Repeated Measurements: Mean and Standard Deviation

Repeated measurements: mean and standard deviation

3 min readUpdated 2026-10-16

Key answer

Take the mean of n repeated readings as the result. The experimental standard deviation s describes how far single readings scatter; s/√n is the standard uncertainty of the mean. Five caliper readings of 20.08, 20.08, 20.06, 20.10 and 20.12 mm give a mean of 20.088 mm, s = 0.0228 mm and s/√5 = 0.0102 mm.

What counts as a repeated measurement?

The VIM calls them measurements under repeatability conditions: the same procedure, the same operators, the same measuring system, the same operating conditions and location, and replicate measurements on the same or similar objects over a short period of time. Readings taken on different days, by different people or with a different caliper belong to other studies (see gauge R&R).

How do you calculate the mean?

Add the n readings and divide by n. For n independent readings that vary randomly, the GUM takes this arithmetic mean as the best available estimate of their expected value. For the five readings above: (20.08 + 20.08 + 20.06 + 20.10 + 20.12) / 5 = 100.44 / 5 = 20.088 mm.

How do you calculate the standard deviation?

Subtract the mean from each reading, square the differences, add them, divide by n − 1 and take the square root. That is the experimental standard deviation s of the GUM.

Reading	Reading − mean	Squared
20.08 mm	−0.008	0.000064
20.08 mm	−0.008	0.000064
20.06 mm	−0.028	0.000784
20.10 mm	+0.012	0.000144
20.12 mm	+0.032	0.001024
Sum		0.002080

s = √(0.002080 / 4) = √0.00052 = 0.0228 mm. It describes the spread of single readings: none of these five lies more than 0.032 mm from the mean, and if the readings are roughly normal, about 95 % of them fall within ±2s ≈ ±0.05 mm.

What is the standard deviation of the mean?

The mean is less uncertain than any single reading. The GUM gives its experimental standard deviation as s/√n and uses it as the Type A standard uncertainty of the mean: 0.0228 / √5 = 0.0102 mm. The degrees of freedom, n − 1 = 4 here, should be stated with it.

More readings shrink s/√n but not s. If s stayed at 0.0228 mm, 20 readings would give 0.0228 / √20 = 0.0051 mm, while single readings would scatter as much as before.

When do these formulas not apply?

They assume independent readings. The GUM warns that when readings are correlated, for example in time, the mean and s/√n may be poor estimates and other methods are needed. A part warming in the hand is one way readings can drift; before averaging, check that they do not climb or fall steadily.

What to do with s next: it becomes the repeatability line of an uncertainty budget, combined with the Type B lines described in measurement uncertainty basics.

Repeated readings in KELIBRON

Not yet. In the lab, placing the caliper again at the same spot gives the same value, because repeated-measurement scatter is not modelled. Identical readings do not mean zero uncertainty: the GUM notes that the resolution of the indication still contributes (F.2.2.1). Practise single readings on the Ø25 shaft for now; a repeated-measurement exercise is planned.

References

JCGM 100:2008 — Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM) · JCGM / BIPM
JCGM 200:2012 — International vocabulary of metrology — Basic and general concepts and associated terms (VIM, 3rd edition) · JCGM / BIPM
D. Flack, Callipers and Micrometers, NPL Measurement Good Practice Guide No. 40, Issue 2, 2014 (first issued 2001) · National Physical Laboratory

How the guides are checked against these sources: About KELIBRON.

Watch it done

Watch in the lab → Follow step by step →

Played back in the lab from domain commands; the reading and verdict come from the measurement model.

Questions

What is the difference between standard deviation and standard deviation of the mean?

s describes how single readings scatter; s/√n describes how well the mean of n readings is known. For five readings with s = 0.0228 mm, s/√5 = 0.0102 mm.

Why divide by n − 1 and not n?

The GUM’s experimental variance uses n − 1, the number of degrees of freedom left after the mean has been estimated from the same readings.

How many repeated measurements should I take?

More readings make the mean better known (s/√n falls), but they do not reduce the scatter s. NPL’s guide notes that with fewer than 10 readings the degrees of freedom should be taken into account in the coverage factor.

Try it yourself

Open the same setup in the 3D lab and take the reading yourself.

Open in the lab →