Measurement Uncertainty Calculator (Type A from repeated readings + Type B, expanded U)

Understanding the measurement uncertainty calculator — concept and principle

Every measurement has some doubt attached. This calculator follows the GUM method used by NIST to combine the scatter of repeated readings (Type A) with the instrument's resolution and other known contributions (Type B) into one expanded uncertainty, so you can report a result as mean ± U.

Scatter and resolution combine in quadrature; doubling gives ±0.015.

How it is calculated

Mean x̄ = Σx ÷ n; sample standard deviation s = √(Σ(x − x̄)² ÷ (n − 1))

Type A: uA = s ÷ √n

Type B from resolution: uR = resolution ÷ (2√3)

Combined: uc = √(uA² + uR² + uB²)

Expanded: U = k × uc; relative U = U ÷ |x̄| × 100 %

Enter readings separated by commas or new lines, the smallest display step and any other Type B standard uncertainty in the same unit. k is the coverage factor.

The principle

Repeated readings scatter around the true value. The standard deviation describes the spread of single readings, but the mean of n readings is more reliable, so its uncertainty shrinks by √n. A digital display cannot show anything finer than one step, so the true value could be anywhere within half a step either side; treating that as a flat (rectangular) distribution gives a standard uncertainty of half-width ÷ √3. Independent sources of error add like perpendicular vectors, so their squares are summed. Multiplying by k widens the interval so that it covers the true value with a stated confidence, about 95 % for k = 2 under near-normal conditions.

Worked example

Default inputs: readings 9.81, 9.79, 9.83, 9.80, 9.82, 9.78, 9.81, 9.84; resolution 0.01; other Type B 0; k = 2.

n = 8; mean = 9.81.
Squared deviations sum to 0.0028; s = √(0.0028 ÷ 7) = 0.02.
uA = 0.02 ÷ √8 = 0.007071.
uR = 0.01 ÷ 3.464 = 0.002887.
uc = √(0.007071² + 0.002887²) = 0.007638.
U = 2 × 0.007638 = 0.01528, or 0.156 % of the mean. Report 9.810 ± 0.015 (k = 2).

Each point is the calculator re-run with only “Coverage factor k” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Coverage factor k	Expanded uncertainty U = k × u_c (±)
0.5	0.003819
1	0.007638
1.5	0.01146
2	0.01528
2.5	0.01909
3	0.02291
4	0.03055

How to read the result

Round U to one or two significant figures and round the mean to the same decimal place. If the relative uncertainty is too large for your purpose, look at which component dominates: here the scatter (Type A) is larger, so more readings would help more than a finer display. Two results agree if their difference is small compared with their combined uncertainties.

Common mistakes

Using the standard deviation s itself instead of s ÷ √n as the uncertainty of the mean.
Dividing by n instead of n − 1 for the sample standard deviation.

Keep in mind: The method assumes independent error sources and a roughly normal distribution. With fewer than about 10 readings, a larger k from the t-distribution is safer; accredited calibrations must follow the full GUM procedures.

Frequently asked questions

What is the difference between error and uncertainty?

Error is the unknown difference from the true value; uncertainty is the quantified doubt about the result.

Why divide the resolution by 2√3?

Half a step is the half-width a of a rectangular distribution, and its standard uncertainty is a ÷ √3.

When should I use k = 3?

When you need higher confidence; k = 3 corresponds to more than 99 % for a normal distribution.

Sources: NIST Technical Note 1297, B. N. Taylor & C. E. Kuyatt, "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results" (1994)

Checked 9 Nov 2026. NIST Technical Note 1297 (B. N. Taylor & C. E. Kuyatt, 1994): Appendix A, Eq. (A-5) Type A standard uncertainty = estimated standard deviation of the mean; Eq. (A-7) and §4.6 rectangular distribution u = a/√3 with a = (a₊ − a₋)/2; §6 expanded uncertainty U = k u_c (k = 2 ≈ 95%, k = 3 > 99%).

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