How is $\pm$ notation taught / actually used by professionals?

Question

When using notation like $5.7 \pm 0.2$ to indicate a measurement and error, it seems like there are many standards to what this could mean. Is there a standard taught to students? Is there a standard used by professionals?

According to this wikipedia $5.7 \pm 0.2$ could mean:

• The value definitely falls within the range $[5.5, 5.9]$
• $.2$ represents one standard deviation of uncertainty, so there's a probability of $.683$ of falling within the range $[5.5,5.9]$
• $.2$ represents two standard deviations of uncertainty, so there's a probability of $.954$ of falling within the range $[5.5,5.9]$

University physics labs I've found online seem to indicate a similar polysemy

initially (page 2) seems to indicate the first of these three, an upper and lower bound on a range of possible values, but then goes on to present techniques for propagation of uncertainties that seem better suited to the cases where the uncertainty represents a number of standard deviations.

I am a mathematics educator trying to put together something about the mathematics of precision and its relation to limits and derivatives in calculus, and want to accurately portray both how this notation is taught and how it is used by professionals.

It makes sense that different situations would call for different uses of the notation, but I can also imagine this being confusing for students and professionals.

• Is there a single standard taught to students at the university level?
• Is there a single standard used by professionals?
• If not, do professionals at least always indicate which standard they're using, or are they sometimes not specified, leading to mix-ups?

Answer 1

The definitive source for determining and reporting uncertainty in measurements is the BIPM's Guide to Uncertainty in Measurement. The BIPM is the international organization that defines and maintains the SI system of units.

In section 7.1.4 they say

when reporting the result of a measurement and its uncertainty, it is preferable to err on the side of providing too much information rather than too little

They give an example of suitable text in 7.2.2

“$m_S = (100,021\ 47 \pm 0,000\ 35) \mathrm{\ g}$, where the number following the symbol $\pm$ is the numerical value of (the combined standard uncertainty) $u_c$ and not a confidence interval.”

Which they immediately follow by

The $\pm$ format should be avoided whenever possible because it has traditionally been used to indicate an interval corresponding to a high level of confidence and thus may be confused with expanded uncertainty

Then in section 7.2.4 they give another example

“$m_S = (100,021\ 47 \pm 0,000\ 79) \mathrm{\ g}$, where the number following the symbol $\pm$ is the numerical value of (an expanded uncertainty) $U = ku_c$, with $U$ determined from (a combined standard uncertainty) $u_c = 0,35 \mathrm{\ mg}$ and (a coverage factor) $k = 2,26$ based on the t-distribution for $\nu = 9$ degrees of freedom, and defines an interval estimated to have a level of confidence of 95 percent.”

So, the use of the $\pm$ symbol does not have a single definite standard meaning. The instructions of the BIPM are to either avoid the symbol entirely or to accompany it with sufficient text to render its meaning clear and unambiguous.

Unfortunately, professionals do not always follow this standard. So there are numerous examples in the literature where the meaning is ambiguous, even in context.

Answer 2

I am only adding things not covered in existing answers.

Unless otherwise indicated, $\pm \sigma$ ought to mean that $\sigma$ is one standard deviation, and thus about 68 percent chance to lie within the range. However the distribution is usually not well approximated by Gaussian. In practice the distribution typically has wider wings, such that the chance of being a few standard deviations from the mean is much higher than it would be for a Gaussian distribution. Also, many experimentalists underestimate the experimental error by a factor $1.5$ to $2$ because they fail to account for the fact that it is so hard to reduce systematic error below the level of the statistical error. They quote a statistical error and just sort of hope for the best about systematic effects. However those working in standards laboratories and the like are much more careful.

Answer 3

At the university level, for majors that are numerically minded, so chemists, statisticians, physicists, engineers, and so on, and in professional use, the only acceptable standard is

$.2$ represents one standard deviation of uncertainty, so there's a probability of $.683$ of falling within the range $[5.5,5.9]$

This is because uncertainty analysis is actually about fitting a Gaußian around the mean value, and reasoning with that. Quite a lot of statistical results are proved only for this case.

In particular, whereas to the beginning student it might make sense to want to have a guarantee like

The value definitely falls within the range $[5.5,5.9]$

but this is impossible in practice for many situations. That makes any standard based upon such a guarantee as to be impractical.

You might be interested in why something like

$.2$ represents two standard deviations of uncertainty, so there's a probability of $.954$ of falling within the range $[5.5,5.9]$

is also ruled out. Well, if we want to discuss such stuff, the correct way is to state a certain confidence interval, whereby then it is customary to state precisely how many percent the confidence interval gives us.

An alternative is the statement of how many $\sigma$ that $\pm$ indicates. The important thing is that any deviations from the standard warrants clear labelling. The standard is too widespread to deviate without good reasons.

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The reason why this is important, is that we want to be able to actually model the real-world uncertainties. Every statistical package allows for the generation of Gaußian distributed random variables, from which then by churning through any numerical process, one can find the empirical spread of said numerical process.

When a known phenomenon is well-approximated by a Gaußian, then it will become possible to detect fraud when a data set fits too well. This means that when an overly-eager beginner attempts to expand the uncertainty estimates on their data set, that can trigger fraud detection, and so it is important not to overstate the uncertainty estimates.

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Have you not ever seen skewed uncertainties? For example, $5.7^{+0.2}_{-0.1}$ for when the uncertainties are not the same positive versus negative? Also, when you have multiple random uncertainties and systematic uncertainties, then having many iterations of the $\pm$ sign used together e.g. $5.7\pm0.2\pm0.3^{+0.2}_{-0.1}$ and so on.

Answer 4

In undergraduate physics labs an expression like $A\pm B$ means that the average or mean value of a set of measurements is equal to $A$ and the standard error of the mean is equal to $B$.

The accuracy of a measured value due to the limits of the experimental apparatus is determined by the number of significant digits to which the data is reported.

Answer 5

A bit of an expanded comment, as the other answers so far don't cover it. In professional (and undergrad lab) evaluations, you often have not only a single kind of uncertainty, but several. And notation like $$ m = (0.0100 \pm 0.0018_\text{sys} \pm 0.0005_\text{stat})\,\mathrm{g}$$ is common.

This separates the systematic contribution (errors due to approximations in the evaluation, or the imperfection of a measurement instrument, so uncertainties that do not average out over multiple measurements) from the statistical contribution (error due to random variation of the measurement process that averages out over repetition, or over a series of measurements with different parameters).

In the following we discuss the simplest kind of analysis to compute such uncertainties. W assume an equation for the result of the measurement of the form: $$ X = f(A, B, C, \ldots) $$

The systematic contribution is then typically handled as a conservative, maximal bound on the error and consequently it is common to assume the worst case of full correlation between the individual systematic components, so the errors add up as the sum of the absolute values: $$ \Delta X_\text{sys} = \left| \Delta A_\text{sys} \frac{\partial X}{\partial A} \right| + \left| \Delta B_\text{sys} \frac{\partial X}{\partial B} \right| + \ldots $$

The statistical error, however, is some confidence interval (commonly 1, 2, 3\sigma) and individual contributions are assumed uncorrelated, so the contributions add up as $\Delta x$: $$ \Delta X_\text{stat} = \sqrt{ \left| \Delta A_\text{stat} \frac{\partial X}{\partial A}\ \right|^2 + \left| \Delta B_\text{stat} \frac{\partial X}{\partial B} \right|^2 + \ldots } $$ (This follows quite easily from the assumption of $A$ and $B$ being Gaussian, uncorrelated random variables. The assumption that they are Gaussian is not too bad in a lot of cases to to the central limit theorem. The statistical uncertainties for these variables are estimated with formulas like $\overline A = \frac 1 N \sum_{i=1}^N A$, $\Delta A = \frac{\sqrt{N}}{\sqrt{N-1}} \sqrt{\overline{A^2} - \overline A^2}$ form a series of $N$ measurements, or from more complex statistical methods like linear regression, which gives uncertainties for the results.)

Of course this is just the very simples kind of analysis. Experimental papers often separate out more kinds of different uncertainties (e.g. different statistical measurement contributions, or different kinds of systematic errors). There may be some correlation between uncertainties (e.g. when using advanced linear regression algorithms, the slope and the offset may have a correlation coeffient).

As the other answers say, what exactly is meant by this must be made explicit in some form. Either because a subfield has an established standard, or you describe it in the text. (Or in the case of undergrad labs, a certain style is taught and used throughout).

Answer 6

This is a very good question, because analysis of uncertainty is tricky, and can be done in many ways. A good practice is to describe clearly how the uncertainties are derived.

Standard deviation is used quite commonly. It can be calculated uniformly for any data set, and most probability distributions. Please note, that it may not correspond to 68% probability for distributions other than Normal.

In particular, flat distribution is not so uncommon. A simple example is a measurement with a ruler. You can determine e.g. that the value is between 10 and 11 cm. Standard deviation of a flat distribution is $$\sigma = (a-b)/\sqrt{12}\approx (a-b)\cdot0.29\,,$$ where $(a-b) = 11-10 = 1$ cm, so the result is $10.5\pm0.3$ cm.

This may seem counterintuitive, but it is a right way to do it, if you need to use the result for further statistical analysis. It may not be appropriate though, if you want to determine e.g. the maximum mechanical tolerance.

Please note how $0.29\cdot2 = 0.58$ is significantly less than $0.68$ for Normal distribution.