1  Introduction

1.1  How Many Digits Should Be Used?

Use many enough digits to avoid unintended loss of significance. Use few enough digits to be reasonably convenient. Keep all the raw data. See section 4.2 for details on how to do things right.

1.2  How Should Uncertainty Be Expressed?

If you want to express the uncertainty, express it separately and explicitly. For example, absolute uncertainty can be properly expressed as 1.234(55) or equivalently 1.234±0.055. Relative uncertainty can be expressed as 2900±13%.

State the form of the distribution, unless this is obvious from context. Examples include Gaussian, square, triangular, et cetera.

1.3  What About Significant Digits?

The whole notion of significant digits is heavily flawed; see section 11 for more on this. Anything that can be done by means of significant digits can be done much better and more easily by other means. People who care about their data don’t use significant digits.

There are plenty of important cases where following the usual “significant figures” rules would introduce large errors into the calculations. See section 3.1 and section 11.3 for simple examples of miscalculating the nominal value. See section 11.8 and section 4.7 for examples of wildly overestimating and wildly underestimating the width of the distribution.

It is not safe to assume that counting the digits in a numeral implies anything about the significance, uncertainty, accuracy, precision, readability, repeatability, tolerance, or anything else.

*   Contents

• 1  Introduction
  • 1.1  How Many Digits Should Be Used?
  • 1.2  How Should Uncertainty Be Expressed?
  • 1.3  What About Significant Digits?
• 2  Fundamental Notions of Uncertainty
• 3  Propagation of Uncertainty
  • 3.1  Simple Example: Reducing the Uncertainty by Data Analysis
  • 3.2  Example: Combatting Lopsided Roundoff to Extract Signal from Noise
  • 3.3  Another Example: Magnesium Mass. Monte Carlo Solution
  • 3.4  Example: Interpolating Between Graduations
  • 3.5  Crank Three Times
  • 3.6  Manual, Algebraic Propagation of Uncertainty
  • 3.7  Example: Manual Propagation
  • 3.8  Amplification of Uncertainty
• 4  How Much Accuracy Is Enough? How Much Data Is Enough?
  • 4.1  Why is this hard?
  • 4.2  Recommendations – How To Do It Right
  • 4.3  Keep All the Raw Data
  • 4.4  Report the Form of the Distribution
  • 4.5  Correlations: Another Reason to Eschew Rounding
  • 4.6  Comparisons, with or without Rounding
  • 4.7  Guard Digits
  • 4.8  Too Many Digits, Oh My!
  • 4.9  How To Avoid Introducing Sig Figs
  • 4.10  Psychological Issues
  • 4.11  How To Survive an Attack of Sig Figs
  • 4.12  Sensitivity Analysis, On-Line Analysis, and Cross-Checking
• 5  “Correctness” versus Agreement
  • 5.1  Your Data is Your Data
  • 5.2  Measurement Errors versus Modelling Errors
• 6  Cumulative Distributions and Convergence
• 7  Samples, Sets, Groups, or Clusters of Observations
  • 7.1  Particles and Clusters
  • 7.2  Estimators
  • 7.3  Terminology
  • 7.4  Mean and Standard Deviation
• 8  Contributions to the Uncertainty
  • 8.1  Statistical Fluctuations
  • 8.2  Roundoff Error
  • 8.3  Series Truncation Error
• 9  Categories of Uncertainty – and Related Notions
  • 9.1  Tolerance
  • 9.2  Significance
  • 9.3  Precision
  • 9.4  Accuracy
  • 9.5  Readability and Reproducibility
  • 9.6  Scatter versus Systematic Error
  • 9.7  “Experimental Error” Deprecated
  • 9.8  Other Limits to the Notion of Uncertainty
• 10  Probability Measure; the Definition of Probability
• 11  Significant Figures are an Abomination
  • 11.1  If You Mean Place Value, Say Place Value
  • 11.2  Observations versus Distributions
  • 11.3  A Simple Example: Calculating Density
  • 11.4  Uncertainty, Insignificance, and Guard Digits
  • 11.5  Rough – Very Rough – Uncertainties
  • 11.6  Sectarian Differences
  • 11.7  Some Answers Are Not Expressible Using Sig Digs
  • 11.8  Some Numbers are Exact
  • 11.9  Units Won’t Solve the Problem
  • 11.10  Decimals Are Exact By Definition
  • 11.11  Ambiguity Is Not an Advantage
  • 11.12  Relative versus Absolute Uncertainty
• 12  Appendix: Place Value and Mantissa Digits
• 13  Appendix: Mean, Standard Deviation, Gaussian, etc.
• 14  References

2  Fundamental Notions of Uncertainty

Best current practice is to speak in terms of the uncertainty. We use uncertainty in a broad sense. Other terms such as accuracy, precision, readability, tolerance, etc. are often used as nontechnical terms ... but sometimes connote various sub-types of uncertainty, i.e. as contributions to the overall uncertainty, as discussed in section 8. In most of this document, the terms “precise” and “precision” will be used as generic, not-very-technical antonyms for “uncertain” and “uncertainty”.

The only way to really understand uncertainty is in terms of probability distributions. In section 10 we will give a deep and formal definition of probability, but in the meantime we will try to skate by using rough 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. If we repeat the experiment many times, ideally we get the probability distribution X shown in figure 1. 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 1: Probability Distribution for a Six-Sided Die

The conventional but somewhat abusive notation for describing such a situation is to write x=3.5±2.5, where x is called an “uncertain quantity”. If you want to know formally and precisely what sort of thing this “x” is, the question is only partially answerable. Obviously the intent of the expression x=3.5±2.5 is to describe the distribution X. However, the form of the expression makes x look like an outcome drawn from X, perhaps some sort of abstract “average” outcome. This is an example of form not following function. The notation is not super-terrible, because the intent is reasonably clear.

It is important to appreciate the distinction between the abstraction x=3.5±2.5 and the real outcomes such as x₁=6 and x₂=3. The outcome x₁ is not an uncertain quantity; it has the value x₁=6 with no uncertainty whatsoever. The so-called uncertain quantity x=3.5±2.5 describes the distribution from which outcomes such as x₁ and x₂ are drawn.

Now suppose we roll a pair of dice. The first time we do the experiment, we observe 8 spots, 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 2. To describe the distribution X, we need to say that the outline of the distribution 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 2: Probability Distribution for a Pair of Dice

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

Any set of experimental observations {x_i} can be considered a probability distribution unto itself. Typically we assign equal weight (i.e. equal measure, to use the technical term) to each of the observations. To visualize such a distribution, you can make a graph that shows how often x_i falls within a given interval. Such a graph is called a histogram.

Oftentimes, given enough observations, the histogram will converge to some well-known theoretical probability distribution. (Or, better, the cumulative distribution will converge, as discussed in section 6.) For example, it is very common to encounter a piecewise-flat distribution as shown by the magenta curve in figure 3. 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, roundoff errors commonly contribute an uncertainty that is uniform over the interval [-0.5, 0.5]; see equation 13. Other quantization errors (such as discrete drops coming from a burette) contribute an uncertainty that is more-or-less uniform over some interval (such as ± half a drop).

It is also very common to encounter a so-called “normal” distribution, also (preferably) called a Gaussian distribution. In figure 3, the black curve is one example of a Gaussian distribution, and the green curve is another example.

Figure 3: Three Probability Distributions

The following table lists a few well-known families of distributions. See section 9.8 for more on this.

Each name in the table 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 requirement that the area under the curve is always 1.0, as it must be for any well-behaved probability distribution.)

In particular, when we write A±B, that means A tells us the center¹ and B tells us something about the halfwidth of the distribution.

We emphasize that B is more closely related to the halfwidth than to the full width, since the expression A±B means A plus-or-minus B, not plus-and-minus.

For a Gaussian distribution, conventionally B represents the standard deviation of the distribution; see section 13 for definitions and useful formulas. In figure 3, the standard deviation of the black curve is 1.0, and is depicted by a blue bar. Meanwhile, the HWHM (half-width at half-maximum) is depicted by a red bar. For any Gaussian, the HWHM is about 15% longer than the standard deviation – i.e. longer by a factor √(2 ln2), to be precise – as you can verify by plugging into equation 30 or equation 31.

For a square distribution, the expression A±B is somewhat ambiguous. In some circles, B denotes the halfwidth of the distribution, while in other circles, B denotes the standard deviation, which is very much shorter than the HWHM – shorter by a factor of √3, as you can verify from the definition, equation 29. You can visualize this in figure 3, since the black curve and the magenta curve have the same standard deviation.

Let’s be clear: An expression of the form A±B only makes sense provided everybody knows what family of distributions you are talking about, and provided it is a well-behaved two-parameter family. 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 3, the black curve and the magenta curve have the same mean and the same standard deviation, but they are certainly not the same curve. Data that is well described by the black curve would not be well described by the magenta curve, nor vice versa.

There are, of course, plenty of cases where writing A±B does make sense.

It is important to keep in mind that A±B does not represent a number, but rather a probability distribution. You learned in grade-school how to add, subtract, multiply, and divide numbers ... but now you are being asked to add, subtract, multiply and divide probability distributions. This requires a tremendously higher level of sophistication.

The “significant figures” method attempts to use a single decimal numeral to express both the center and the halfwidth of a distribution: the nominal 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.

• For starters, the “significant digits rules” cannot represent the uncertainty more accurately than the nearest power of ten. For example, they represent the quantity 45±3 in exactly the same way as the quantity 45±1, but as we can see in figure 3, these are markedly different quantities. In the figure, the heavy black curve represents 45±1 while the thin green curve represents 45±3. These curves certainly look different. In this example the uncertainties differ by a factor of three; if the difference had been closer to a factor of ten the contrast would have been even more extreme.
• What’s often worse, the “significant figures rules” make it impossible to keep guard digits, which are important for reasons discussed in section 3.2.
• Additional reasons why “significant figures” are deprecated can be found in section 11.

Professionals avoid these problems by avoiding the whole idea of “significant figures”. Instead, they use two separate numerals, expressing the nominal value and the standard deviation separately. For example, NIST (reference 1) reports the charge of the electron as

which can equally well be written as

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

This is as it should be. People who know what they’re doing know that the “significant digits rules” are absurd.

For specific recommendations on what you should do, see section 4.2.

3  Propagation of Uncertainty

In a moment (section 4.2) we will discuss how to express a given amount of uncertainty. Before we get there, though, we must face a bigger challenge, namely figuring out how much uncertainty there is. Sometimes you are given some quantities (called inputs), and asked to calculate other quantities (called outputs). If you know the uncertainties of the inputs you can calculate the uncertainties of the outputs, by a process known as propagation of uncertainty.

To say the same thing in other words, we are learning a new type of arithmetic, i.e. performing computations on probability distributions rather than on simple numbers.

Ideally, the task of figuring out how much uncertainty there is should be independent of how you express the uncertainty. But in practice the two issues are sometimes related, because bad methods of expressing the uncertainty (e.g. the “sig figs rules”) commonly contribute to making the uncertainty worse. And at a more superficial level, people who don’t think clearly about how to express uncertainty are unlikely to think clearly about propagation of uncertainty.

However, we should not blame the “sig figs rules”, however bogus they may be, for all the world’s problems. There are plenty of other things that can contribute to making the uncertainty worse.

3.1  Simple Example: Reducing the Uncertainty by Data Analysis

This subsection demonstrates why there cannot possibly be a good “all-purpose” rule for rounding off numbers.

Let’s start with an ultra-simple example

where each of the addends has an uncertainty of ±10%, normally 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 3.3 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!

The basic problem in this case is that the “significant digits rules” demand grossly too much roundoff.

Rounding off always introduces some error. This is called roundoff error or quantization error. Keeping one or two guard digits reduces the roundoff error by one or two orders of magnitude.

Let’s be clear: When there is noise (i.e. uncertainty) in your raw data, guard digits don’t make the raw noise any smaller ... but they do make the roundoff errors smaller.

See section 8 for more discussion of various contributions to the uncertainty.

3.2  Example: Combatting Lopsided Roundoff to Extract Signal from Noise

The example in equation 3 is certainly not the only example where the uncertainty in the final answer is less than the uncertainty in the raw data.

We now turn to a somewhat trickier example, where the nature of the problem is not quite so obvious. Again, for simplicity, let’s assume the data is normally distributed and uncorrelated. As we shall see shortly, roundoff errors can be quite serious even in this case.

Suppose each of the raw data points is uncertain at the 0.01 level. If we average 100 such points, the mean value will be uncertain at the 0.001 level. More generally, if we average N points, the mean will be less uncertain than the raw data by a factor of √N.

We denote the ith raw data point by A(i) ± σ_A(i), where σ_A(i) is the noise. This noise is already present in the raw data.

Next, we round off each data point. That leaves us with something like A(i) ± R_A(i) ± σ_A(i), where R_A(i) is the roundoff error. It is easy to fall into situations where even though the σ_A(i) are independent and normally distributed, the R_A(i) have a viciously lopsided non-normal distribution.

Just because the raw data is normally distributed doesn’t mean the roundoff errors will be normally distributed!

This is illustrated by the example worked out in the “roundoff” spreadsheet (reference 2), 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±0.018 and you already see that we are departing from the usual “significant figures” nonsense. There is considerable uncertainty in the second decimal place, so you may be wondering why I am recording the data to three decimal places.

Answer: as will become clear very soon, it is important to keep that third decimal place. 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. 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). Your typical GPS receiver deals with even worse SNRs – and the stuff that JPL puts in the Deep Space Network is mind-boggling. 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, because a digit that is uncertain (and many digits to the right of 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. 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.

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 programs² 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 error 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. But 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.) However, you should not imagine that this will solve all your problems. In any situation where fiddling with the roundoff rules makes any significant difference in the results, you are in serious trouble. The situation is overly burdened by roundoff errors, and fiddling with the roundoff rules will only affect the tip of the iceberg. The only real solution is to use more precision (more guard digits) during the calculation. 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 2) 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 create Gaussian random numbers. 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 4.2 and reference 3.

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

   a)

The brute-force way to see if you are carrying enough guard digits in the intermediate steps is to ask the question: If I did this twice, rounding up in one case and rounding down in the other case, would both cases give the same answer, to an acceptable approximation? If not, you need to carry more digits.

   b)

In the case where the roundoff errors are small, that brute force method can be well approximated by looking at the first derivative of the final answer with respect to whatever quantity you want to round off.

   c)

In a multi-step or iterated calculation where many roundoff operations occur, you need to worry about accumulation of errors.

3.3  Another Example: Magnesium Mass. Monte Carlo Solution

This is based on question 3:21 on page 122 of reference 4.

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 5.

The textbook provides the raw data shown in table 1.

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

Table 1: Isotopes of Magnesium, Rough Raw Data

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

The approach in the textbook has at least three problems:

• The primary problem is that the textbook uses methods that vastly overestimate the halfwidth of the distribution.
• Secondarily, 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, so now the nominal value and the halfwidth are messed up.
• Thirdly, the textbook expresses the uncertainty using the dreaded “sig digs” rules, which give only a crude indication of the halfwidth of the distribution.

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 if we had solved the primary problem (getting a good estimate of the uncertainty) then the “sig digs” rules would not have called for such drastic rounding. So the propagation-of-error issue really is primary.

Let’s deal with the tertiary problem first. Let’s convert to a more reasonable notation for expressing the uncertainty. The problem is that the “sig digs” notation in table 1 gives us only the crudest idea of the uncertainty ... is it one count in the last decimal place, or two, or many? If we use only the numbers presented in the textbook, we have to guess. Let’s 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 2.

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

Table 2: Isotopes of Magnesium, Rough Data with Explicit Uncertainty

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.

It is important to notice that 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.

If we look deeper, we discover another interesting point that is absolutely necessary if we are to benefit from the aformentioned “ballpark” property: it is important that the percentage numbers are in fact percentages, so they add up to 100%. We say there is a sum rule.

That means the uncertainty in any one of the abundance numbers is strongly anticorrelated with the uncertainty in the other two. The usual elementary “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 500 ppm uncertainty, but the sum rule tells us they actually add up to 100 with some utterly negligible uncertainty:

because if you add up all the possibilities you have to get 100%. Maybe there is some fourth, hitherto-unknown isotope that makes the LHS of equation 7 less than 100%, but any such contribution is utterly negligible relative to the other uncertainties in the problem.

There are various ways to alleviate this problem.

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 2, 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 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 is a good trick, but (a) it requires understanding the problem to some extent, (b) it only works if the problem is linear, and (c) it doesn’t entirely solve the problem, because it doesn’t fully exploit the sum rule.

We now turn to a completely different technique, namely Monte Carlo.

This 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 explicitly.

Remember (as mentioned in section 2) that an uncertain quantity is really a probability distribution. 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 list of observations drawn according to the distribution.

The representation in terms of a huge sample is often 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 6.

The strategy is to treat each of the uncertain quantities in table 3 as a probability distribution, and to represent each distribution by 100 observations. Using these observations, we make 100 independent trial calculations of the average mass, and then compute the mean and standard deviation of these 100 trial values.

Actually, if we’re going to go to all that trouble, we might as well use the best available data, taken from reference 7, as shown in table 3. (Actually, using the rough textbook data – i.e. table 2 – would lead to nearly the same results, as you can verify by plugging it into the spreadsheet.)

isotope	molar mass / amu	abundance
24Mg	23.9850423(8)	78.99(4)%
25Mg	24.9858374(8)	10.00(1)%
26Mg	25.9825937(8)	11.01(3)%

Table 3: Isotopes of Magnesium, IUPAC Data

In the trial calculations on the spreadsheet, all of the points are drawn independently, except that the numbers representing the ²⁴Mg abundance are not independent at all, but instead are calculated from the other two abundance numbers using the sum rule.

The final answer appears in cells H5 and H6, namely 24.3050(6).

Technical notes:

• In the spreadsheet, “automatic recalculation” is turned off. To recalculate, you have to hit the F9 button. Note that this gives you a new batch of random numbers.
• Once again Box-Muller transforms are used to generate a sample of a Gaussian normal distribution. This is implemented by columns I through T 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 against the IUPAC standard value given in reference 8, you find that the Monte Carlo calculation gives the same nominal value and the same standard deviation. That’s encouraging.

*   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 2. 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.

*   Discussion

To summarize: As mentioned near the top of section 3.3, the textbook has at least three problems: Primarily, 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 amu, with whatever degree of uncertainty is implied by that number of “sig digs”.

We now compare that with the our preferred answer, 24.3050(6) amu. Our standard deviation is 25 ppm; theirs is something like one part per thousand (although we can’t be sure). In any case, their uncertainty is about 40 times worse than ours.

Their nominal value differs from our nominal value by something like eight times the length of our error bars. Actually the factor is somewhere between 2.5 and 25; we can’t be sure because of the crudity of the halfwidth information in the textbook data (table 1). In any case, it’s a huge discrepancy.

3.4  Example: Interpolating 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 4: Length of Blocks, Raw Data

and somewhere beside the column is a notation that all the numbers are good to 1/4th of a millimeter.

If we worshipped the “sig digs rules” we would say that that the first number (40) had one “sig dig” and therefore had an uncertainty of a few dozen units. But that would be wrong. The actual uncertainty is a hundred times smaller than that. The lab book says the uncertainty is 1/4th 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 it is accurate to a few percent of a millimeter. The actual uncertainty is ten times larger than that. The lab book says that all the numbers are good to 1/4th of a millimeter, and that’s the end of the story.

The numbers in table 4 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 20% of its half-width. Twenty percent seems like a lot. It may or may not be harmless. In contrast, writing 3/4ths as .75 cannot cause problems, might be better than rounding off, and costs nothing.

Bottom line: Paying attention to the “sig digs rules” is unnecessary at best. 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 raw data. See section 4.2 for more details.

Even more-extreme examples can be found. Many rulers are graduated in 1/8ths of an inch. When measuring things to the nearest 1/8th, it is often convenient and sensible to write things to three decimal places, e.g. 4.375 in. If we worshipped the “sig digs rules” we might think such a number was accurate with a few thousandths of an inch, but that would be completely wrong. The actual uncertainty is one or two orders of magnitude larger.

Any time your measurements are quantized with a step-size that doesn’t divide 10 evenly, the “sig digs rules” will mess things up.

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. There is no reasonable basis for any such rule. For obvious reasons, instruments are typically calibrated in conventional units (e.g. SI units) times some power of ten. If the readability and/or precision of the instrument happens to coincide with some unit times a power of ten, it’s probably a coincidence.

Usually, the fundamental limit of readability is set by some sort of noise, fluctuations, or fuzz. That makes sense, because if the reading were not fuzzy, you could just apply some magnification and get more accuracy for free.

3.5  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.

Set up the calculation. Do it once in the usual way, using the nominal, best-estimate values. Then pick one input variable that you reckon makes the dominant contribution to the uncertainty of the result. Then re-do the calculation with this one variable at the top end of its error bar. Then do it again at the bottom end of the error bar.

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

Equation 8 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%.

The Crank-Three-Times method is a type of “what if” analysis. We can also consider it a simple example of an iterative method of estimating the uncertainty (in contrast to the algebraic methods described in section 3.6). This simple method is a nice lead-in to fancier iterative methods such as Monte Carlo, as discussed in section 3.3.

This 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).

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, 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 more than one variable at a time takes on a non-nominal value, the sig figs approach is hopeless, and algebraic methods are painfully impractical. Your only reasonable option is Monte Carlo, as discussed in section 3.3.

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

Equation 9 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. They can arise whenever nonlinearities are involved. Note that the Crank-Three-Times method can handle nonlinearities just fine. This is vastly superior to the manual, algebraic methods discussed in section 3.6, which treat everything as approximately linear. That is, they effectively expand everything in a Taylor series, and keep only the zeroth-order and first-order terms.

3.6  Manual, Algebraic Propagation of Uncertainty

People often ask for rules for calculating uncertainty by hand. In general, I recommend against it, because in almost all cases you’re better off using an iterative approach: perhaps the Crank-Three-Times method discussed in section 3.5, or if that’s not good enough, the Monte Carlo method as discussed in section 3.3. Remember, you don’t have to re-invent all the Monte Carlo technology on your own; just copy the existing spreadsheet (reference 6) and re-jigger it to do what you want.

However ... if you insist on doing things without a computer, I will now provide some rough-and-ready techiques you can use. 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.

1. Addition and Subtraction: If you are calculating C := A + B or C := A − B, and provided the uncertainties are uncorrelated and normally distributed, then the absolute uncertainties add in quadrature. That is: σ_C² = σ_A² + σ_B².
2. Multiplication and Division: If you are calculating C := A × B or C := A / B, and provided the uncertainties are uncorrelated and normally distributed, and provided the relative uncertainties are small (compared to unity), then the relative uncertainties add in quadrature. That is: (σ_C/C)² = (σ_A/A)² + (σ_B/B)².
3. Powers: If you are calculating B := A^N, where N is exact, provided the uncertainties are uncorrelated and normally distributed, and provided the relative uncertainty is small, then the relative uncertainty grows in proportion to N. That is, σ_B/B = |N| σ_A/A. 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.
4. Other operations: There are no simple, general rules. Yes, I know there are lots of references that claim to provide such rules ... but they’re not reliable. 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 typical simple rules would have you believe. One valid way to approach it would be to study the rate of convergence of the Taylor series ... but oftentimes it’s much easier to give up on the manual approach and just do the Monte Carlo.

Remarks:

• There are a lot of provisos in these rules, and they must be taken seriously. Otherwise you may seriously underestimate or overestimate the uncertainty.
• Please remember that the rules given above 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 4.2.
• In many cases, a propagation-of-uncertainty calculation is not the best way to find the uncertainty of the final result, perhaps because there are sources of uncertainty not accounted for in the inputs to the calculation, or perhaps because of unsuspected violations of the provisos (independence, normal distribution, et cetera).
• Still the propagation-of-uncertainty calculation remains valuable. Among other things it will give you some guidance as to what is required so that your calculational process does not introduce any additional uncertainty.

3.7  Example: Manual Propagation

Suppose somebody who adheres to the sig-digs cult asks you to work the following problem³

Each of the three quantities involved has some uncertainty, so your first task is to figure out how much uncertainty. Another way of saying it is that each of those quantities looks like a number but is really a probability distribution in disguise, and you have to figure out the width of the distribution. One semi-reasonable guess is that each quantity has about three counts of uncertainty in the last digit. But it could be a lot more, or a lot less ... you never know, which is one of the abominable things about significant figures.

So let’s make that guess, and restate the problem as 4.4±.3 × 2.617±.003 − 9.064±.003.

Using the usual precedence rules, we do the multiplication first. According to the propagation rules in section 3.6, we will need to convert the absolute uncertainties to relative uncertainties. That gives us: 4.4±6.82% × 2.617±0.1%. When we carry out the multiplication, the result is 11.5136±6.82%. 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 11.5136±0.785 − 9.064±.003 = 2.4496±0.789.

Now we have to decide how to present this result. One reasonable possibility would be to round it to 2.45±0.79 or equivalently 2.45(79). One could also justify heavier rounding, to 2.4(8). Note that this version differs from the previous version by only 6% of an error bar, so the extra rounding wasn’t particularly disastrous.

Trying to express the foregoing result using sig digs would be a nightmare, as discussed in more detail in section 11.7.

3.8  Amplification of Uncertainty

Note that this problem is what we call a noise amplifier. We started with three numbers, one of which had about 7% relative uncertainty, and the others much less. We ended up with about 32% relative uncertainty. Any time you compute a small difference between large numbers, the relative uncertainty will be magnified.

It appears that the uncertainty grew during the calculation, but you should not blame the calculation. The calculation did not cause the uncertainty; it merely made manifest the uncertainty that was inherent in the original data.

If you need more precision in the final answer, you will need to make a more precise measurement of the raw data, or you will have to redesign the experimental procedure, so that subtracting off such a large “baseline” number is not required.

4  How Much Accuracy Is Enough? How Much Data Is Enough?

4.1  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. But 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 3.2) or because of anticorrelated errors (section 3.3). 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).

4.2  Recommendations – How To Do It Right

There are several equally good ways of expressing a number along with an explicit uncertainty. It usually doesn’t matter whether the uncertainty is expressed in absolute or relative terms, so long as it is expressed clearly. The following express absolute uncertainty, and are synonymous:

and the following expresses relative uncertainty:

There are special rules for raw data, as described in section 4.3. Otherwise, all these recommendations apply equally well to measured quantities and calculated quantities.

If you have a long list of numbers, you may be able to save yourself some writing by “distributing out” the statement of uncertainty, e.g. by writing a note that applies to the whole list, saying that the list elements are all ±0.055, or all ±4%, or all limited by roundoff. That way you don’t need to attach the uncertainty directly to each element of the list.

You should report the form of the distribution, as discussed in section 4.4. Once we know the form of the distribution, if it is a two-parameter distribution, then either of the expressions in equation 11 gives us a complete description of the distribution.

In the rather common case where roundoff error is the dominant contribution to the uncertainty, this can be expressed using a slash in parentheses:

That can be viewed as shorthand for 0.087(½) i.e. an uncertainty of half a count in the last place, but it also conveys the fact that the distribution of roundoff errors is usually highly non-Gaussian, usually closer to a flat distribution. In particular, the standard deviation may be markedly smaller than the halfwidth, as discussed in connection with figure 3.

These recommendations to 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 find it convenient to round that off to 250 mm, ± 10%, but I am not required to do so.

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 for qualitative, at-a-glance comparisons, as discussed in section 4.6.

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.⁴ 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.

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. 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 3.2. Additional discussion of roundoff procedurs can be found in reference 3.

The cost of carrying one or two guard digits more 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.

4.3  Keep All the Raw Data

When you are making observations, the rule is that you should record all the raw 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 9.2.)
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.⁵
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.

4.4  Report the Form of the Distribution

Whenever you report an uncertain quantity, keep in mind that you are describing some sort of probability distribution.

Therefore it is important to report 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 clear basis for such an assumption, so if your data is known to be – or even suspected to be – non-Gaussian and/or non-IID, it is important to point this out explicitly. See section 9.8 for more on this.

4.5  Correlations: Another Reason to Eschew Rounding

Consider the following example: Newton’s constant of universal gravitation, G, is known to about 100 ppm. The mass of the earth, M, is also known to about 100 ppm. So far so good. The tricky thing is that the product GM is known to about 2 parts per billion.

Now suppose you have a value of G and a value of M that are consistent in the sense that when multiplied together, they give the correct value of GM, accurate to 2 ppb. If you round off G and/or M to four or five digits, attempting to match the sig figs of each quantity to the uncertainty of that quantity, you will very seriously degrade the accuracy of the product GM.

One way to visualize this situation is shown in figure 4. (In general, you can always visualize a probability distribution in terms of a scatter plot). In the figure, the the abscissa represents G and its standard deviation is shown by the magenta bar. The ordinate represents M and its standard deviation is shown by the blue bar. The standard deviation of the product GM is shown by the yellow bar.

Figure 4: Correlated Uncertainties

In this figure, the amount of correlation has been greatly de-emphasized for clarity. The uncertainty of the product is 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.)

In the case of a simple multi-dimensional Gaussian, the contours of constant probability are ellipses when we plot the probability as in figure 4. If the variables are highly correlated, the ellipses are highly elongated, and the axes of the ellipse 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. Figuring out what’s safe and what’s not is often quite difficult.

4.6  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. It is easier to compare rounded-off numbers, since rounding makes the similarities and differences more immediately apparent to the eye:

However, roundoff to facilitate comparison is definitely not the best procedure. 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. In other situations you can get a false match. So, again we see that significant figures are a convenient way of getting the wrong answer.

It is far more sensible to subtract the numbers at full precision (as in equation 16), 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 look at the deltas without rounding. 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 more informative to look for large-percentage variations in the deltas than to look for small-percentage variations in the original values.

4.7  Guard Digits

The need for guard digits is intimately connected to the fact that uncertainty is not the same as insigificance. See section 3.2, section 11.4, and section 8 especially figure 7 in section 9.2.

As a first example, uncertain digits can be significant in signal averaging, as discussed in section 3.2.

As another example, guard digits are often necessary when there are correlated uncertainties. Such correlations are commonly encountered in situations where there is a small difference between large numbers. As an illustration, suppose A and B are two points on the number line. Suppose we know that A = 30.050±3.0 km, while B = 30.040±3.0 km. As you can see, both of these numbers have roughly 10% uncertainty. If that were the whole story, you might feel justified in rounding them off quite a bit. However, in this case it turns out that the difference vector (A−B) known – somehow – to be 10±1 meters. Note that the absolute uncertainty in (A−B) is measured in meters, not kilometers. In a situation like this, if you are storing the A and B positions in a database, you might very well decide to store them with ±1 meter precision or better, so that you can faithfully represent the information you have about (A−B).

4.8  Too Many Digits, Oh My!

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

“What if a student divides distance by time 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 answer is always the same: No, there is no problem.

To me, that nine-digit number doesn’t 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. But there are plenty of cases where the uncertainty is not (yet) known, and the only smart thing to do is to write down loads of guard digits.

Suppose we later discover the 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 kill you.

4.9  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 3.5. (Apply it to selected problems now and then, not every problem.) It requires less sophistication and produces better results than 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.

4.10  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’s 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. For more on this, see reference 9.

  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.

• Some students think that being wrong is Wrong, with a capital W, in the same way that lying and cheating 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.

  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.