Expanded Uncertainty Calculator
The mean of repeated measurements with its standard uncertainty and the expanded uncertainty at a chosen coverage factor — the ± you report.
Repeat a measurement n times: the standard deviation describes the scatter of single readings, and the standard uncertainty of the mean is that divided by √n (a Type A evaluation in the GUM).
How the expanded uncertainty calculator works
Repeat a measurement n times: the standard deviation describes the scatter of single readings, and the standard uncertainty of the mean is that divided by √n (a Type A evaluation in the GUM). Multiply by a coverage factor k — 2 for about 95% confidence, 3 for 99.7% — for the expanded uncertainty you quote: result = mean ± U. Add Type B components (calibration, resolution) in quadrature if you have them.
Formula: u = s ÷ √n; U = k × u; result = mean ± U
Worked examples
| Inputs | Expanded uncertainty U | Note |
|---|---|---|
| Eight readings around 10.00, k = 2 | 0.0189 | 10.006 ± 0.019 |
| With a Type B component of 0.01 | 0.0275 | ± 0.028 |
| Three readings, k = 3 | 0.1732 | 5.2 ± 0.173 |
FAQFrequently asked questions
What is the difference between standard deviation and uncertainty of the mean?
The standard deviation says how much one more reading would scatter; the uncertainty of the mean says how well you know the average, and it shrinks with √n. Ten readings pin the mean about three times better than one.
What coverage factor should I use?
k = 2 (about 95%) is the default in calibration certificates and most reports. For very few readings the t-distribution gives a larger k — about 2.6 for n = 5 — which the simple k = 2 understates.
What are Type B components?
Uncertainties you do not get from repeats: the instrument’s calibration certificate, its resolution (half the last digit ÷ √3), temperature effects. Convert each to a standard uncertainty and combine in quadrature; the calculator takes one combined figure.
Does more repeats always help?
Only for the random part. If a Type B component dominates — a balance calibrated to ±0.1 g — averaging a hundred readings will not get you below it.
Where these figures come from
- IUPAC — Standard atomic weights (2021 conventional values) — the molar-mass table
- NIST — CODATA 2018 fundamental physical constants — Avogadro constant, gas constant, speed of light
- NIST Chemistry WebBook — thermochemical data
- CSIRO — Australia's national science agency
Last checked: September 2026. Atomic masses are the IUPAC conventional values; constants are CODATA 2018; equations are the standard textbook forms.