The standard error of a statistic is the standard deviation (or an estimate thereof) of its sampling distribution. Why is this called an error? In metrology, measurement error is the difference between the (unknown) true value and the measured value; the measurement uncertainty is an estimate of the dispersion of many measurements around the (unknown) true value. If that dispersion is normally distributed, and it often is, then the standard deviation of many measurements may be used as an estimate for the standard uncertainty. Wikipedia notes that the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, but that sounds more like a standard uncertainty than to a standard error. This is confusing.

Why is it called the standard “error” rather than standard “uncertainty”?


The question "why is this term used, rather than this other term" is, like much terminology, a matter of historical happenstance. Sometimes the outcome is felicitous and sometimes less so. I think that with a little context it makes reasonable sense in this case.

Yule used it in 1897$^{[1]}$, in the context of a particular phrase that I think makes its intent pretty clear:

"We see that $\sigma_1\sqrt{1 - r^2}$ is the standard error made in estimating $x$"

[This is in turn quoted in the Oxford English Dictionary and is mentioned in the Standard Error entry (by John Aldrich) in $[2]$.]

Here it is with a little context (NB the journal is long out of copyright):

Referring again to the expressions (8), we see that σ1√(1−r^2) is the standard error made in estimating x from the characteristic x = r σ1 / σ2 . y, and similarly σ2√(1−r^2)is the standard error made in estimating y from the second characteristic y = r σ2 / σ1 . x .

Yule later extended that use to estimating other quantities.

I think "the standard error-made-in-estimating" a quantity is clear enough, and once the origin is clear, the shorthand standard error isn't so obscure.

Of course, like the term or not, once people start to treat such a term as conventional, like the QWERTY keyboard, it's entrenched; you're pretty much stuck with it.

$[1]$ Yule, G.U. (1897), "On the Theory of Correlation," Journal of the Royal Statistical Society, 60, 812-854

$[2]$ Miller, J. "Earliest Known Uses of Some of the Words of Mathematics".

  • 1
    $\begingroup$ +1. In teaching I always underline that error quantifies being erratic rather than being erroneous. In this I draw upon characteristically dry, wry and sharp comments in Harold Jeffreys, Theory of Probability. Mistakes are included in many datasets, but error means uncontrolled variability rather than mistakes. $\endgroup$ – Nick Cox Nov 4 '17 at 10:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.