Type A uncertainty calculator
Paste a column of repeated readings to get the mean, the sample standard deviation, the standard uncertainty of the mean, the degrees of freedom, and the expanded uncertainty at your chosen coverage factor. This is the GUM (JCGM 100) Type A evaluation. Everything runs in your browser.
Your readings
Enter at least two readings of the same quantity taken under repeatability conditions. Paste straight from Excel.
k = 2 gives about 95% only with large degrees of freedom. For a few readings, the Student t option is the honest choice.
Result
Method: GUM (JCGM 100:2008) clause 4.2, Type A evaluation. Standard deviation uses the n − 1 divisor; the standard uncertainty of the mean is s / √n; degrees of freedom are n − 1. Figures are rounded for display.
Type A uncertainty, in plain terms
What a Type A evaluation is
Measure the same thing several times and the readings will not all agree. That scatter is real, and a Type A evaluation is how you turn it into a number. The GUM (JCGM 100) clause 4.2 defines it as the evaluation of a standard uncertainty by the statistical analysis of a series of observations. You take the mean as your best estimate and use the spread of the readings around that mean to say how well the mean is known. It is the most direct uncertainty component there is, because you observed it rather than assumed it.
Standard deviation, and the n minus 1 divisor
The experimental standard deviation s measures the scatter of the individual readings. You sum the squared deviations of each reading from the mean and divide by n minus 1, not by n, then take the square root. The n minus 1 is the Bessel correction: because the deviations are measured from the sample mean you calculated, dividing by n would bias the estimate low, and n minus 1 corrects for it. That same n minus 1 is the degrees of freedom of the estimate, which matters when you choose a coverage factor.
A single reading versus the mean
The experimental standard deviation s describes how much any one reading scatters. If you report a single reading, s is its standard uncertainty. But averaging cancels random scatter, so if you report the mean of n readings its standard uncertainty is smaller: s divided by the square root of n, the experimental standard deviation of the mean (GUM clause 4.2.3). Ten readings do not make one reading ten times better, but the mean of them is about three times better, because the improvement goes with the square root of n.
Expanded uncertainty and coverage
A standard uncertainty is one standard deviation. To state an interval that should contain the value most of the time, you multiply by a coverage factor k to get the expanded uncertainty U = k times u. A factor of k = 2 is the usual shorthand for about 95%, but that holds only when the degrees of freedom are large. With just a handful of readings the degrees of freedom are small, and k = 2 gives less than 95% coverage. The honest fix is a Student t factor t(nu, 95%), which is larger for small samples and settles toward 2 as the sample grows (GUM Annex G). This tool offers both so the interval matches your real sample size.
A Type A evaluation covers only the scatter of these repeats. A complete measurement also carries Type B components such as the reference standard, resolution, and drift. Combine the standard uncertainty from here in quadrature with those in themeasurement uncertainty budget calculatorfor the full budget. References: GUM (JCGM 100:2008); ISO/IEC 17025:2017.
Common questions
What is a Type A evaluation of uncertainty?
A Type A evaluation estimates a standard uncertainty statistically, from a set of repeated measurements. You take several readings of the same quantity, compute the mean and the experimental standard deviation, and derive the standard uncertainty from that scatter. It is defined in the GUM (JCGM 100) clause 4.2, and it is the component that captures the random spread you see when you measure the same thing more than once.
What is the difference between Type A and Type B uncertainty?
The two labels describe how a component is evaluated, not what kind of quantity it is. Type A is evaluated statistically, from repeated observations, which is exactly what this tool does. Type B is evaluated by any other means: a calibration certificate, a manufacturer specification, a resolution limit, an assumed distribution, or engineering judgement. Both produce a standard uncertainty in the same units, and both are combined the same way when you build the full budget.
Why divide by n minus 1 instead of n?
You are estimating the spread of the population from a finite sample whose own mean you had to calculate first. Dividing the sum of squared deviations by n minus 1 (Bessel correction) removes the bias that dividing by n would introduce, since the deviations are taken from the sample mean rather than the true mean. The result, s, is the experimental standard deviation the GUM uses. The n minus 1 is also the degrees of freedom of the estimate.
Do I report the standard uncertainty of a single reading or of the mean?
It depends on what value you report. If your reported result is the mean of the n repeats, the relevant standard uncertainty is the standard uncertainty of the mean, s divided by the square root of n, which is smaller because averaging suppresses the random scatter. If your reported result is a single reading, the standard uncertainty is s itself. This calculator shows both and lets you pick which one is expanded.
Does a coverage factor of k = 2 always give about 95%?
Only when the degrees of freedom are large. k = 2 gives roughly 95% coverage for an approximately normal result with many effective degrees of freedom. When you have only a few readings, the degrees of freedom (n minus 1) are small and k = 2 understates the interval. The more honest choice for small samples is a Student t factor t(nu, 95%), which is larger. This tool offers that option so the expanded uncertainty matches your actual sample size.
What coverage factor does JCGM 100:2008 recommend?
JCGM 100:2008 (the GUM, clause 6.3) does not mandate a single value. It notes that k is typically in the range 2 to 3, and that k = 2 gives an interval with a level of confidence of approximately 95 percent when the result is approximately normal with large effective degrees of freedom. That combination is why calibration certificates overwhelmingly report expanded uncertainty at k = 2, approximately 95 percent coverage.
How does this feed the full uncertainty budget?
The standard uncertainty from a Type A evaluation is one line in a larger budget. You combine it in quadrature (root sum of squares) with every Type B component, such as the reference standard, resolution, and drift, to get the combined standard uncertainty, then expand that with a coverage factor. Take the standard uncertainty this tool reports into the measurement uncertainty budget calculator to combine it with the rest.
This calculator performs the GUM (JCGM 100) Type A statistics on the readings you enter. It estimates one uncertainty component, the scatter of your repeats, and is not a complete uncertainty budget. It does not replace the GUM, ISO/IEC 17025, or your laboratory’s own procedures. A full evaluation adds every significant Type B contributor for your specific measurement. Review any figure against your quality system before you report it.
Keep the uncertainty with the record
Axiospec keeps every calibration, uncertainty, and certificate in one tamper-evident ledger built for ISO/IEC 17025. See it on real calibration data, no signup.