follows: The accepted values for e and m are correlated. Virtually 100% correlated.

Simple recommendation: If you want to calculate e/m, don’t just look up the values for e and m separately. Use the NIST website to look them up jointly along with the correlation coefficient.

Before we go on, lets try to understand the physics that produces the high correlation between e and m. It’s an interesting story: You could measure the mass of the electron directly, but there’s not much point in doing so, because it turns out that indirect methods work much better. It’s a multi-step process. The details are not super important, but here’s a slightly simplified outline of the process.

A) The fine structure constant is measured to 0.32 ppb relative uncertainty.

B) The Rydberg constant is measured to 0.005 ppb.

C) The Rydberg constant is equal to m e⁴ / 8 є₀² h³ c and the fine-structure constant is e² / 2 є₀ h c.

Combining α³/Ry gives e²/m to 0.96 ppb. It hardly matters whether they are correlated or not, since the uncertainty is dominated by the uncertainty in α³. Note that the speed of light is exact, by definition, so it does not contribute to the uncertainty.

D) The charge on the electron is measured to 22 ppb.

E) If you want the e/m ratio, divide e²/m by e. The uncertainty in e/m is dominated by the uncertainty in e.

F) To find the mass, calculate e² (using the measured charge directly) then divide by the e²/m value obtained in item (c) above. The uncertainty is 44 ppb, dominated by the uncertainty in e².

Bottom line: Whenever you have two randomly-distributed quantities and you want to combine them – by adding, subtracting, multiplying, dividing, or whatever – you need to find out whether they are correlated. Otherwise you will have a hard time calculating the combined uncertainty.

7.8  Example: Solving a Quadratic Polynomial for the pH

Figure 30 shows pH as a function of concentration, for various pK_a values, including weak acids and strong acids, as well as intermediate-strength acids, which are particularly interesting.

Figure 30: pH versus Concentration for Various pK_a Values

This is obviously not a contrived example. There are plenty of good reasons for preparing a plot like this. For present purposes, however, we are not particularly interested in the meaning of this figure, but rather in the process of computing it. (If you are interested in the meaning, please see reference 13.)

For simplicity, we temporarily restrict attention to the parts of figure 30 that are not too near the top. That is, we focus attention on solutions that are definitely acidic, with a pH well below the pH of water. (This restriction will be lifted in section 7.9.)

In this regime, the relevant equation is:

Equation 14 is a quadratic polynomial, where the coefficients are:

It has one positive root and one negative root, as we shall see. For more on where this comes from and what it means, see reference 13 and references cited therein.

Let’s plug in the numbers for our dilute solution of a strong acid:

Let’s use the numerically stable version of the quadratic formula, as discussed in reference 14:

where sgn_L(x) is the left-handed sign-function, which is defined to be −1 whenever x is less than or equal to zero, and +1 otherwise. In most computer languages it can be implemented as (2*(x>0)-1). (Do not use the regular sgn() function, which is zero at zero.) The names “small” and “large” are based on the absolute magnitude of the roots.

That gives us:

You can see that this is definitely a “big root / small root” situation, so you need to use the smart version of the quadratic formula, for reasons explained in reference 14.

Only the positive root in equation 18 makes sense. Taking the logarithm, we find

Note that the “small root” here is not some minor correction term; it is the entire answer.

For a discussion of the lessons we can learn from this example, see section 7.11.

We revisit this example again in section 7.23, in connection with the rules for step-by-step first-order propagation of uncertainty.

7.9  Example: Solving a Cubic Polynomial for the pH

We now consider the full pH versus concentration diagram, without the restrictions on strength and/or concentration imposed in section 7.9.

The full curves in figure 30 were computed by solving the following equation.

That’s a cubic, with one positive root and two negative roots. For more on where this comes from and what it means, see reference 13.

It is easy to solve the equation with an iterative root-finding algorithm.

In contrast, beware that standard “algebraic” formulas for solving the cubic can give wrong answers in some cases. Depending on details of the implementation, the formulas can be numerically unstable. That is to say, the result gets trashed by roundoff errors. Specifically: I tried using the standard library routine gsl_poly_complex_solve_cubic() and it failed spectacularly for certain values of pK_a and pC_HA. Some of the alleged results were off by multiple orders of magnitude. Some of the alleged results were complex numbers, even though the right answers were real numbers. It might be possible to rewrite the code to make it behave better, but that’s not a job I’m eager to do.

For a discussion of the lessons we can learn from this example, see section 7.11.

7.10  Another Example: Multi-Step Relativity

7.10.1  Correct Direct Calculation

Once upon a time, at Acme Anvil company, there was an ensemble of particles. The boss wanted a relativistically-correct calculation of the kinetic energy. He especially wanted the mean and standard deviation of the ensemble of kinetic-energy values.

The boss assigned two staffers to the task, Audrey and Alfred. Audrey worked all morning computing the total energy E(v) and the rest energy E(0) for each particle. Then Alfred worked all afternoon, subtracting these two quantities to find the kinetic energy for each particle.

In all cases, Audrey and Alfred used the relativistically correct formulas, namely

The following data describes a typical particle in the ensemble:

For this particle, Audrey calculated the following results:

where both of those numbers are repeating decimals.

Later, Alfred subtracted those numbers to obtain

which is again a repeating decimal.

After calculating the kinetic energy for all the particles, Alfred calculated the mean and standard deviation, namely:

which is in fact the correct answer.

7.10.2  Unsuccessful Double-Precision Direct Calculation

Meanwhile, across the street at Delta Doodad Company, they needed to do the exact same calculation. The boss assigned Darla and Dave to do the calculation.

Darla calculated E(v) and E(0) using a spreadsheet program, which represents all numbers using IEEE double-precision floating point. For the typical particle described in equation 22, she obtained:

These numbers cannot be represented to any greater accuracy using IEEE double precision.

When Dave subtracted these numbers, he found the kinetic energy was zero. In fact the apparent kinetic energy was zero for all particles. When he calculated the mean and standard deviation, they were both zero. Alfred suspected that 0±0 was not the correct answer, but given what he had to work with, there was no way for him to compute a better answer.

The problem is that IEEE double precision can only represent about 16 decimal digits, whereas at least 20 digits are needed to obtain a useful answer in this case. If you use less than 20 digits, the roundoff error will be unacceptably large. (By way of contrast, across the street, Audrey used 25 digits just to be on the safe side.)

7.10.3  Gross Failure: Sig Figs

Meanwhile, down the street at General Gadget Company, they needed to do the same calculation. The boss was a big fan of sig figs. He demanded that everybody adhere to the sig figs rules.

The boss assigned Gail and Gordon to the task. In the morning, Gail calculated the total energy and rest energy. She noticed that there was some uncertainty in these numbers. The relative uncertainty was about 0.5%. So for the typical particle described in equation 22, she obtained:

In accordance with the usual sig figs rules, Gail rounded off these numbers, as follows:

Gail’s reasons for rounding off included:

1. She felt obliged to communicate the uncertainty to Gordon. Writing down a large number of digits (as Audrey did in section 7.10.1) would “imply” – via the sig-figs rules – a very small uncertainty, which in this case would be quite wrong. It would be downright dishonest.
2. Equation 28 “looks nicer” than equation 27.
3. She knew the boss would get angry and call call her “numerically ignorant” if she wrote down a bunch of trailing digits, i.e. uncertain, irreproducible digits.

All in all, it was “obvious” to Gail that equation 28 was the right way to express things.

In the afternoon, Gordon subtracted these numbers. He found that every particle had zero kinetic energy.

Based on the uncertainty in the numbers he was given, he tried to apply the propagation-of-error rules. Since Gail did not report any correlations, he assumed all her results were uncorrelated, so that the rules presented in section 7.20 could be applied. On this basis, he estimated that the uncertainty in the difference was about ± 1×10¹⁵. So Gordon could have reported his result as 0± 1×10¹⁵ joule.

That’s the wrong answer. Gordon’s estimate of the mean is wrong by about 200 standard deviations. That’s a lot. Gordon’s estimate of the standard deviation is also off by about seventeen orders of magnitude. That’s a lot, too.

One problem is that Gail didn’t feed Gordon enough digits. She actually calculated enough digits, but she felt obliged to round off her results, in accordance with the sig figs rules. This illustrates a general principle:

Another problem is that for each particle, Gail’s numbers for E(v) and E(0) have very highly correlated uncertainties. Therefore Gordon’s application of the propagation-of-error rules was invalid.

Thirdly, just to add insult to injury: The sig-figs method does not provide any way to represent 0 ± 1×10¹⁵, so Gordon could not find any way to report his results at all. The boss wanted a sig-figs representation, but no such representation was possible.

7.10.4  Algebraic Simplification

Meanwhile, across town at Western Widget Company, yet another company was faced with the same task. At this company, they noticed that equation 21 implies that:

where on the second line we have used some trigonometric identities. Both lines in equation 29 share an important property: the factor in square brackets is a purely mathematical function. The function can be defined in terms of a subtraction that involves no uncertainty of any kind. In contrast, if you were to multiply through by m c² before subtracting, you would then face the problem of subtracting two things that not only have some uncertainties (because of the uncertainty in m) but would have highly correlated uncertainties.

It must be emphasized that equation 29 is relativistically correct; no approximations have been made (yet).

Since the task at hand involves ρ values that are very small compared to 1, the following approximations are good to very high accuracy:

You can check that these approximations are consistent with each other to third order in ρ or better, in the sense that they uphold the identities tanh= sinh/cosh and cosh² − sinh² = 1.

Plugging into equation 29 we find that, with more than enough accuracy,

which allows us to calculate the kinetic energy directly. No subtractions are needed, and ordinary floating-point arithmetic gives us no roundoff-error problems. The next term in the series is smaller than the E_kin by a factor of v²/c², as you can easily verify.

We apply this formula to all the particles, and then calculate the mean and standard deviation of the results. The answer is E_kin = 1.481(7) joule, which is identical to the result obtained by other means in section 7.10.1.

7.11  Discussion: Loss of Significance

The pH examples in section 7.8 and section 7.9 are obviously real-world examples. They are typical of examples that come up all the time, in many different situations, ranging from astronomy to zoology.

The relativity example in section 7.10 is a bit more contrived, but it illustrates an important theoretical point about the relationship between special relativity and classical dynamics. It is representative of a wider class of problems ... just simplified for pedagogical purposes.

There are a number of lessons we can learn from these examples:

1. Something that purports to be an “exact” closed-form solution is not exact at all if you have to evaluate it using floating point numbers or other rounded-off numbers. Ironically, in practice, an approximate and/or iterative solution might be much more accurate than the purportedly “exact” formula.
2. In section 7.8, even the lame “textbook” version of the quadratic formula would have worked if all of the calculations had been done using double precision. We only got into real trouble when we copied down the numbers and rounded them off to some «common sense» number of digits.

   Therefore: When using a calculator or any kind of computer, it is good practice to leave the numbers in the machine (rather than writing them down and keying them in again later). Learn how to use the STORE and RECALL functions on your calculator. Most machines use at least 15 digits, which is usually more than you need, but since keeping them is just as convenient as not keeping them, you might as well keep them all. (In contrast, writing the numbers down and keying them in again is laborious and error-prone. You will be tempted to round them off. Even the effort of deciding how much roundoff is tolerable is more work than simply leaving the numbers in the machine.)

   In the spirit of “check the work”, it is reasonable to write down intermediate results, but you should leave the numbers in the machine also. When you recall a number from storage, you can check to see that it agrees with what you wrote down.

3. On the other hand, the library function used in section 7.8 fails, even though it is using IEEE double precision.

   Double precision is not infinite precision.

4. These failures serve as a reminder of the difference between uncertainty and significance. The internal calculations, if they are to have any hope of working, require a large number of digits, out of all proportion to the uncertainty of the inputs and/or the required tolerance on the outputs.

   To put it bluntly: If you see an expression of the form:

   X   =   ⎛ ⎜ ⎝   1.497925297894696 ...  ± 0.01 ⎞ ⎟ ⎠       (incomplete)

   (32)

   you should not assume it is safe to round things off. It may be that such a number already has too few digits. It may already have been rounded off too much.

   Equation 32 is marked “incomplete” for the following reason: Suppose you need to write down something to represent the distribution X. The problem is, because of the correlations, it is not sufficient to report the variance; you need to report the covariances as well. The equation as it stands is not wrong, but without the covariances it is incomplete and possibly misleading.

   Not that the ± notation can only represent the variance (or, rather, the square root thereof), not the covariances, so it cannot handle the task when there are nontrivial correlations.

5. One way a loss of significance can occur is via accumulation of small errors in a multi-step calculation, as in section 7.12.
6. Another very common way a loss of significance can occur is via “small differences between large numbers”. More specifically:
   • If you have a small difference between large numbers and the fluctuations are uncorrelated, you suffer from a noise amplifier.

     In the relativity example considered in section 7.10, E(v) is in fact highly correlated with E(0). I know (based on how the particles were prepared) that there is some uncertainty in the mass of the particle. A factor of mass is common to both of the terms that are being subtracted. The uncertainty in the particle velocity is relatively small, so all in all there is nearly 100% correlation in the uncertainties. (There is of course no uncertainty in the speed of light, since it is 299792458 m/s by definition.)

   • If you have a small difference between large numbers and the fluctuations are highly correlated, then the intrinsic noise is smaller than it looks, but you suffer from a roundoff error amplifier.

   It is all-too-common to find expressions for the roots of a polynomial that depend on subtracting numbers that are highly correlated.

7. The technique of restructuring a calculation so as to avoid a loss of significance falls under the heading of “numerical methods”. There are entire books devoted to the subject, e.g. reference 15.

   The same idea can be applied to experiments, not just calculations. For example, to avoid a problem with small differences between large numbers, you can use null measurements, differential measurements, bridge structures (such as a Wheatstone bridge), et cetera.

8. Expanding things to lowest order is one of the most commonly used tools in the scientist’s toolbox.
9. It must be emphasized that when Audrey wrote down her numbers for E(v) and E(0), she did not know the uncertainty. This is typical of a great many real world situations: Often you need to write down a number when the uncertainty is not known ... and may not be know until weeks or months later, if at all.

   As mentioned in item 4, my advice is: If you have a number that ought to be written down, write it down. Just write it down already. You can worry about the uncertainty later, if necessary. Write down plenty of guard digits. The number of digits you write down does not imply anything about the uncertainty, precision, tolerance, significance, or anything else.

10. Contrary to what what Gail’s boss was telling her, you are not obliged to attach an implicit (or explicit) uncertainty to numbers you write down. If you have an ensemble of numbers, you might be able to summarize it in terms of a mean and a standard deviation, but you might not ... and even if you are able to summarize it, you are not obliged to. The ensemble speaks for itself, better than any summary ever could. Adding the width of the error bars to the width of the ensemble makes things very much worse, as discussed in section 5.2 and reference 2. In section 7.10.1 Alfred was able to calculate the ensemble of kinetic energy values just fine without assigning any uncertainty to Audrey’s E(v) and E(0) numbers.

    Indeed, in section 7.10.3, Gail’s uncertainty numbers were in some hyper-technical sense correct, but they were highly misleading. They were worse than nothing, because the correlations were not taken into account.

11. It really makes me cringe when students get points taken away and get called “numerically ignorant” for doing exactly the right thing, i.e. keeping plenty of guard digits.

7.12  Example: Signal Averaging: Extracting a Signal from Noise

There are lots of situations where the uncertainty in the final answer is less than the uncertainty in the raw data.

This can be understood in terms of “signal to noise” ratio. When we process lots of data, if we do things right, the signal will accumulate faster than the noise. (Conversely, if we don’t do things right, the accumulated errors can rapidly get out of hand.)

We now consider an example that illustrates this point. For simplicity, we assume the raw data is normally distributed and uncorrelated, as shown in figure 31. The spreadsheet for creating this figure is in reference 16. In this section we assume the analysis is done correctly; compare section 7.13.

Figure 31: Extracting a Signal from Noisy Data

Specifically, each data point is drawn from a Gaussian distribution that has a width of 0.018 units. Suppose we run the experiment many times. On each run, we take the average of 100 points. We know the average much more accurately than we know any particular raw data point. In fact, if we look at all the runs, the averages will have a distribution of their own, and this distribution will have a width of only 0.0018 units, ten times narrow than the distribution of raw data points. The distribution of averages is represented by the single black point with error bars at the top of figure 31. (This is a cooked data point, not a raw data point.)

We can say the same thing using fancy statistical language. Each run is officially called a sample. Each sample contains N raw data points. We assume the points are IID, normally distributed. We compute the mean of each sample. Theory tells us that the sample means behave as if they were drawn from a Gaussian distribution, which will be narrower than the distribution of raw data, narrower by a factor of √N.

7.13  Example: The Effect of Roundoff Error

Let’s re-analyze the data from section 7.12. In particular, let’s consider the effect of roundoff errors that occur while we are calculating the average. Even though the raw data is normally distributed and IID, the roundoff errors will not be normally distributed, and if we’re not careful this can lead to serious problems.

We denote the ith raw data point by a_i. It is drawn from a distribution A that has some uncertainty σ_A.

Next, we round off each data point. That leaves us with some new quantity b_i. These new points behave as if they were drawn from some new distribution B.

The new uncertainty σ_B will be larger than σ_A, but we don’t know how much larger, and we don’t even know that distribution B can be described as a Gaussian (or any other two-parameter model). It may be that B is a viciously lopsided non-normal distribution (even though A was a perfectly well-behaved normal distribution).

This is illustrated by the example worked out in the “roundoff” spreadsheet (reference 16), as we now discuss. The first few rows and the last few rows of the spreadsheet are reproduced here. The numbers in red are seriously erroneous.

The leftmost column is a label giving the row number. The next column is the raw data. You can see that the raw data consists of numbers like 0.048. As usual, the raw data points have no width whatsoever. However, the distribution from which these numbers were drawn has a width of 0.018. You can see that we are already departing from the usual “significant figures” hogwash. If you believed in sig figs, you would attribute considerable uncertainty to the second decimal place in each raw data point, and you would not bother to record the data to three decimal places.

In contrast, in reality, it is important to keep that third decimal place, for reasons that will become clear very soon. We are going to calculate the average of 100 such numbers, and the average will be known tenfold more accurately than any of the raw inputs.

To say the same thing in slightly different terms: there is in fact an important signal – a significant signal – in that third decimal place. The signal is obscured by noise; that is, there is a poor signal-to-noise ratio. Your mission, should you decide to accept it, is to recover that signal.

This sort of signal-recovery is at the core of many activities in real research labs, and in industry. On ordinary GPS receiver depends on signals that are hundreds of times less powerful than the noise (SNR on the order of -25 dB). The second thing I ever did in a real physics lab was to build a communications circuit that picked up a signal that was ten million times less powerful than the noise (SNR = -70 dB). The JPL Deep Space Network deals with SNRs even worse than that. Throwing away the signal at the first step by “rounding” the raw data would be a Bad Idea.

Take-home message #1: Signals can be dug out from the noise. Uncertainty is not the same as insignificance. A digit that is uncertain (and many digits to the right of that!) may well carry some significance that can be dug out by techniques such as signal-averaging. Given just a number and its uncertainly level, without knowing the context, you cannot say whether the uncertain digits are significant or not.

Take-home message #2: An expression such as 0.048 ± 0.018 expresses two quantities: the value of the signal, and an estimate of the noise. Combining these two quantities into a single numeral by rounding (according to the “significant figures rules”) is highly unsatisfactory. In cases like this, if you round to express the noise, you destroy the signal.

Now, returning to the numerical example: I assigned three students (Alice, Bob, and Carol) to analyze this data. In the data table, the first column under each student’s name is a running sum. The second column is a running estimate of the uncertainty of the running sum.

Alice didn’t round any of the raw data or intermediate results. She got an average of

and the main value (0.0435) is the best that could be done given the points that were drawn from the ensemble. (The error-estimate is a worst-case error; the probable error is somewhat smaller.)

Meanwhile, Bob was doing fine until he got to row 31. At that point he decided it was ridiculous to carry four figures (three decimal places) when the estimated error was more than 100 counts in the last decimal place. He figured that if rounded off one digit, there would still be at least ten counts of uncertainty in the last place. He figured that would give him not only “enough” accuracy, but would even give him a guard digit for good luck.

Alas, Bob was not lucky. Part of his problem is that he assumed that roundoff errors would be random and would add in quadrature. In this case, they aren’t and they don’t. The errors accumulate linearly (not in quadrature) and cause Bob’s answer to be systematically high. The offset in the answer in this case is slightly less than the error bars, but if we had averaged a couple hundred more points the error would have accumulated to disastrous levels.

Carol was even more unlucky. She rounded off her intermediate results so that every number on the page reflected its own uncertainty (one count, possibly more, in the last digit). In this case, her roundoff errors accumulate in the “down” direction, with spectacularly bad effects.

The three students turned in the following “bottom line” answers:

  
Note that Alice, Bob, and Carol are all analyzing the same raw data; the discrepancies between their answers are entirely due to the analysis, not due to the randomness with which the data was drawn from the ensemble.

Alice obtains the correct result. This is shown by the single black point with error bars at the top of figure 31. Bob’s result is slightly worse, but similar. Carol’s result is terrible, as shown by the red point with error bars at the top of figure 31.

Take-home message #3: Do not assume that roundoff errors are random. Do not assume that they add in quadrature. It is waaaay too easy to run into situations where they accumulate nonrandomly, introducing a bias into the result. Sometimes the bias is obvious, sometimes it’s not.

Important note: computer programs⁶ and hand calculators round off the data at every step. IEEE 64-bit floating point is slightly better than 15 decimal places, which is enough for most purposes but not all. Homebrew numerical integration routines are particularly vulnerable to serious errors arising from accumulation of roundoff errors.

One of the things that contributes to Bob’s systematic bias can be traced to the following anomaly: Consider the number 0.448. If we round it off, all at once, to one decimal place, we get 0.4. On the other hand, if we round it off in two steps, we get 0.45 (correct to two places) which we then round off to 0.5. This can be roughly summarized by saying that the roundoff rules do not have the associative property. If you have this problem, you might find it amusing to try the round-to-even rule: round the fives toward even digits. That is, 0.75 rounds up to 0.8, but 0.65 rounds down to 0.6. There are cases where this is imperfect (e.g. 0.454) but it’s better overall, it’s easy to implement, and it has a pleasing symmetry. (This rule has been invented and re-invented many times; I re-invented it myself when I was in high school.) Alas, it is not really an improvement in any practical sense.

The important point is this: If fiddling with the roundoff rules produces a non-negligible change in the results, it means you are in serious trouble. It means the situation is overly burdened by roundoff errors, and fiddling with the roundoff rules is just re-arranging deck chairs on the Titanic. Usually the only real solution is to use more precision (more guard digits) during the calculation ... or to use a different algorithm, so that fewer steps (hence fewer roundings) are required. If the rounding is part of a purely mathematical exercise, keep tacking on guard digits until the result is no longer sensitive to the details of the roundoff rules. If the rounding is connected to experimental data, consider redesigning the experiment so that less rounding is required, perhaps by nulling out a common-mode signal early in the process. This might be done using a bridge, or phaselock techniques, or the like.

You can play with the spreadsheet yourself. For fun, see if you can fiddle the formulas so that Bob’s bias is downward rather than upward. Save the spreadsheet (reference 16) to disk and open it with your favorite spreadsheet program.

Notes:

1. I’ve got automatic recalculation turned off; you can either turn it back on, or push your spreadsheet’s “recalculate” button (F9 or some such) when necessary.
2. Hiding in columns R, S, and T is a Box-Muller transformation to draw numbers randomly from a Gaussian distribution. You might think any decent spreadsheet would have a built-in function to generate a normal distribution, but some versions of Excel don’t. (Sometimes it’s provided by an add-in.) In any case, it’s good to know the Box-Muller trick.

Additional constructive suggestions and rules of thumb:

• Remember, uncertainty is not the same as insignificance.
• It is always safer to have too many digits than too few.
• Suppose you know the Nth digit is uncertain, for physics-related reasons. Any digits to the right of that are guard digits. They can’t hurt unless they become unduly laborious – but now in the era of hand calculators and spreadsheet programs it’s often easier to carry huge numbers of digits than it is to figure out exactly how many digits are needed at each step. The purpose of the guard digits is to ensure that accumulated roundoff errors remain smaller than the physics-related errors.
• Before you decide to round off, you must do some sort of theoretical and/or operational check to ensure that rounding doesn’t introduce a serious error.
• Beware that there are a lot of pseudo-experts and textbooks out there who have no understanding of signal recovery, and blindly put complete faith in some “sig digs rules”. Just because they do it doesn’t make it right.
• If you have something worth saying, don’t say it in terms of “significant digits”. There are better ways. For instance, rather than saying “this is good to 5 sig digs”, it would be better to say “this is accurate to 10 ppm”.
• See also section 8.2, reference 8, and reference 10.

There exist very detailed guidelines for rounding off if that turns out to be necessary.

   a) Here is a crude way to check whether we are carrying enough guard digits in the intermediate steps. Ask the question: If we did this twice, rounding up at every step in one case and rounding down at every step in the other case, would both cases give the same answer, to an acceptable approximation? If not, we need to carry more digits.

   b) Here is a more reliable check: Ask the question: If we ran the calculation N times, randomly rounding up or down at each step in each run, would every run give the same answer, to an acceptable approximation? If not, we need to carry more digits.

   c) In the case where the roundoff errors are small, you might be able to get away with an analytic approach. This involves looking at the first derivative of the final answer with respect to whatever quantity you want to round off.

This is risky in a multi-step or iterated calculation where many roundoff operations occur. That’s because you need to worry about accumulation of errors.

The main advantage is that if you have a problem and are trying to fix it, the analytic approach will probably tell you where to focus your attention. Very commonly, some steps require extra digits while other steps do not.

7.14  Crank Three Times™

Here’s a simple yet powerful way of estimating the uncertainty of a result, given the uncertainty of the thing(s) it depends on.

Here’s the procedure, in the simple case when there is only one input variable with appreciable uncertainty:

1. Set up the calculation. Do it once in the usual way, using the nominal, best-estimate values for all the input variables.
2. Then re-do the calculation with the uncertain variable at the end of its upper error bar.
3. Then re-do the calculation with the uncertain variable at the end of its lower error bar.

I call this the Crank Three Times™ method. Here is an example:

Equation 35 tells us that if x is distributed according to x = 2±.02 then 1/x is distributed according to 1/x = .5±.005. Equivalently we can say that if x = 2±1% then 1/x = .5±1%. We remark in passing that the percentage uncertainty (aka the relative uncertainty) is the same for x and 1/x, which is what we expect provided the uncertainty is small.

The Crank Three Times™ method is a type of “what if” analysis. We can also consider it a simple example of an iterative numerical method of estimating the uncertainty (in contrast to the step-by-step first-order methods described in section 7.20). This simple method is a nice lead-in to fancier iterative methods such as Monte Carlo, as discussed in section 7.16.

The Crank Three Times™ method is by no means an exact error analysis. It is an approximation. The nice thing is that you can understand the nature of the approximation, and you can see that better and better results are readily available (for a modest price).

One of the glories of the Crank Three Times™ method is that in cases where it doesn’t work, it will tell you it isn’t working, provided you listen to what it’s trying to tell you. If you get asymmetrical error bars, you need to investigate further. Something bad is happening, and you need to check closely to see whether it is a little bit bad or very, very bad.

As far as I can tell, for every flaw that this method has, the sig-figs method has the same flaw plus others ... which means Crank Three Times™ is Pareto superior.

This method requires no new software, no learning curve, and no new concepts beyond the concept of uncertainty itself. In particular, unlike significant digits, it introduces no wrong concepts.

Crank Three Times™ shouldn’t require more than a few minutes of labor. Once a problem is set up, turning the crank should take only a couple of minutes; if it takes longer than that you should have been doing it on a spreadsheet all along. And if you are using a spreadsheet, Crank Three Times™ is super-easy and super-quick.

If you have N variables that are (or might be) making a significant contribution to the uncertainty of the result, the Crank Three Times™ method could more precisely be called the Crank 2N+1 Times™ method. Here’s the procedure: Set up the spreadsheet and wiggle each variable in turn, and see what happens. Wiggle them one at a time, leaving the other N−1 at their original, nominal values.

If you are worried about what happens when two of the input variables are simultaneously at the ends of their error bars, you can check that case if you want. However, beware that if there are many variables, checking all the possibilities is exponentially laborious. Furthermore, it is improbable that many variables would simultaneously take on extreme values, and checking extreme cases can lead you to overestimate the uncertainty. For these reasons, and others, if you have numerous variables and need to study the system properly, at some point you need to give up on the Crank Three Times™ method and do a full-blown Monte Carlo analysis.

In the rare situation where you want a worst-case analysis, you can move each variable to whichever end of its error bar makes a positive contribution to the final answer, and then flip them all so that each one makes a negative contribution. In most cases, however, a worst-case analysis is wildly over-pessimistic, especially when there are more than a few uncertain variables.

Remember: there are many cases, especially when there are multiple uncertain variables and/or correlations among the variables and/or nonlinearities, your only reasonable option is Monte Carlo, as discussed in section 7.16. The Crank Three Times™ method can be considered an ultra-simplified variation of the Monte Carlo method, suitable for introductory reconnaissance.

Here is another example, which is more interesting because it exhibits nonlinearity:

Equation 36 tells us that if x is distributed according to x = 2±.9 then 1/x is distributed according to 1/x = .5(+.41−.16). Equivalently we can say that if x = 2±45% then 1/x = .5(+82%−31%). Even though the error bars on x are symmetric, the error bars on 1/x are markedly lopsided.

Lopsided error bars are fairly common in practice. Sometimes they are merely a symptom of a harmless nonlinearity, but sometimes they are a symptom of something much worse, such as a singularity or a branch cut in the calculation you are doing.

This is vastly superior to the step-by-step first-order methods discussed in section 7.20, which blissfully assume everything is linear. That is to say, in effect they expand everything in a Taylor series, and keep only the zeroth-order and first-order terms. In cases where this is not a good approximation, you are likely to get wrong answers with little or no warning.

Here is yet another example, which is interesting because it shows how to handle correlated uncertainties in simple cases. The task is to calculate the molar mass of natural bromine, given the nuclide mass for each isotope, and the corresponding natural abundance.

The trick here is to realize that the abundances must add up to 100%. So if one isotope is at the low end of its error bar, the other isotope must be at the high end of its error bar. So the abundance numbers are anticorrelated. This is an example of a sum rule. For more about correlations and how to handle them, see section 7.16.

(The uncertainties in the mass of each nuclide are negligible.)

So by comparing the three columns (light case, nominal case, and heavy case), we find the bottom-line answer: The computed molar mass of natural bromine is 79.90361(52). This is the right answer based on a particular sample of natural bromine. The usual “textbook” value is usually quoted as 79.904(1), which has nearly twice as much uncertainty, in order to account for sample-to-sample variability.

Note that if you tried to carry out this calculation using “significant figures” you would get the uncertainty wrong. Spectacularly wrong. Off by two orders of magnitude. The relative uncertainty in the molar mass is two orders of magnitude smaller than the relative uncertainty in the abundances.

7.15  Another Example: Magnesium Mass, Preliminary Attempt

This is based on question 3:21 on page 122 of reference 17.

Suppose we want to calculate (as accurately as possible) the molar mass of natural magnesium, given the mass of the various isotopes and their natural abundances.

Many older works referred to this as the atomic mass, or (better) the average atomic mass ... but the term molar mass is strongly preferred. For details, see reference 18.

The textbook provides the raw data shown in table 7.

---

isotope	molar mass / dalton	abundance
24Mg	23.9850	78.99%
25Mg	24.9858	10.00%
26Mg	25.9826	11.01%

Table 7: Isotopes of Magnesium, Rough Raw Data

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The textbook claims that the answer is 24.31 dalton and that no greater accuracy is possible. However, we can get a vastly more accurate result.

The approach in the textbook has multiple problems:

• The textbook uses methods that vastly overestimate the halfwidth of the distribution. This damages the estimated uncertainty.
• The textbook fails to take into account the fact that the abundance variables are highly correlated, as they must be, since they always sum to 100%. This further damages the estimated uncertainty.
• The textbook expresses the uncertainty using the dreaded “sig digs” rules, which give only a crude indication of the halfwidth of the distribution. This does yet more damage to the estimated uncertainty.
• The textbook rounds off the answer according to the usual foolish “sig digs” rules, which don’t permit guard digits. The roundoff error damages the nominal value.

It is tempting to blame all the problems on the “sig digs” notation, but that wouldn’t be fair in this case. The primary problem is mis-accounting for the uncertainty, and as we shall see, we are still vulnerable to mis-accounting even if the uncertainty is expressed using proper notation.

Similarly note that even if we did manage to get good estimate of the uncertainty, the “sig digs” rules would not have called for such drastic rounding. So the propagation-of-error issues really are primary.

Let’s make a preliminary attempt to figure out what’s going on. If we clean up the notation, it will facilitate understanding and communication. In particular, it will expose a bunch of problems that the text sweeps under the rug.

We can start by re-expressing the textbook data so as to make the uncertainties explicit. We immediately run into some unanswerable questions, because the “sig digs” notation in table 7 gives us only the crudest idea of the uncertainty ... is it half a count in the last decimal place? Or one count? Or more??? If we use only the numbers presented in the textbook, we have to guess. Let’s temporarily hypothesize a middle-of-the-road value, namely three counts of uncertainty in the last decimal place. We can express this in proper notation, as shown in table 8.

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isotope	molar mass / dalton	abundance
24Mg	23.9850(3)	78.99(3)%
25Mg	24.9858(3)	10.00(3)%
26Mg	25.9826(3)	11.01(3)%

Table 8: Isotopes of Magnesium, Rough Data with Explicit Uncertainty

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This gives the molar mass of the ²⁵Mg isotope with a relative accuracy of 12 parts per million (12 ppm), while the abundance is given with a relative accuracy of 3 parts per thousand (3000 ppm). So in some sense, the abundance number is 250 times less accurate.

If you think about the data, you soon realize that the abunance numbers are in percentages, and must add up to 100%. We say there is a sum rule.

The sum rule means the uncertainty in any one of the abundance numbers is strongly anticorrelated with the uncertainty in the other two. The widely-taught pseuo-sophisticated “propagation of uncertainty” rules don’t take this into account; instead, they rashly assume that all errors are uncorrelated. If you just add up the abundance numbers without realizing they are percentages, i.e. without any sum rule, you get

with (allegedly) 500 ppm uncertainty, even though the sum rule tells us they actually add up to 100 with essentially no uncertainty:

Even if you imagine that equation 38 is not perfectly exact – perhaps because it fails to account for some fourth, hitherto-unknown isotope – the sum must still be very nearly 100%, with vastly less uncertainty than equation 37 would suggest.

To say the same thing another way, we are talking about three numbers (the percent abundance of the three isotopes). Taken together, these numbers specify a point in some abstract three-dimensional space. However, the valid, physically-significant points are restricted to a two-dimensional subspace (because of the sum rule).

Here’s another fact worth noticing: All three isotope masses are in the same ballpark. That means that uncertainties in the abundance numbers will have little effect on the sought-after average mass. Imagine what would happen if all three isotopes had the same identical mass. Then the percentages wouldn’t matter at all; we would know the average mass with 12 ppm accuracy, no matter how inaccurate the percentages were.

There are various ways to take the “ballpark” property into account.

One method, as pointed out by Matt Sanders, is to subtract off the common-mode contribution by artfully regrouping the terms in the calculation. That is, you can subtract 25 (exactly) from each of the masses in table 8, then take the weighted average of what’s left in the usual way, and then add 25 (exactly) to the result. The differences in mass are on the order of unity, i.e. 25 times smaller than the masses themselves, so this trick makes us 25 times less sensitive to problems with the percentages. We are still mis-accounting for the correlated uncertainties in the percentages, but the mis-accounting does 25 times less damage.

The idea of subtracting off the common-mode contribution is a good one, and has many applications. The idea was applied here to a mathematical calculation, but it also applies to the design of experimental apparatus: for best accuracy, make a differential measurement or a null measurement whenever you can.

To summarize, subtracting off the common-mode contribution is a good trick, but (a) it requires understanding the problem and being somewhat devious, (b) in its simplest form, it only works if the problem is linear, (c) it doesn’t entirely solve the problem, because it doesn’t fully exploit the sum rule.

7.16  Magnesium Mass, Monte Carlo Solution

The situation described in section 7.15 has so many problems that we need to start over.

For one thing, if we’re going to go to the trouble of calculating things carefully, we might as well use the best available data (rather than the crummy data given in the textbook, i.e. table 8). A secondary source containing mass and abundance data for the isotopes of various elements can be found in reference 19. We can use that for our mass data. Another secondary source is reference 20.

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isotope	molar mass / dalton
24Mg	23.9850423(8)
25Mg	24.9858374(8)
26Mg	25.9825937(8)

Table 9: Isotopes of Magnesium, IUPAC Mass Data

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Reference 19 appears to be taking its magnesium abundances from reference 21, and it is always good to look at the primary sources if possible, so let’s do that.

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		abundance
isotope pair		ratio		95% confidence
25Mg/24Mg		x = 0.12663	±	0.00013
26Mg/24Mg		y = 0.13932	±	0.00026

Table 10: Isotopes of Magnesium, NBS Abundance Data

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The first thing you notice is that that the scientists to did the work report their results in the form 0.12663 ± 0.00013 at 95% confidence. The uncertainty is clearly and explicitly stated. People who care about their data don’t use sig figs. (Beware that the 95% error bar is two standard deviations, not one.)

Another thing you notice is that they report only two numbers for the abundance data. They report the ratio of ²⁵Mg abundance to ²⁴Mg abundance, and the ratio of ²⁶Mg abundance to ²⁴Mg abundance. They report the uncertainty for each of these ratios. These two numbers are just what we need to span the two-dimensional subspace mentioned in section 7.15. The authors leave it up to you to infer the third abundance number (by means of the sum rule). Similarly they leave it up to you to infer the uncertainty of the third number ... including its correlations. The correlations are important, as we shall see.

To find the percentages in terms of the ratios (x and y) as defined in table 10, we can use the following formulas:

You can easily verify that the abundances add up to exactly 100%, and that the ratios are exactly x and y, as they should be.

The smart way to deal with this data, including the correlations, is to use the Monte Carlo technique. As we shall see, this is simultaneously easier and more powerful than the textbook approach.

Monte Carlo has many advantages. It is a very general and very powerful technique. It can be applied to nonlinear problems. It is flexible enough to allow us to exploit the sum rule directly. Relatively little deviousness is required.

As mentioned in section 1.2 and section 5, we must keep in mind that there is no such thing as an “uncertain quantity”. There is no such thing as a “random number”. Instead we should be talking about probability distributions. There are many ways of representing a probability distribution. We could represent it parametrically (specifying the center and standard deviation). Or we could represent it graphically. Or (!) we could represent it by a huge sample, i.e. a huge ensemble of observations drawn from the distribution.

The representation in terms of a huge sample is sometimes considered an inelegant, brute-force technique, to be used when you don’t understand the problem ... but sometimes brute force has an elegance all its own. Doing this problem analytically requires a great deal of sophistication (calculus, statistics and all that) and even then it’s laborious and error-prone. The Monte Carlo approach just requires knowing one or two simple tricks, and then the computer does all the work.

You can download the spreadsheet for solving the Mg molar mass question. See reference 22.

The strategy goes like this: As always, whenever we see an expression of the form A±B we interpret it as a probability distribution. We start by applying this rule to the mass data in table 9 and the abundance-ratio data in table 10. This gives a mathematical distribution over five variables. Then we represent this distribution by 100 rows of simulated observations, with five variables on each row, all randomly and independently drawn from the mathematical distribution. This gives us another representation of the same distribution, namely a sampled representation. Using these observations, on each row we we make an independent trial calculation of the average mass, and then compute the mean and standard deviation of these 100 trial values.

On each row of the spreadsheet, the five raw observations are drawn independently. The three percentage abundance numbers are not raw data, but instead are calculated from the two abundance ratios. The means the three percentage abundance numbers are not independent. They exhibit nontrivial correlations.

The final answer appears in cells M10 and M12, namely 24.30498(18), where our reported uncertainty represents the one-sigma error bar (unlike reference 21, which reported the two-sigma error bar).

Technical notes:

• In the spreadsheet, I have the “automatic recalculation” feature turned off. You can re-run the calculation – using a new set of random numbers – by hitting the F9 button.
• Once again Box-Muller transforms are used to generate a sample of a Gaussian normal distribution. This is implemented by columns O through Z of the spreadsheet. You are not expected to remember the details of this trick ... but you should remember that it exists, and can google Box-Muller to find out the details whenever you need a supply of normally-distributed numbers.

If you compare my value for the average mass against the value quoted in reference 21, you find that the nominal value is the same, but the estimated uncertainty is slightly less. There are a couple of explanations for this. For one thing, they make an effort to account for some systematic biases that the Monte Carlo calculation knows nothing about. Also, at one point they add some uncertainties linearly, whereas I suspect they should have added them in quadrature. Futhermore, it’s not clear to what extent they accounted for correlated uncertainties.

7.17  Exercise

Pretend that we didn’t have a sum rule. That is, pretend that the abundance data consisted of three independent random variables, with standard deviations as given in table 8. Modify the spreadsheet accordingly. Observe what happens to the nominal value and the uncertainty of the answer. How important is the sum rule?

Hint: There’s an entire column of independent Gaussian random numbers lying around unused in the spreadsheet.

7.17.1  Discussion: Mg Mass

To summarize: As mentioned near the top of section 7.15, the textbook approach has multiple problems: For one thing, it does the propagation-of-uncertainty calculations without taking the sum rule into account (which is a huge source of error). Then the dreaded “sig digs” rules make things worse in two ways: they compel the non-use of guard digits, and they express the uncertainty very imprecisely.

The textbook answer is 24.31 dalton, with whatever degree of uncertainty is implied by that number of “sig digs”.

We now compare that with the our preferred answer, 24.30498(18) dalton. Our standard deviation is less than 8 ppm; theirs is something like one part per thousand (although we can’t be sure). In any case, their uncertainty is more than 100 times worse than ours.

Their nominal value differs from our nominal value by something like 27 times the length of our error bars. That’s a lot.

Last but not least, note that this whole calculation should not be taken overly seriously. The high-precision abundance-ratio data we have been using refers to a particular sample of magnesium. Magnesium from other sources can be expected to have a different isotope ratio, well outside the error bars of our calculation.

7.18  Reporting Correlated Uncertainties

In this section, we are interested in the isotope abundance percentages (not just the average molar mass).

Recall that reference 21 reported only the two abundance ratios. In contrast, the text reported three abundance percentages, without mentioning the sum rule, let alone explaining how the sum rule should be enforced. So the question arises, if we wanted to report the three abundance percentages, what would be the proper way to do it?

The first step toward a reasonable representation of correlated uncertainties is the covariance matrix. This is shown in cells Q3:S5 in the spreadsheet (reference 22), and shown again in equation 40

For uncorrelated variables, the off-diagonal elements of the covariance matrix are zero. Looking at the matrix in our example we see that the off-diagonal elements are nonzero, so we know there are correlations. Of course we knew that already, because the sum rule guarantees there will be correlations.

Alas, it is not easy to understand the physical significance of a matrix by looking at its matrix elements. For example, it may not be obvious that the matrix in equation 40 is singular ... but if you try to invert it, you’re going to have trouble.

Ideally, if we could represent the matrix in terms of its singular value decomposition (SVD), its meaning would become considerably clearer. Since the matrix is symmetric, the SVD is identical to the eigenvalue decomposition (EVD).

There exist software packages for calculating the SVD. If the matrix is larger than 3×3, it is generally not practical to calculate the SVD by hand.

• For 2×2, find the large eigenvectors by applying the power method. Find the small eigenvector by applying the power method to the inverse matrix.
• For 3×3, find the large and small eigenvectors using the power method, as above. Then find the third eigenvector by taking a cross product. (A spreadsheet formula for calculating the cross product can be found in reference 23.)

Once you have the eigenvectors, it is trivial to get the eigenvalues.

Even in situations where you cannot readily obtain the exact SVD, you can still make quite a lot of progress by using an approximate SVD, which I call a ballpark decomposition (BPD). This is shown in cells Q9:AA11 in the spreadsheet and shown again in equation 41.

where R is a unitary matrix and S is “almost” diagonal. Specifically, R consists of a set of approximate eigenvectors of the covariance matrix, considered as column vectors, normalized and stacked side-by-side. The approximate eigenvalues of the covariance matrix appear on the diagonal of S.

The approximate eigenvalues can be figured out using the following reasoning: It is a good guess that [1, 1, 1] or something close to that is the most-expensive eigenvalue of the covariance matrix, because if you increase all three abundance percentages, you violate the sum rule. Secondly, if you check this guess against the computed covariance matrix, equation 40, it checks out, in the sense that it is an eigenvector with zero eigenvalue. Thirdly, if you look at the definition of the covariance matrix and apply a little algebra, you can prove that [1, 1, 1] is exactly (not just approximately) an eigenvector with zero eigenvalue.

Meanwhile, the cheapest eigenvector must be [1, 0, −1] or something like that, because that corresponds to increasing the amount of ²⁴Mg and decreasing the amount of ²⁶Mg, which is cheap (in terms of Mahalanobis distance) because of the relatively long error bar on the ²⁶Mg/²⁴Mg ratio as given in table 10.

The third approximate eigenvector is determined by the requirement that it be perpendicular to the other two. (You might guess that it would be something like [1, −1, 0], but that wouldn’t be perpendicular.) In general, you can take a guess and then orthogonalize it using the Gram-Schmidt process. In the particular case of D dimensions where D−1 of the vectors are known, you can take the cross product (or its higher-dimensional generalization). In the present example, the third member of the orthogonal set is [1, −2, 1]. This is middle eigenvector, neither the cheapest nor the most expensive.

We interpret this as follows: Since the off-diagonal elements in the S-matrix in equation 41 are relatively small, we can say that the uncertainties in the eigenvalues are almost uncorrelated. The eigenvalues are a good (albeit not quite exact) indication of the variance associated with the corresponding eigenvector. Take the square root of the variance to find the standard deviation.

For what it’s worth, equation 42 gives the actual SVD. You can see that it is not very different from the ballpark decomposition in equation 41.

In C++ the armadillo package can be used to perform SVD. In python the numpy package knows how to do SVD.

7.19  Another Example: Solving a Quadratic via Monte Carlo

Consider the following scenario. Suppose we are given that:

The variable x behaves as if it were drawn from some distribution X, and our goal is to find a description of this distribution.

It suffices to treat this as a mathematical puzzle unto itself, but if you would prefer to have some physical interpretation, context, and motivation, we remark that equations like this (and even nastier equations) arise in connection with:

• trajectory of a particle subject to uniform acceleration;
• wave propagation (reflected wave, transmitted wave);
• optimization problems (finding the shortest path);
• finding pH as a function of concentration;
• et cetera.

We can solve this equation using the smart version of the quadratic formula, as explained in reference 14.

We can get a feel for the two variable coefficients (b and c) by making a two dimensional scatter plot. The result is a sample drawn from a two-dimensional Gaussian distribution, as shown in figure 32.

Figure 32: Coefficient b versus Coefficient c

The two-dimensional Gaussian distribution from which this sample was drawn has the following properties: The probability density is highest near the nominal value of (b, c) = (−2.08, 1.08). The density tails off from there, gradually at first and then more quickly.

Let’s see what we can learn by using the Crank Three Times™ method. In this case it will actually require five turns of the crank, since we have two uncertain coefficients to deal with.

The first crank, as always, involves setting the coefficients a, b, and c to their nominal values and solving for x. When we do this, we find two solutions, namely x=1.00 and x=1.08. In some sense these x values are “centered” on the point x=1.04. We shall see that x=1.04 is a point of pseudo-symmetry for this system, and we shall call it the “nominal” x-value.

In figure 32 the region with the tan background corresponds to points (b, c)-space where the discriminant b²−4ac is positive, resulting in a pair of real-valued solutions for x. Meanwhile, the region with the gray background corresponds to points where the discriminant is negative, resulting in a conjugate pair of complex-valued solutions.

There is zero probability of a point falling exactly on the boundary. This would result in a double root. For example, the point (b, c) = (−2.08, 1.0816) would produce a double root at x=1.04. Since this is vanishingly unlikely, we will have nothing further to say about it, and will speak of the roots as occurring in pairs.

For present purposes, we will keep all the x-values we find, including both elements of each pair of roots, and including complex as well as real values. (In some situations there could be additional information that would allow us to discard some of the solutions as unphysical, but for now it is easier and more informative to consider the most general case, and just keep all the solutions.)

If we (temporarily!) consider just the real-valued solutions, we find that x has lopsided error bars. This means it is not safe to describe the x-distribution in terms of some nominal value plus-or-minus some uncertainty. Lopsided error bars are a warning, telling us to investigate more closely, to see whether the problem is just a mild nonlinearity, or whether something very very bad is going on.

When we take into account the complex-valued solutions, we immediately discover that the situation falls into the very very bad category. The Crank Three Times™ method has given us a valuable warning, telling us that it cannot give us the full picture. To get the full picture, we need to do a full-blown Monte Carlo analysis. The result of such an analysis can be presented as a scatter plot in the complex plane, as shown in figure 33.

Figure 33: Pitchfork : x-values for Δb=1.0, Δc=0.05

The distribution of x-values can be plotted in the complex plane, as shown in figure 33. This distribution does not even remotely resemble a two-dimensional Gaussian. It looks more like some sort of diabolical pitchfork.

The probability density actually goes to zero at the nominal point x=1.04.

Sprouting out from the nominal x-value are four segments, shown using four different colors in the diagram. These correspond to whether we take the plus or minus sign in front of the ± square root, and whether the discriminant (b²−4ac) is positive or negative. (The sign of the discriminant depends on the luck of the draw, when we draw values for the coefficients b and c. The ± sign does not depend on the luck of the draw, because except in the case of a double root, for every point in (a,b,c)-space we get two points in x-space.)

This diagram is more-or-less equivalent to something that in another context would be called a root locus plot or root locus diagram.

In the interests of simplicity, let us consider a slightly different version of the same problem. The statement of the problem is the same as before, except that there is less uncertainty on the coefficients. Specifically, we wish to describe the distribution X that models the behavior of the variable x, given that:

The scatter plot for the coefficients (b, c) is shown in figure 34.

Figure 34: Coefficient b versus Coefficient c

The corresponding scatter plot for the solutions x in the complex plane is shown in figure 35. The pitchfork shape is less evident here. It looks more like a Greek cross. The curvature of the upper and lower segments is barely visible. Compared to figure 33, this is similar except more “zoomed in”; that is, all the points now lie closer to the nominal x-value. The probability density is still zero at the nominal point, so the nominal solution is by no means the best solution. It is arguably not even a solution at all.

Figure 35: x-values for Δb=Δc=0.01

Mathematically speaking, it is straightforward to calculate the sample mean, i.e. the mean of the points shown in figure 35. It comes out to very nearly the nominal x-value, namely x=1.04.

Also mathematically speaking, it is straightforward to calculate the variance and the standard deviation of the sample points. The standard deviation is essentially the RMS distance of the points from the mean value. Actually I prefer to call it the RMAS, for root-mean-absolute-square, since technically speaking we want the absolute square |x|² rather than the plain old square x². It comes out to be about 0.11 for this sample.

I emphasize that calculating these numbers is easier than assigning any useful meaning to the numbers. Specifically, it would be grossly misleading to describe this distribution in terms of its mean and standard deviation. That is, it would be grossly misleading to write x=1.04±0.11 without stating the form of the distribution. This distribution is about as non-Gaussian as anything I can imagine. For figure 35, it might make sense to describe the mean and standard deviation of each of the four segments separately ... but for figure 33, not even that would do a good job of describing the overall x-distribution.

Note that if we – hypothetically and temporarily – pretend the RMAS is a useful measure of the uncertainty, then the relative uncertainty on x is almost 11 percent, which is more than an order of magnitude larger than the uncertainty in either of the coefficients. Non-hypothetically speaking, keep in mind that the RMAS barely begins to describe what we know (and don’t know) about the distribution of x-values.

These examples illustrate the importance of plotting the data and looking at it, rather than relying on mathematical abstractions such as mean and standard deviation. If you just blithely calculated numerical values for the mean and standard deviation, you would come nowhere near understanding this system.

These examples also illustrate the tremendous power of the Monte Carlo method. It works when other methods fail.

• For the scenario given in equation 45 the Crank Three Times™ method fails. It fails gracefully, in the sense that when you try it, you get very peculiar results, including complex numbers, so you know you that three cranks will be nowhere near sufficient. You need many, many cranks, i.e. a full-blown Monte Carlo.
• The step-by-step first-order propagation approach described in section 7.20 also fails. It is guaranteed to fail, and extending it to second order (or higher order) won’t help. That’s because the square-root function is not differentiable at any point where its argument goes to zero, i.e. for any point along the tan/gray boundary in figure 34. This guarantees we will have trouble constructing a Taylor series. If we pick any point in the (b, c) plane and construct a Taylor series there, the radius of convergence cannot extend across the tan/gray boundary. Therefore neither the nominal point nor any other point in the plane will give us a radius of converence large enough to encompass all the points in the sample. It should be obvious that there is no circle that encompasses all the points without crossing the tan/gray boundary, since the points themselves sit on both sides of the boundary.

  In the introductory texts, when they lay down “rules” for propagating the uncertainty step-by-step, they often neglect to mention that you need to systematically check the radius of convergence at every step. If you fail to check, convergence problems will go unnoticed, and you will get seriously wrong answers. Unfortunately, this sort of checking is quite laborious, so it is seldom done, and serious errors are common.

  Remember that there are three problems layered on top of each other: Misrepresentation, Malexpansion, and Correlation. This is discussed in section 7.1.

Bottom line: In this example, and in many similar examples, if you want a good, simple, quantitative answer for the nominal value and uncertainty of the distribution X, you’re out of luck. There is no such thing. We need to ask a different question, such as “How can we understand what’s going on in this system?”

Looking at a scatter plot such as figure 35 is a good starting point for understanding what is going on.

7.20  Step-by-Step First-Order Propagation of Uncertainty

Suppose we have a procedure, consisting of one or more steps. We start with a_i and then calculate b_i and then c_i et cetera. Here a_i is an observation drawn from some distribution A. We assume the distribution A can be represented by a blob of the form ⟨A⟩±[A] where ⟨A⟩ is the mean and [A] is the standard deviation.

The hallmark of step-by-step propagation is that at each step in the calculation, rather than keeping track of plain old numbers such as a_i, b_i et cetera, we keep track of the corresponding distributions, by means of the blobs ⟨A⟩±[A], ⟨B⟩±[B], et cetera.

This approach suffers from three categories of problems, namely misrepresentation, malexpansion, and correlation.

7.20.1  Disclaimers

People often ask for some mathematical rules for keeping track of the uncertainty at each step in a long calculation, literally “propagating’ the uncertainty on a step-by-step basis. This approach works fine in a few simple, ideal cases. Perhaps the biggest advantage of the step-by-step approach is that thinking about the logic behind the rules helps give you a feel for what’s going on, and allows you to predict which steps are likely to make the largest contributions to the overall uncertainty.

On the other hand, beware: The step-by-step first-order approach is subject to many provisos that often make it inapplicable to practical problems. (If you ignore the provisos, you will get wrong answers – often with little or no warning.)

In a complicated multi-step problem, you may find that step-by-step first-order propagation works fine everywhere except for one or two steps. Alas, a chain is only as strong as its weakest link, so the method fails to solve the overall problem. The quadratic formula in section 7.19 serves as an example of just such an overall failure, even though the method worked for every step except one, i.e. except for the step that called for extracting the square root.

Also beware that even in cases where the step-by-step method is applicable, it can become quite laborious. For example, when stepping through the quadratic formula (as in equation 43 for example), there is a product, then a sum, then a square root, then another sum, and then a division. This requires repeated conversion between absolute uncertainty and relative uncertainty. In this case, calculating the uncertainty requires about three times as many arithmetical operations as calculating the nominal value. You can reduce the workload by using ultra-crude approximations to the uncertainty (such as sig figs), but this gives you the wrong answer. There is no advantage to having an easy way of getting the wrong answer.

Generally speaking, when dealing with messy, complicated, practical cases you’re better off letting a computer do the work for you. You can start with the Crank Three Times™ method discussed in section 7.14, and if that’s not good enough, you can use the Monte Carlo⁷ method as discussed in section 7.16.

7.20.2  Step-by-Step Propagation Rules

These rules have some advantage and disadvantages. In situations where they are valid, they are very convenient. For example, if you know that a certain distribution has a mean of 19 and a relative uncertainty of 10%, then if you double every element of the ensemble you get a new distribution with a mean of 38 and the same relative uncertainty, namely 10%. This is easy and intuitive, and gets the right answer in this situation. You don’t need to understand any calculus, you don’t need to worry about the radius of convergence, and you hardly need to do any work at all.

However, beware that a collection of anecdotes is not a proof. These rules work in certain selected situations, but they fail miserably in other situations.

I assume you already know how to add, subtract, multiply, and divide numbers, so we will now discuss how to add, subtract, multiply, and divide probability distributions, subject to certain restrictions.

Each of the capital-letter quantities here (A, B, and C) is a probability distribution. We can write A := m_A±σ_A, where m_A is the mean and σ_A is the standard deviation.

The best way to explain where these rules come from is to use calculus, but if you don’t know calculus you can (a) start by accepting the rules as plausible hypotheses, and then (b) checking them for consistency. More specifically, calculus is needed for any serious understanding of the limitations of the rules.

1. Addition and Subtraction: If you are calculating C := A + B or C := A − B, provided that A and B are uncorrelated, then the absolute uncertainties add in quadrature. That is:

   σC2 = σA2 + σB2              (46)

2. Multiplication and Division: If you are calculating C := A × B or C := A / B, provided that A and B are uncorrelated, and provided the relative uncertainties are small (compared to unity), then the relative uncertainties add in quadrature. That is:

   (σC/mC)2 = (σA/mA)2 + (σB/mB)2              (47)

3. Powers: If you are calculating B := A^N, provided N is an exact integer, and provided that A has only a small relative uncertainty, then the relative uncertainty grows in proportion to N. That is:

   σB/mB = |N| σA/mA              (48)

   Note that you cannot get this result by applying the product rule. The product rule is not applicable, since taking powers involves multiplying quantities with correlated uncertainties.

   If N is not an integer, equation 48 is not reliable. It might work, or it might not. For example, consider the case where N=½. Suppose we know x² = y and the distribution on y is 81±1ppm. The problem is, we don’t know whether x ≈ 9 or x ≈ −9, so we might need to write x = 0±9, in which case the uncertainty on x is incomparably more than the uncertainty on y. For more on this, see section 7.19.

4. Functions and other operations: The general rule, roughly speaking, is to expand the function in a first-order Taylor series, and then apply the first rule above, i.e. the “addition” rule. This assumes that the function be well approximated by a first-order Taylor series, which is sometimes a very bad assumption. As an illustration, suppose you want to calculate the tangent of 89±2 degrees. The uncertainty in the result is for all practical purposes infinite, far in excess of what the first-order approximation would have you believe. You could imagine trying to improve the results by using a higher-order Taylor series, but (a) that’s laborious, and (b) it doesn’t fully solve the underlying problem, because the Taylor series might not converge at all. Sometimes you can figure out the radius of convergence of the Taylor series – as we do for instance in section 7.19 – but oftentimes it’s much easier to give up on the step-by-step approach entirely, and just do the Monte Carlo.

7.20.3  More Disclaimers

• There are a lot of provisos in the rules in section 7.20.2. The provisos must be taken seriously. Otherwise you may seriously underestimate or overestimate the uncertainty.
• Please remember that the rules in section 7.20.2 are not sig-figs rules. They are not round-off rules. They allow you to calculate the standard deviation σ_A ... but knowing σ_A does not oblige you to round off m_A to any particular number of digits. The guidelines for rounding off are given in section 8.2.
• The step-by-step approach is very laborious. It requires extra work – fussy, sophisticated work – at every step in the calculation. This stands in contrast to the Crank Three Times™ approach and the Monte Carlo approach, where all you need to worry about are the top-line inputs and the bottom-line outputs (unless there are correlations or other complications). All the intermediate steps are taken care of by the computer, and you can calculate the uncertainties using the same program that you were already using for calculating the nominal values.
• Despite all the limitations, the step-by-step propagation approach can sometimes be made to work. Thinking about the logic behind the rules helps give you a feel for what’s going on, and allows you to predict which steps are likely to make the largest contributions to the overall uncertainty. This in turn may suggest ways of redesigning the whole experiment, so as to make it less sensitive to noise.

Bottom line: As a practical matter, step-by-step “algebraic” propagation of uncertainty calculation is usually not the best approach. Usually Monte Carlo is both better and easier. The more steps in the calculation, the more you gain from the Monte Carlo approach.

7.21  OK Example: Step-by-Step Propagation

Here is an example where the propagation rules give the correct answer. For a counterexample, see section 7.23.

Suppose somebody asks you to carry out the computation indicated on the RHS of equation 49. If you wish, for concreteness you may imagine that the first number is a raw observation, the second number is some scale factor or conversion factor, and the third number is some baseline that must be subtracted off.

As always, the [⁄] indicates that the uncertainty results from roundoff, and is a half-count in the last decimal place. That means we can restate the problem as 4.4±.05 × 2.617±.0005 − 9.064±.0005, with due regard for the fact that roundoff errors are never Gaussian distributed. In this example, for simplicity, we assume the roundoff errors follow a rectangular distribution.

Using the usual precedence rules, we do the multiplication first. According to the propagation rules in section 7.20, we will need to convert the absolute uncertainties to relative uncertainties.

That gives us: 4.4±1.14% × 2.617±0.02%. When we carry out the multiplication, the result is 11.5148±1.14%. Note that the uncertainty in the product is entirely dominated by the uncertainty in the first factor, because the uncertainty in the other factor is relatively small.

Next we convert back from relative to absolute uncertainties, then carry out the subtraction. That results in 11.5148±0.131 − 9.064±.005 = 2.4508±0.131.

Now we have to decide how to present this result. One reasonable possibility would be to round it to 2.45±0.13 or equivalently 2.45(13). One could maybe consider heavier rounding, to 2.5(1). Note that this version differs from the previous version by 39% of an error bar, which seems like a nasty thing to do to your data.

Trying to express the foregoing result using sig digs would be a nightmare, as discussed in more detail in section 17.5.4. Expressing the result properly, e.g. 2.45(13), is no trouble at all.

7.22  Amplification of Uncertainty

The calculation set forth in equation 49 is an example of what we call a noise amplifier. We started with three numbers, one of which had about 1% relative uncertainty, and the others much less. We ended up with more than 5% relative uncertainty.

This is not a problem with the step-by-step approach; Monte Carlo would have given you the same result.

It appears that the uncertainty grew during the calculation, but you should not blame the calculation in any way. The calculation did not cause the uncertainty; it merely made manifest the uncertainty that was inherent in the situation from the beginning.

As a rule of thumb: Any time you compute a small difference between large numbers, the relative uncertainty will be magnified.

If you have a noise amplifier situation that results in unacceptable uncertainty in the final answer, you will need to make major changes and start over. In some cases, it suffices to a more precise measurement of the raw data. In other cases, you will need to make major architectural changes in the experimental apparatus and procedures, perhaps using some sort of “null” technique (electrical bridge, acoustical beats, etc.) so that subtracting off such a large “baseline” number is not required.

7.23  Counterexample: Step-by-Step Propagation

Let’s carry out the calculation of the pH along the lines suggested in section 7.8. We assume a dilute solution of a weak-ish acid:

We can find the pH by direct application of the lame “textbook” version of the quadratic formula. If you understand what’s going on, you know that the actual relative uncertainty in the pH is one percent. The Crank Three Times™ method gives the correct answer, namely one percent.

In this section we will compare the correct result with the result we get from propagating the uncertainty step-by-step, using the rules set forth in section 7.20.2 ... except that we will not pay attention to the provisos and limitations that are contained in the rules.

Here is a snapshot of the spreadsheet (reference 24) used to carry out the calculation. The final pH has a calculated uncertainty, highlighted with boldface, that is off by about three orders of magnitude. The explanation is that in one of the steps, we subtracted two numbers with highly correlated uncertainties, violating one of the crucial provisos.

There are two parts to the lesson here:

• The step-by-step propagation rules contain lots of provisos that must be taken seriously.
• In practice, if the provisos are being violated, it is not necessarily easy to notice until it is too late.

  In this example, the problem is so large as to be obvious. However, beware that in other situations, you could easily make a mistake that is not quite so conspicuous ... just wrong enough to be fatal, but not wrong enough to be noticeable until it is too late.

Hint: If you want to see some less-obvious mistakes, try modifying this example by increasing the concentration and/or decreasing the uncertainty on the concentration.

Note that the more numerically-stable version of the quadratic formula, equation 17, does slightly better, but still does not play nicely with the step-by-step propagation rules. It gets an uncertainty that is off by “only” about one order of magnitude.

Also keep in mind that no matter what you are doing, you can always make it worse by using sig figs. Section 7.8 shows how sig figs can do insane amounts of damage to the quadratic formula in general and pH calculations in particular.

7.24  Curve Fitting – Least Squares and Otherwise

The basic scenario goes like this: We start with some raw data. The distribution over raw data has some uncertainty. We choose a model that has some adjustable parameters. We run the data through the curve-fitting process. This gives us a set of best-fit parameters. There will be some uncertainty associated the parameters.

There are methods for estimating the uncertainty, based on what we know about the model and the distribution of raw data. This can be considered a form of step-by-step analytic propagation of the kind considered in section 7.20. As such, it might work or it might not. It is, as the saying goes, a checkable hypothesis. After doing the calculation, it is rather easy to wiggle the parameters and confirm that the fitted model is behaving in a way that is consistent with the estimated uncertainties.

For the next level of detail on this, see reference 25.

7.25  Choosing a Method of Propagation

There are some simple situations where simple approaches provide accurate propagation and/or provide useful insight. In these situations the simple approaches should be used and fancier methods would be a waste of effort. For example, as mentioned in section 7.20.2, if you know that a certain distribution has a mean of 19 and a relative uncertainty of 10%, then if you double every element of the ensemble you get a new distribution with a mean of 38 and the same relative uncertainty, namely 10%. This is easy and intuitive and gets the right answer in this situation.

Consider the following multi-way contrast:

A) If you are dealing with pointlike raw data points, you shouldn’t be doing any propagation anyway. The raw data points never had any error bars to begin with, as discussed in section 4.3.

In this case, the right answer is less laborious than step-by-step propagation, by at least a factor of 2.

B) Now suppose we are dealing with a cooked data blob of the form A±B.

B1) In cases where step-by-step first-order propagation is valid and seems convenient, go ahead and use it.

However, there are lots of situations where the hard part is checking the validity. After you figure that out, the calculation is probably easy ... but you have to account for all the work, not just the calculational crank-turning work.

If you skip the validation step, you are very likely to get the wrong answer with no warning.

B2) There are some cases where an exact analytic solution exists, and you might as well use it. For example, we didn’t need to do a Monte Carlo to find the Maxwell-Boltzmann distribution, because Mr. M. and Mr. B. have already worked it out for us.

Even when an analytic solution exists, it might be a good idea to check it against the Monte Carlo solution. Analytic calculations are not infallible.

• It is altogether too easy to drop a minus sign or a factor of two.
• If you grab the Maxwell-Boltzmann equation for the speed when you wanted the energy (or vice versa) you will get the wrong answer.
• If you grab the Maxwell-Boltzmann equation for 3D and apply it in 2D (or vice versa) you will get the wrong answer.

Errors of this kind can be exceedingly hard to catch. However, the Monte Carlo solution provides a very powerful check.

B3) There are plenty of cases where Monte Carlo is just plain easier. You only need one equation, namely the equation for analyzing an individual data point.

This contrasts with the step-by-step approach, where (at a minimum) you need two equations: one equation for the nominal value ⟨X⟩ and another very-different equation for the uncertainty [X]. Just not having to derive (and check!) this second equation may be a significant savings. The fact that you need 1000 iterations to collect the Monte Carlo statistics is a negligible cost, because you don’t do that work yourself; the computer does it.

Last but not least, there are plenty of situations where Monte Carlo is the only option.

8  How Much Accuracy Is Enough? How Much Data Is Enough?

8.1  Why is this hard?

Suppose you are taking data. How many raw data points should you take? How accurately should you measure each point? There are reliable schemes for figuring out how much is enough. However, the reliable schemes are not simple, and the simple schemes are not reliable. Any simple rule like “Oh, just measure everything to three significant digits and don’t worry about it” is highly untrustworthy. Some helpful suggestions will be presented shortly, but first let’s take a moment to understand why this is a hard problem.

First you need to know how much accuracy is needed in the final answer, and then you need to know how the raw data (and other factors) affect the final answer.

Sometimes the uncertainties in the raw data can have less effect than you might have guessed, because of signal-averaging or other clever data reduction (section 7.12) or because of anticorrelated errors (section 7.16). Conversely, sometimes the uncertainties in the raw data can be much more harmful than you might have guessed, because of correlated errors, or because of unfavorable leverage, as we now discuss.

As an example of how unfavorable leverage can hurt you, suppose we have an angle theta that is approximately 89.3 or 89.4 degrees. If you care about knowing tan(theta) within one part in a hundred, you need to know theta within less than one part in ten thousand.

Whenever there is a singularity or near-singularity, you risk having unfavorable leverage. The proverbial problem of small differences between large numbers falls into this category, if you care about relative error (as opposed to absolute error).

8.2  How To Do It Right – Basic Recommendations

If you are recording some points:

If you are describing a distribution, and you think it can be described in terms of its center and halfwidth:

There are several equally good ways of expressing the mean and halfwidth of a distribution. It usually doesn’t matter whether the uncertainty is expressed in absolute or relative terms, so long as it is expressed clearly. For example, here is one common way to express the relative uncertainty of a distribution:

Meanwhile, there are multiple ways to express the absolute uncertainty of a distribution. The following are synonymous:

Another way of expressing absolute uncertainty is:

The “interval” or “range” notation in equation 53 has the connotation that the probability is flat and goes to zero outside the stated interval. A flat distribution can result from roundoff, or from other quantization phenomena such as discrete drops coming out of a burette. You could use either of the forms in equation 52 for such a distribution, but then there would be questions as to whether the stated error bars represented the HWHM or the standard deviation.

Sometimes the uncertainty can be expressed indirectly, for example by giving a rule that applies to a whole family of distributions. See section 6.1 for an example.

There are a couple of additional special rules for raw data, as described in section 8.4. Otherwise, all these recommendations apply equally well to measured quantities and calculated quantities.

Remember that a distribution has width, but an individual point sampled from that distribution does not. For details on this, see section 5.2 and reference 2.

Therefore, if you are recording a long list of points, there is normally no notion of uncertainty attached to the individual points, so the the question of how to express uncertainty on a per-point basis does not arise. If you want to describe the distributional properties of the whole collection of points, do that separately. Note the contrast:

Note that there is a distinction between the mean and standard deviation of the sample, and the sample-based estimate of the mean and standard deviation of the population. For an explanation of this, see reference 2.

You should report the form of the distribution, as discussed in section 8.5. Once the form of the distribution is known, if it is a two-parameter distribution, then any of the expressions in equation 51 or equation 52 or perhaps equation 53 suffice to complete the description of the distribution.

Returning to the basic recommendations given at the start of this section: These recommendations do not dictate an “exactly right” number of digits. You should not be surprised by this; you should have learned by now that many things – most things – do not have exact answers. For example, suppose I know something is ten inches long, plus or minus 10%. If I convert that to millimeters, I get 254 mm, ± 10%. I might choose to round that off to 250 mm, ± 10%, or I might choose not to. In any case I am not required to round it off.

Keep in mind that there are plenty of numbers for which the uncertainty doesn’t matter, in which case you are free to write the number (with plenty of guard digits) and leave its uncertainty unstated. For example, an experiment might involve ten numbers, one of which makes an obviously dominant contribution to the uncertainty, in which case you don’t need to obsess over the others.

When comparing numbers, don’t round them before comparing, except maybe for qualitative, at-a-glance comparisons, and maybe not even then, as discussed in section 8.7.

When doing multi-step calculations, whenever possible leave the numbers in the calculator between steps, so that you retain as many digits as the calculator can handle.⁸ Leaving numbers in the calculator is vastly preferable to copying them from the calculator to the notebook and then keying them back into the calculator; if you round them off you introduce roundoff error, and if you don’t round them off there are so many digits that it raises the risk of miskeying something.

Similarly: When cut-and-pasting numbers from one program to another, you should make sure that all the available digits get copied. And again similarly: When a program writes numbers to a file, to be read back in later, it should ordinarily write out all the available digits. (In very exceptional cases where this would incur unacceptable inefficiency, some sort of careful data compression is needed. Simple rounding does not count as careful data compression.)

Note that the notion of “no unintended loss of significance” is meant to be somewhat vague. Indeed the whole notion of “significance” is often hard to quantify. You need to take into account the details of the task at hand to know whether or not you care about the roundoff errors introduced by keeping fewer digits. For instance, if I’m adjusting the pH of a swimming pool, I suppose I could use an analytical balance to measure the chemicals to one part in 10⁵, but I don’t, because I know that nobody cares about the exact pH, and there are other far-larger sources of uncertainty.

When thinking about precision and roundoff, it helps to think about the same quantity two ways:

• From an operational point of view, counting digits is where the roundoff rubber means the road. You have direct control of how many digits you keep.
• From a conceptual, data-centric point of view, it is natural to think about roundoff errors in terms of percent or ppm or the like. Talking about 10 ppm or 100 ppm is often vastly more expressive than talking about 5 decimal places – not least because you can nicely express 30 ppm if you need to, whereas writing five and half digits is remarkably ugly.

Therefore it makes sense to use a two-step process: First figure out how much roundoff error you can afford, and then use that to give you a lower bound on how many digits to use.

Beware that the terminology can be confusing here: N digits is not the same as N decimal places. Let’s temporarily focus attention on numbers in scientific notation (since the sig-digs rules are even more confusing otherwise). A numeral like 1.234 has four digits, but only three decimal places. Sometimes it makes sense to think of it in four-digit terms, since it can represent 10⁴ different numbers, from 1.000 through 9.999 inclusive. Meanwhile it sometimes makes sense to think of it in three-decimal-place terms, since the stepsize (stepping from one such number to the next) is 10⁻³.

If you want to keep the roundoff errors below one part in 10 to the Nth, you need N decimal places, i.e. N+1 digits of scientific notation. For example numbers near 1.015 will be rounded up to 1.02 or rounded down to 1.01. That is, the roundoff error is half a percent.

Also beware that roundoff errors are not normally distributed, as discussed in section 8.3. In multi-step calculations, roundoff errors accumulate faster than normally-distributed errors would. Details on this problem, and suggestions for dealing with it, can be found in section 7.12. Additional discussion of roundoff procedures can be found in reference 8.

The cost of carrying more guard digits than are really needed is usually very small. In contrast, the cost of carrying too few guard digits can be disastrously large. You don’t want to do a complicated, expensive experiment and then ruin the results due to roundoff errors, due to recording too few digits.

8.3  Indicating Roundoff and Truncation

In the not-too-unusual situation where the uncertainty of a distribution is dominated by roundoff error or some similar quantization error, the situation can be expressed using a slash in square brackets:

This can be viewed as shorthand for 0.087[½] i.e. a roundoff error of at most half a count in the last place. Although it is tempting to think of this as roughly equivalent to 0.0870(5), you have to be careful, because the distribution of roundoff errors is nowhere near Gaussian, and roundoff errors are often highly correlated.

• If you start with an exact number such as 5.432 and round it to one decimal place, the roundoff error is the same every time.
• If you have relatively narrow Gaussian distribution such as 5.432(3), if you round it to one decimal place, the roundoff error will be nearly the same almost every time.
• At the other extreme, if you have a very noisy distribution such as 5.432(99) and round it to two decimal places, the distribution of roundoff errors is very nearly a flat distribution. Beware that the standard deviation will markedly smaller than the halfwidth, as discussed in connection with figure 19.

Similarly, if the uncertainty is dominated by a one-sided truncatation error (such as rounding down), this an be expressed using a plus-sign in square brackets:

It is tempting to think of this as roughly equivalent to 0.0875(5), but you have to be careful, as discussed above.

If you have a situation where there is some combination of more-or-less Gaussian noise plus roundoff error, there is no simple way to describe the distribution.

8.4  Keep All the Original Data

When you are making observations, the rule is that you should record all the original data, just as it comes from the apparatus. Do not make any “mental conversions” on the fly.

1. Don’t round off readings. If you think the last-place digit is insignificant, record it anyway. One reason is that there is too much chance of mistakes during mental roundoff. Another reason is that there is the chance that roundoff could throw away some useful information. (Remember, uncertainty is not the same as significance, as discussed in section 14.)
2. Don’t convert scale factors. For example, if the instrument is showing 12.34 millivolts, record the reading as 12.34 mV. The reason is that if you try to make a mental conversion from mV to V, there is too much chance of mistakes.
3. Be sure to write down the units (such as mV) explicitly. This is particularly important with auto-ranging meters. That’s because later, during the analysis phase, it is really embarrassing to see an entry of 12.34 and not be sure whether it is in V or mV. (If there is a large group of readings all with the same units, you can save some writing if you omit the units from individual readings, provided you include a clear annotation stating the units for the group. This uses the same principle as the distributive law of algebra.)
4. Record the non-varying quantities as well as the varying quantities. For example, if you are measuring the (I,V) characteristic of the collector on a bipolar transistor, don’t just record collector current versus collector voltage; you need to record things like the base current. If it’s the same for all (I,V) readings, you only need to record it once, but you need to record it.
5. Don’t discard readings just because you “think” they won’t be needed.⁹
6. Never, never, never discard readings that you think are “wrong”. If a reading seems wild, record it anyway. Record it as is. (Mark it with a “?” or a “???” if you wish, but record the value as is.) If you think the sample is contaminated, make a note to that effect in the logbook, but record the reading as is.
7. Never, never, never erase or obliterate readings from the log book. If you decide a reading is wrong, add a note explaining why it is wrong, but leave the raw data as is. The reason is that you might change your mind about what’s right and what’s wrong.
8. If you are measuring a peak that sits on a baseline, don’t just record the peaky part of the peak; include enough of the wings so that you will be able to confidently establish the baseline.

We are making a distinction between the raw data and the calculations used to analyze the data. The point is that if you keep all the raw data, if you discover a problem with the calculation, you can always redo the calculation. Redoing the calculation may be irksome, but it is usually much less laborious and much less costly than redoing all the lab work.

There is a wide class of analog apparatus – including rulers, burettes, graduated cylinders etc. – for which the following rule applies: It is good practice to record all of the certain digits, plus one estimated digit. For example, if the finest marks on the ruler are millimeters, in many cases you can measure a point on the ruler with certainty to the nearest millimeter … and then you should try to estimate how far along the point is between marks. If you estimate that the point is halfway between the 13 mm and 14 mm marks, record it as 13.5 mm. This emphatically does not indicate that you know the reading is exactly 13.5 mm. It is only an estimate. You are keeping one guard digit beyond what is known with certainty, to reduce the roundoff errors. You don’t want roundoff errors to make any significant contribution to the overall uncertainty of the measurement. [Also, if possible, include some indication of how well you think you have estimated the last digit: perhaps 13.5(5)mm or 13.5(3)mm or even 13.5(1)mm if you have really sharp eyes.]

There is a class of instruments, notably analog voltmeters and multimeters, where in order to make sense of the reading you need to look at the needle and at the range-setting knob. (This is in contrast to digital meters, where the display often tells the whole story.) I recommend the following notation:

which is to be interpreted as follows:

Note that both of the aforementioned readings correspond to 0.288 volts.

There are two things going on here: First of all, converting on-the-fly from what the scale says (2.88) to SI units (0.288) is too error prone, so don’t do it that way; record the 2.88 as is, and do the conversion later. Secondly, there are two ways of getting this reading, either most of the way up on the 300mV scale (the first line in the table above) or partway up on the 1V scale (the second line). It is important to record which scale was used, in case the two scales are not equally well calibrated.

Note that the notation “/3*300mV” also tells you the algebraic operations needed to convert the raw data to SI units: in this case divide by 3, and multiply by 300mV.

8.5  Report the Form of the Distribution

Whenever you are describing a distribution, it is important to specify the form of the distribution, i.e. the family from which your distribution comes. For instance if the data is Gaussian and IID, you should say so, unless this is obvious from context. Only after the family is known does it make sense to report the parameters (such as position and halfwidth) that specify a particular member of the family.

On the other side of the same coin, people have a tendency to assume distributions are Gaussian and IID, even when there is no reasonable basis for such an assumption. Therefore if your data is known to be – or even suspected to be – non-Gaussian and/or non-IID, it is doubly important to point this out explicitly. See section 13.8 for more on this.

8.6  The Effect of Rounding

As mentioned in section 2.1, whenever you write down a number, you have to round it to “some” number of digits. As mentioned in section 1.1, you should keep many enough digits so that roundoff error does not cause any unintended loss of significance. Therefore, we need to understand the effect of roundoff error.

• Some numbers can be represented exactly and conveniently using decimal notation. For example, the number of items in a dozen is 12.
• Some numbers cannot be represented exactly in decimal notation, so any decimal representation must involve some amount of roundoff error. For example, the reciprocal of 12 has no exact decimal representation. We can approximate it by 0.08333[⁄].

8.6.1  Rounding Off a Gaussian

Figure 36 shows how a Gaussian distribution is affected by roundoff. It shows an “original” distribution and two other distributions derived from that by rounding off, as follows:

Figure 36: The Effect of Rounding Off

Obviously, the blue curve is the best. It is the most faithful representation of the real, original distribution.

As I see it, the dashed yellow curve is not better, but it’s not much worse than the original. Its Kullback-Leibler information divergence (relative to the original) is about 0.0003. You can see that even if you keep more digits than are called for by the sig-figs rules, the roundoff error is not entirely negligible.

The dotted red curve is clearly worse. You can see at a glance that it represents a different distribution. It’s K-L information divergence (relative to the original) is more than 0.05. You can see that following the sig-figs rules definitely degrades the data.

8.6.2  Rounding Off a Histogram

To show the effect of rounding, let’s do the following experiment, which can done using nothing more than a spreadsheet program: We draw a sample consisting of N=100 numbers, drawn from a source distribution, namely a Gaussian centered at 1.17 with a standard deviation of 0.05.

As usual, the first thing to do is look at a scatter plot of the data, as shown in figure 37. We calculate a mean of 1.164 and a standard deviation of 0.0510, so the sample is not too dissimilar from the source distribution.

Figure 37: Scatter Plot of Raw Data

Next we round each data point to the nearest 0.01, and histogram the results. This is shown in figure 38.

Figure 38: Histogram of Lightly Rounded Data

Next we round off this data to the nearest 0.1 units and histogram the results. This is shown in figure 39. The mean and standard deviation of the rounded data are 1.157 and 0.0624 ... which means that the roundoff has increased the spread of the data by more than 20%.

Figure 39: Histogram of Heavily Rounded Data

Rather than plotting the probability density, which is what these histogram are doing, it is often smarter to plot the cumulative distribution. This is generally a good practice when comparing two distributions, for reasons discussed in reference 2. This is shown in figure 40. The green curve is the theoretical distribution, namely the integral of a Gaussian, which we recognize as a scaled and shifted error function, erf(...), as discussed in reference 2.

You can see that the raw data (shown in black) does a fairly good job of sticking to the theoretical distribution. The data that has been rounded to the nearest 0.01 (shown in blue) does a slightly worse job of sticking to the theoretical curve, and the data that has been rounded to the nearest 0.1 (shown in red) does a much, much worse job.

Figure 40: Cumulative Distribution, With and Without Rounding (100 data points)

Now let’s see what this looks like if we use a larger sample, namely N=1000 points, as shown in figure 41. You can see that the raw data (shown in black) is smoother, and sticks to the theoretical curve more closely.

In the limit, by using ever-larger samples, we can make the black curve converge to the green curve as closely as desired. The convergence works like this: Each of the N raw data points in figure 37 can be considered a delta function with measure 1/N. When we integrate to get the cumulative distribution, as in figure 40 or figure 41, each data point results in a step, such that the black curve rises by an amount 1/N. If you look closely, you can see 100 such steps in figure 40. For arbitrarily large N, the steps become arbitrarily small.

In contrast, the rounded data will always be a series of stair-steps, due to the rounding, and the steps do not get smaller as we increase N. In this example, the red curve will never be much better than a two-step approximation to the error function, and the blue curve will never be much better than a 20-step approximation. The only way to get the rounded data to converge would be to use less and less rounding, i.e. more and more digits.

Figure 41: Cumulative Distribution, With and Without Rounding (1000 data points)

8.6.3  Nonuniform Relative Error

If we think in terms of relative error, aka percentage error, we see that roundoff does not affect all numbers the same way. Figure 42 shows the percentage error introduced by rounding X to one significant digit, plotted as a function of X. The function is periodic; each decade looks the same.

For numbers near 150, the roundoff error is 33%. For numbers near 950, the roundoff error is barely more than 5%.

Figure 42: Rounding to 1 Sig Fig

The situation does not improve when the number of digits gets larger, as you can see from figure 43. For numbers near 105, the roundoff error is 5%. Meanwhile, for numbers near 905, the roundoff error is an order of magnitude less.

Figure 43: Rounding to 2 Sig Figs

When some quantity has been observed repeatedly and the ensemble of observations has an uncertainty of 1%, there is an all-too-common tendency for people to say the measurement is “good to two significant figures”. This is a very sloppy figure of speech, and should be avoided.

As always, the rule should be: Say what you mean, and mean what you say.

• If you mean that a measurement is good to 1%, say it is good to 1%. (This is not the same as two sig figs, or any other number of sig figs.)
• If you mean a certain voltage measurement has been rounded off to the nearest 0.01 volts, say it has been rounded off to the nearest 0.01 volts. (This should not be expressed in terms of significant figures, because when the voltage is small, rounding to the nearest .01 volts will have fewer significant digits than when the voltage is larger.)

As a rule, whenever you are tempted to say anything in terms of significant digits, you should resist the temptation. There is almost certainly a better way of saying it.

8.6.4  Roundoff Error is Not Necessarily Random

Note the following contrast:

As a slight digression, let us look at some random data (figure 44). We shall see that it does not look anything like roundoff errors (figure 42 or figure 43).

Suppose we conduct an experiment that can be modeled by the following process: For a given value of λ, we construct a Poisson random process with expectation value λ. We then draw a random number from this process. We calculate the residual by subtracting off the expected value. We then express the residual in relative terms, i.e. as a percentage of the expected value. All in all, the normalized residual is:

For selected values of λ we collect ten of these normalized residuals, and plot them as a function of λ, as shown in figure 44. The magenta curves in the figure represent ±σ, where σ is the standard deviation of the normalized residuals.

Figure 44: Poisson Normalized Residuals

Our purpose here is to compare and contrast two ideas:

• roundoff error, as in figure 42, and
• experimental uncertainty, as in figure 44.

In both cases, the ordinate in the figure is the percentage “discrepancy”. The style of representation is the same, to facilitate comparing the two ideas.

Now, when we make the comparison, we find some glaring dissimilarities.

1. Figure 42 shows a function. That is, there is a unique ordinate for every abscissa. That is because every time we round off a given number, it rounds off the same way.

   Roundoff errors are not uniformly random. Sometimes they’re not random at all.

   In contrast, the random data plotted in figure 44 is not a function. There are ten different residuals (the ordinate) for each value of λ (the abscissa).

2. The roundoff error incurred when rounding off X is a periodic function of log₁₀(X). In contrast, the relative uncertainty in a Poisson process is a smooth monotone decreasing function of λ.

8.6.5  Correlations

Here is a good estimate for the mass of the earth, as discussed in section 9.3:

Looking at this value, you might be tempted to think that the nominal value has several insignificant digits, five digits more than seem necessary, and six or seven digits more than are allowed by the usual idiotic sig figs rules. It turns out that we will need all those “extra” digits in some later steps, including forming products such as GM_⊕ and ratios such as M_⊕/M_⊙, as discussed in section 9.

Part of the fundamental problem is that the uncertainty indicated in equation 57 only tells us about the variance, and doesn’t tell us about the covariance between M_⊕ and other things we are interested in.

Indeed, the whole idea of associating a single uncertainty with each variable is Dead on Arrival, because when there are N variables, we need on the order of N² covariances to describe what is going on.

Indeed, decent terminology allows us take a multi-step approach, which is usually preferable: First, write down M_⊕ = 5.9725801308 ×10²⁴ kg, with no indication of uncertainty. Similarly, write down all the other quantities of interest, with no indication of uncertainty. In a later step, write down the full covariance matrix, all in one place.

It is permissible to write something like M_⊕ = (5.9725801308 ± 0.00071)×10²⁴ kg, but indicating the uncertainty in this way is possibly misleading, and at best redundant, because you are going to need to write down the covariance matrix eventually. The variances are the diagonal elements of the covariance matrix, and this is usually the best way to present them.

In the exceptional case where all the variables are uncorrelated, the covariance matrix is diagonal, and we can get away with using simple notions of “the” uncertainty “associated” with a particular variable.

See section 9.

8.7  Comparisons, with or without Rounding

One of the rare situations where rounding off might arguably be helpful concerns eyeball comparison of numbers. In particular, suppose we have the numbers

and we are sure that a half-percent variation in these numbers will never be significant. From that we conclude that on the first line there is no significant difference between a and b, while on the second line there is. Superficially, it seems “easier” to compare rounded-off numbers, since rounding makes the similarities and differences more immediately apparent to the eye:

However, rounding is definitely not the best way to facilitate comparisons. Rounding can get you into trouble. For example, if 3.4997 gets rounded down to 3 and 3.5002 gets rounded up to 4, you can easily get a severely false mismatch. On the other side of the same coin, if 3.5000 gets rounded up to 4, and 4.4997 gets rounded down to 4, you get a false match. Once again, we find that aggressive rounding produces wrong answers. Note that the sig-figs rules require aggressive rounding.

It is far more sensible to subtract the numbers at full precision, tabulate the results (as in equation 60), and then see whether the magnitude of the difference is smaller than some appropriate amount of “fuzz”.

If you are doing things by computer, computing the deltas is no harder than computing the rounded-off versions, and you should always write programs to display the deltas without rounding. (Here “delta” is shorthand for the difference b−a.) While you are at it, you might as well have the computer display a flag whenever the delta exceeds some configurable threshold.

Compared to equation 58 or even equation 59, the advantage goes to equation 60. It makes it incomparably less likely that important details will be overlooked.

Even if you are doing things by hand, you should consider calculating the deltas, especially if the numbers are going to be looked at more times than they are calculated. It is both easier and less error-prone to look for large-percentage variations in the deltas than to look for small-percentage variations in the original values.

8.8  Guard Digits

Guard digits are needed to ensure that roundoff error does not become a significant contribution to the overall uncertainty. An introductory example is discussed in section 7.3. The need for guard digits is also connected to the fact that uncertainty is not the same as insignificance. The distinction between significance, overall uncertainty, and roundoff error is well illustrated by examples where there are uncertain digits whose significance can be revealed by signal averaging, such as in section 7.12, section 17.4.4, section 12, and especially figure 51 in section 14.

Another phenomenon that drives up the need for guard digits involves correlated uncertainties. A familiar sub-category comprises situations where there is a small difference between large numbers. As an example in this category, suppose we have a meter stick lying on the ground somewhere at NIST, in Gaithersburg, oriented north/south. We wish to record this in a Geospatial Information System (GIS). Let point A and point B represent the two ends of stick. We record these in the database in the form of latitude and longitude (in degrees), as follows:

The uncertainty of ± 0.002 represents the fact that the location of the stick is known only approximately, with an uncertainty of a couple hundred meters.

You may be wondering why we represent these numbers using nine decimal places, when the sig-figs doctrine says we should use only three. The answer is that the difference between these two vectors is known quite accurately. The difference |A−B| is 0.000 009 0075(90) degrees of latitude, i.e. one meter, with an uncertainty of ± 1 millimeter or less.

We emphasize that the absolute uncertainty in A−B is on the order of a millimeter or less, whereas the uncertainty in A or B separately is several orders of magnitude greater, on the order of hundreds of meters.

Remember: As mentioned in section 2.1, section 6.3, section 8.8, and section 17.1, roundoff error is only one contribution to the overall uncertainty. The uncertainty in A or B separately is on the order of 0.002, but that does not tell you how much precision is needed. The sig figs approach gets the precision wrong by a factor of a million. Situations like this come up all the time the real world, including GIS applications and innumerable other applications.

8.9  «Final» Results : Guard Digits; Actual Significance

There are two situations that must be considered. In one case your best efforts are required, and in the other case maybe not.

• We start by considering the case where your best efforts are required. For example, suppose you are a metrologist, and your job is to measure this-or-that fundamental constant to high precision. You wouldn’t be bothering to do that if the published “handbook” value was good enough. You are part of a team, and the downstream members of the team need all the precision you can give them.

  Here’s another scenario that leads to the same conclusion: Sometimes you measure something before you know what it’s going to be used for. Many fundamental constants are in this category. Again, common sense says you should report your best results; you should not degrade your results by rounding. In other words, your final results should have plenty of guard digits.

  Suppose you have a calculation with a great many intermediate steps. This this is quite common, especially when using an iterative algorithm. In this case you may need an extra-large number of guard digits on the intermediate results, to prevent an accumulation of roundoff error. You still need some guard digits on the bottom-line result, but perhaps not quite so many.

  Hypothetically, sometimes people imagine they can quote their «final» result using sig figs (even though they used plenty of guard digits on the intermediate results).

  In reality, you have to assume somebody is going to use your result. Therefore your “final output” is somebody else’s input. An example of this can be seen in the teamwork scenario in section 7.10.3. In any case, from an overall point of view, all results are intermediate results, and all of them need guard digits.

  Applying sig figs to the supposedly «final» result is a blunder. It does horrendous damage to the «final» result (both the nominal value and the uncertainty). Don’t do it.

  Hypothetically, if you tried to make guard digits compatible with sig figs, you would need to invent some new notation so that at each step of the calculation you could distinguish the so-called significant digits from the guard digits.

  In reality, I’ve never seen anybody try to distinguish guard digits from other digits. It’s too much work for too little benefit. Anybody who cares enough to go to that much trouble presumably knows about easier and better methods.

  In reality, you do not need to keep track of exactly how many guard digits there are, so long as there are enough.

  Hypothetically, sometimes people imagine the following excuse for rounding off the «final» answer: Suppose there is an academic busywork assignment, where nobody really cares about the answer. The teacher unwisely decides that it is OK for everybody to get an unrealistic answer, so long as everybody gets the same answer. In this situation, conformity is more important than integrity.

  In reality, this is a terrible lesson. Don’t do it. Instead, accept the fact that real-world numbers have guard digits, and the guard digits will be noisy. Accept the fact that not all correct answers will be numerically identical. Make academic exercises as authentic as possible. Insist on integrity in all that you do.

• Here’s a slightly artificial scenario: Suppose you work in the Quality Assurance department in a large manufacturing plant. A batch of widgets has arrived from your supplier. You test them to see whether they conform to specifications. The contract gives you only two options: Accept or reject the batch. Pass or fail. Green light or red light. In other words, after all your measurements, the final result necessarily gets rounded to a single bit ... not even one decimal digit, just one bit. Any additional resolution would be insignificant in the strict sense, i.e. it would be immaterial.

  In this scenario, the appropriate roundoff is determined by what happens downstream of your decision. This stands in stark contrast to the “propagation of error” techniques that are used in conjunction with sig figs, where the amount of rounding is determined by what happens upstream of your result. The fig-figs minions refer to this as «significance» but that’s an abuse of the word; when you calculate the uncertainty using propagation-of-error (or Crank Three Times™ or any other method), that does not tell you whether or not the uncertainty is significant in the strict sense. Real significance depends details of what happens downstream.

  This QA scenario is slightly artificial, for the following reason: If the supplier had any sense, they would negotiate a better contract. They would ask you to report your testing results in detail, in addition to the pass/fail grade. This is particularly important in the case of a fail or a marginal pass, to help the supplier tighten up their process.

  You can argue both sides of this forever:

  • Sometimes there are categorical, discrete decisions: pass or fail, ball or strike, rocket-launch go or no-go, et cetera.
  • Even so, in the case of a close call, it is worth recording in detail the factors that went into the decision, for later analysis.

    Furthermore, even in situations that appear discrete, it is sometimes necessary to have tie breakers. For example, things like graduation or promotion to a higher rank are to a first approximation yes/no decisions. However, if there are two military officers with the same rank, the one who has held that rank longer is senior to the other ... and in rare occasions this actually matters.

Bottom line: In most cases, you should record your final answer with plenty of guard digits, to protect it from roundoff error. If there is the slightest doubt, keep plenty of guard digits.

In other words: sometimes quantizing the «final» result is the right thing to do ... but sometimes it isn’t. Do not make a habit of throwing away the guard digits.

It should go without saying that sig figs is never the right reason or the right method for rounding your results. If/when you need to state the uncertainty, state it separately and explicitly, perhaps using the notation 1.234(55) or 1.234±0.055 or the like.

8.10  Too Many Digits, Oh My!

I often get questions from people who are afraid there will be an outbreak of too many insignificant digits. A typical question is:

“What if a student divides 10 meters by 35 seconds and reports the result as 0.285714286 m/s? Isn’t that just wrong? In the absence of other information, it implies an uncertainty of 0.0000000005 m/s, which is a gross underestimate, isn’t it?”

My reply is always the same: No, those «extra» digits are not wrong, and they do not imply anything about the uncertainty.

Yes, I see nine digits, but no, that doesn’t tell me the uncertainty. The uncertainty might be much greater than one part in 10⁹, or it might be much less. If the situation called for stating the uncertainty, I might fault the student for not doing so. However, there are plenty of cases where the uncertainty does not need to be expressed, and may not even be knowable, in which case the only smart thing to do is to write down plenty of guard digits.

Suppose we later discover that the overall relative uncertainty was 10%. Then I interpret 0.285714286 as having eight guard digits. Is that a problem? I wish all my problems were as trivial as that.

If you think excess digits are a crime, we should make the punishment fit the crime. Let’s do the math:

• Black&white printing costs about a penny a page;
• There are about 500 words to a page;
• Each over-long numeral such as 0.285714286 means there will be an entire word less that fits on the page.
• Therefore each student who perpetrates such a crime will be fined 20 microbucks for each offense.
• I figure if a student does that twice a day, every day, all term, at the end of the term he will have to give me one gummi-bear, to pay off the accumulated fines.

My time is valuable. The amount of my time wasted by people who are worried about the «threat» of excess digits greatly exceeds the amount of my time wasted reading excess digits.

My advice: Breathe in. Breathe out. Relax already. Excess digits aren’t going to hurt you. They might even help you.

8.11  How To Avoid Introducing Sig Figs

In an introductory course, the most sensible approach is to adopt the following rules:

• Record the observed raw data to full precision.
• Unless/until you are told otherwise, in this course it suffices to:

  – do all intermediate calculations to 6-digit precision or better

  – round the final answer to 3 digits, as the very last step.

This is much simpler than dealing with sig figs. It also more honest. Reporting no information about the uncertainty is preferable to reporting wrong information about the uncertainty (which is what you get with sig figs).

If the students are “mathematically challenged” and even “reading challenged”, it is a safe bet that they are not doing multi-digit calculations longhand. And they probably aren’t using slide rules either. So let’s assume they are using calculators. Therefore the burden of keeping intermediate results to 6-digit precision or better (indeed much better) is negligible. It has the advantage of getting them in the habit of keeping plenty of guard digits.

Yes, some of those digits will be insignificant. So what? Extra digits will not actually kill anybody.

At some point in the course, we want the students to develop “some” feeling for uncertainty. So let’s do that. We can do it easily and correctly, using the Crank Three Times™ method as described in section 7.14. (Apply it to selected problems now and then, not every problem.) It requires less sophistication, requires less effort, and produces better results – compared to anything involving sig figs.

Using sig figs is like trying to eat a bowlful of clear soup using a fork. It’s silly, especially since spoons are readily available. Even if somebody has a phobia about spoons, the fork is still silly; they’d be better off throwing it away and using no utensil at all.

8.12  Psychological Issues

In an introductory course, some students (especially the more thoughtful students) will be appalled by the crudity and unreliability of the sig figs doctrine, and will appreciate the value of guard digits.

On the other hand, there will also be some students (especially the more insecure students) for whom various psychological issues make it hard to appreciate the necessity for guard digits. These issues include the following:

• First, there is something I call barnyard ethology. One of the rules of the barnyard is to never admit weakness. For example, an injured sheep will go to amazing lengths to conceal its injury. This makes sense, because if a sheep is seen to be injured, not only will the wolves pick on it, the other sheep will pick on it. (This leads to the old farmer’s saying that “a sick sheep is a dead sheep”. That saying arises because by the time it becomes obvious that a sheep is ill, it is very gravely ill.)

  This rule of barnyard ethology applies to some spheres of human activity, including lawyering, politics, and military combat. Never admit weakness, and never admit uncertainty.

  However ... students need to realize that science is not like lawyering, or politics, or combat. Scientists do admit uncertainty. The surest way to be recognized as a non-scientist is to pretend to be certain when you’re not.

  It may seem ironic or even paradoxical, but it is true: One of the most basic steps toward reducing uncertainty is to admit that there is some uncertainty, and to account for it. For example, it would always be wrong to say that the true voltage is 1.23 volts, whereas we might be quite confident that the true voltage is in the range bewteen 1.22 and 1.24 volts. For more on this, see reference 26.

  Being able to admit uncertainty requires some emotional maturity, some emotional security, some grownupness. This is an important part of why students go to school, to learn such things.

• As mentioned in section 5.6, some students think that being wrong is Wrong with a capital W, in the same way that lying and stealing are Wrong, i.e. sinful. They tell themselves, perhaps unconsciously, that if they write down a number with guard digits, there will be uncertainty in the guard digits, which is to say those digits might be wrong, i.e. Wrong, i.e. sinful. Therefore they refuse to write into their logbooks anything that has the slightest uncertainty. They adore the sig figs doctrine, especially the half-count sect, because it allows them to keep rounding until all the uncertain digits have been eliminated.

  This is spectacularly unscientific. By rounding off the number to the point where it is not fluctuating, they have arranged to get the same number every time ... but it is wrong every time. It is wrong because of excessive roundoff error. Evidently they would rather be wrong with certainty than right with uncertainty.

• Some students have been trained to not say anything or write anything unless they are sure that it is “true”.

  They need to realize that when they write down raw observations, with or without guard digits, they are recording the indicated values, not the true values. The indicated value represents the range of true values, but it is not the same thing.

  When describing a distribution, don’t worry about the fact that the description is non-unique. There are lots of ways of describing the same distribution. If it makes you feel better, first write down the width of the distribution, and then write down the nominal value. If the distribution has a half-width of ±7%, it doesn’t matter whether you express the nominal value as 51, or 51.13, or 51.1394744. The fact that the trailing digits are uncertain and non-unique doesn’t make these numbers wrong. They are all equivalent, for almost all practical purposes.

  If you were to claim that any number such as 51, or 51.13, or 51.1394744 (with or without guard digits) represented an exact measurement, that would be wrong. So don’t pretend it’s exact. Say it has an uncertainty of ±7%. Once you’ve said that, you are free to write down as many guard digits as you like. (You need at least some uncertain digits, to guard against roundoff errors.)

• Students want certainty. Indeed, everybody wants certainty. Alas, you can’t always get what you want.

  The real world does not offer certainty. Students should not blame themselves for uncertainty, and should not blame the teacher. We live in an uncertain world. The goal is not to eliminate all uncertainty; the goal is to learn how to live in an uncertain world.

  One of the crucial techniques for dealing with uncertainty is to represent things as distributions rather than as plain numbers.

• We turn now from the normal range of uncertainty to outright mistakes.

  The goal is not to avoid all mistakes. Everybody makes mistakes. Students are expected to make more mistakes than professionals, but even professionals make mistakes. The goal is to (a) minimize the cost of the mistakes, and (b) learn from the mistakes. For example, real-world engineers commonly build pilot plants and/or carry out pilot programs, so they can learn from mistakes relatively cheaply, before they commit to a multi-billion-dollar full-scale program. For more along this line, see section 8.14.

  I have seen students go to great lengths to avoid having the slightest imperfection in their lab books. These students need to realize that real science involves approximation, including what we call successive refinement. That is, we first make a rough measurement, and then based on what we just learned, we make successively more refined measurements. If the first measurement were perfect, we wouldn’t need the later measurements. Learning is not a sin.

8.13  How To Survive an Attack of Sig Figs

There are two issues: writing sig figs, and reading sig figs.

If you ever feel you need to write something using sig figs, you should lie down until the feeling goes away. Figure out what you are trying to say, and find a better way of saying it. If you are going to express the uncertainty at all, express it separately. See also section 8.11.

The rest of this section is devoted to reading sig figs. That is, suppose you are given a bunch of numbers and are required to interpret them as having significant digits.

If that’s all you have to go on, it is not necessary – and not possible – to take the situation seriously. If the authors had intended their uncertainties to be taken seriously, they would have encoded the data properly, not using significant digits.

Sometimes, though, you do have more information available.

One good strategy, if possible, is to simply ask the authors what they think the data means. If the data is from a book, there may be a statement somewhere in the book that says what rules the authors are playing by. Along similar lines, I have seen blueprints where explicit tolerance rules were stated in the legend of the blueprint: one example said that numbers with 1, 2, or 3 decimal places had a tolerance of ±0.001 inches, while numbers with 4 decimal places had a tolerance of ±0.0001 inches. That made sense.

Another possibility is to use your judgment as to how much uncertainty attaches to the given data. This judgment may be based on what you know about the source of the data. For instance, if you know that the data results from a counting process, you might decide that 1100 is an exact integer, even though the sig figs rules might tell you it had an uncertainty of ±50 or even ±500 or worse.

As a next-to-last resort, you can try the following procedure. We need to attribute some uncertainty to each of the given numbers. Since we don’t know which sect of the sig-digs cult to follow, we temporarily and hypothetically make the worst-case assumption, namely just shy of ten counts of uncertainty in the last place. For example, 1.23 becomes 1.23±0.099, on the theory that 1.23±0.10 would have been rounded to 1.2 according to the multi-count sect. (The multi-count sect is generally the worst case when you are decoding numbers that are already represented in sig-figs notation. Conversely, the half-count sect is generally the worst case when you are encoding numbers into the sig-figs representation, because it involves the greatest amount of destructive rounding.)

Now turn the crank. Do the calculation, using plenty of guard digits on the intermediate results. Propagate the uncertainty using the methods suggested in section 7.

Now there are two possibilities:

• If you are lucky, you will find that the uncertainty in the final result is so small as to be immaterial, i.e. insignificant in the given context, for the given application of the result. That is, if the data is “good enough” by a wide margin, you don’t care exactly how wide the margin is. This takes the pressure off you, meaning you don’t need to bother re-examining the worst-case assumptions that we used in getting to this point. You don’t really know the uncertainty – you merely hypothesized something – but you don’t need to know it. (This is another illustration of the fundamental difference between uncertainty and insignificance.)
• The other possibility is that the calculated uncertainty is significant. Then you must do a great deal of additional work. Start by seeing which of the input variables make the most contribution to the uncertainty of the result. (If there are many input variables, typically only a few make large contributions.) Then you need to get better information about those variables. Perhaps you can find out which sect’s sig-dig rules to apply, thereby eliminating one source of uncertainty about the uncertainty. Or, in the spirit of the Crank Three Times™ method, you could consider the whole range of sectarian differences, and then choose a middle-of-the-road value, as was done in going from table 7 to table 8 in section 7.15. This choice may be partly guided by your opinion as to whether overestimating the uncertainty is worse than underestimating it, or vice versa.

  At some point you might well decide that the given data is inadequate for the purpose. Go back to Square One and obtain some better data.

I categorically decline to suggest an explicit convention as to what sig figs “should” mean. There are two reasons for this: First of all, the sectarian differences are too huge; anything I could say would be wildly wrong, one way or the other, according to one sect or another. Secondly, as previously mentioned, what’s safest when writing sig figs is not what’s safest when reading and trying to interpret sig figs. Last but not least, sig figs “should” not be used at all; I don’t want to say anything that could be misinterpreted as endorsing their use.

8.14  Sensitivity Analysis, On-Line Analysis, and Cross-Checking

Spreadsheets are great. You need to analyze the data one way or another, so you might as well do it on a spreadsheet. This gives you a big bonus: you can do some “what-if” analysis. You don’t need to do a full-blown Monte Carlo analysis as in section 7.16; instead just wiggle a few of your data points to see how that affects the final answer. The same goes for other quantities such as calibration factors: find out how much of a perturbation is needed to significantly affect the final answer.

If good-sized changes in a data point have negligible effect on the final answer, it means you can relax a bit; you don’t need to drive yourself crazy measuring that data point to extreme precision. Conversely, if you find that smallish changes in a single data point have a major effect on the answer, it tells you that you’d better measure each such data point as accurately as you can, and/or you’d better take a huge amount of data (so you you can do some signal-averaging, as discussed in section 7.12). You can also consider upgrading the apparatus, perhaps using more accurate instruments, and/or redesigning the whole experiment to give you better leverage.

There is a lesson here about procedures: It is a really bad idea to take all your data and then do all your analysis. Take some data and do some analysis, so you can see whether you’re on the right track and so you can do the sensitivity analysis we just discussed. Then take some more data and do some more analysis. This is called on-line analysis.

This is quite important. As mentioned in section 8.12, real-world engineers commonly build pilot plants and/or carry out pilot programs, so they can learn what the real issues are before they commit to full-scale production. Once the program is in operation, they do a lot of trend monitoring, so that if a problem starts to develop about it they learn about it sooner rather than later.

You should also find ways to make internal consistency checks. If there are good theoretical reasons why the data should follow a certain functional form, see if it does. Exploit any sum rules or other constraints you can find. Make sure there is enough data to overconstrain the intended interpretation. By that I mean do not rely on two points to determine a straight line; use at least three and preferably a lot more than that, so that there will be some internal error checks. Similarly, if you are measuring something that is supposed to be a square, measure all four sides and both diagonals if you can. Measure the angles also if you can.

There are few hard-and-fast rules in this business. It involves tradeoffs. It involves judgment. You have to ask: What is the cost of taking more data points? What is the cost of making them more accurate? What is the cost of a given amount of uncertainty in the final answer?

Additional good advice can be found in reference 27.

9  Correlation and Covariance

9.1  Example: Electron Charge-to-Mass Ratio

If you want to calculate the electron e/m ratio, correlations must be taken into account. This is discussed in section 7.7.

9.2  Example: Common Mode versus Differential Mode

Consider the simplified ohmmeter circuit shown in figure 45

Figure 45: Common-Mode and Differential-Mode Signals

In such a circuit, it would not be uncommon to find the following voltages:

The question arises, what is the differential-mode signal V_A − V_B? If you thought V_A and V_B were uncorrelated, you would calculate

However, in the real world, with a little bit of work you could probably arrange for V_A and V_B to be very highly correlated. It might turn out that

and with extra work you could do even better. There is no way to calculate the result in equation 64, not without a great deal of additional information, but that’s not the point. The point is that assuming the voltages are uncorrelated would be a very very bad assumption. The physics of the situation is that the stray time-dependent magnetic flux φ‌· affects both V_A and V_B in the same way, to an excellent approximation. Communications equipment and measuring instruments depend on this. It’s not something that happens automatically; you make it happen by careful engineering.

9.3  Example: Mass and Gravitation (I)

Let’s do an example involving Newton’s constant of universal gravitation (G), the mass of the earth (M_⊕), and the product of the two (GM_⊕).

In order to speak clearly, we introduce the notation D(M_⊕) to represent a direct measurement of M_⊕. We use the unadorned symbol M_⊕ to represent our best estimate of M_⊕. If necessary, we can use T(M_⊕) to represent the true, ideal, exact value, which will never be known by mortal man.

The last time I checked,

You could obtain an estimate of M_⊕ from geology and seismology, but even that wouldn’t count as a “direct” measurement, and more importantly it wouldn’t be particularly helpful, since it would not be anywhere near as accurate as D(GM_⊕)/D(G).

Here are the actual nominal values and absolute uncertainties, from reference 28 and reference 29:

Looking at the value for M_⊕ in equation 66, you might be tempted to think that the nominal value has several insignificant digits, five digits more than seem necessary, and six or seven digits more than are allowed by sig figs doctrine. However, it would be a Bad Idea to round off this number. Note the contrast:

The fundamental issue here is the fact that M_⊕ is highly correlated with G. They are correlated in such a way that when you multiply them, the uncertainty of the product is vastly less than the uncertainty in either one separately.

To better understand this situation, it may help to look at the diagram shown in figure 46. Recall from section 5.2 that fundamentally, an “uncertain quantity” such as G or M_⊕ is really a probability distribution. Also recall that as a general principle, you can always visualize a probability distribution in terms of a scatter plot. In this case, it pays to plot both variables jointly, as a two-dimensional scatter plot. In figure 46, G is plotted horizontally and its standard deviation is shown by the magenta bar. Similarly M_⊕ is plotted vertically its standard deviation is shown by the blue bar. The standard deviation of the product GM_⊕ is represented – loosely – by the yellow bar.

Figure 46: Correlated Uncertainties

In this figure, the amount of correlation has been greatly de-emphasized for clarity. The uncertainty of the product is portrayed as only six times less than the uncertainty of the raw variables. (This is in contrast to the real physics of mass and gravitation, where the uncertainty of the product is millions of times less than the uncertainty of the raw variables.)

If the probability distribution is a two-dimensional Gaussian, the contours of constant probability are ellipses when we plot the probability as in figure 46. If the variables are highly correlated, the ellipses are highly elongated, and the principal axes of the ellipse are nowhere near aligned with the axes of the plot. (Conversely, in the special case of uncorrelated variables, the axes of the ellipse are aligned with the axes of the plot, and the ellipse may or may not be highly elongated.)

This example serves to reinforce the rule that you should not round off unless you are sure it’s safe. It’s not always easy to figure out what’s safe and what’s not. When in doubt, keep plenty of guard digits.

9.4  Dealing with Correlations

To make progress, we need to construct the covariance matrix. It is defined as: