Type A uncertainty from repeated readings
Paste a column of repeated measurements and get n, the mean, the sample standard deviation, the standard uncertainty of the mean and the degrees of freedom — laid out in the fields the uncertainty budget calculator asks for, including the ν that most budgets quietly drop.
The two answers differ by √n and the data cannot tell them apart, so this has to be your answer rather than an assumption.
Straight into the uncertainty budget
One row, field by field, as the uncertainty budget calculator asks for them. The value is already a standard uncertainty, so it is entered against the standard-uncertainty distribution with no divisor — picking a normal distribution and k = 2 here would halve the contribution.
Component: Repeatability, mean of 10 readings (Type A) Value: 0.00687184 °C Distribution: Standard uncertainty (Type A or already u) k: not used — the value above is already a standard uncertainty c_i: 1 dof: 9
The degrees of freedom matter as much as the value. Leaving ν blank tells a budget that this component is known perfectly, which produces k = 2 where 10 readings actually earn k = 2.32 — the same data, reported 16 % smaller.
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What this is and is not
- This is repeatability, not reproducibility. Ten readings taken one after another by one person on one day, without touching anything, measure the narrowest kind of scatter there is. The variation between operators, between days, after a re-fit or a re-immersion is larger and is a different experiment — and it is the one that usually dominates a real budget.
- It is one line of a budget. The reference standard’s certificate, its drift since calibration, resolution, the gradient between reference and item, and every other Type B contribution still have to be added. A Type A figure alone is not a measurement uncertainty.
- Nothing was rejected. No outlier rule is applied, because discarding a reading needs a criterion written down before the data was seen. The furthest reading is marked so you can see it; whether it stays is a decision with a signature on it, and it gets recorded either way.
- The readings have to be of the same thing. A Type A evaluation assumes independent repeats under the same conditions. If the item was still equilibrating, if the reference was settling, or if the readings were taken as a ramp, the scatter describes that movement instead and the figure understates what a real repeat would show.
The arithmetic follows JCGM 100:2008 (the GUM) clause 4.2: the arithmetic mean as the best estimate, the experimental standard deviation with divisor n − 1, and the standard uncertainty of the mean as s/√n with ν = n − 1 degrees of freedom. The coverage factor shown for that ν comes from GUM Table G.2 at p = 95.45 %, and it is there to show what the degrees of freedom cost — the budget applies its own, from the effective degrees of freedom across every component.
How the calculation works
- 01The arithmetic follows JCGM 100:2008 (the GUM) clause 4.2: the arithmetic mean as the best available estimate, and the experimental standard deviation of the observations computed with divisor n − 1.
- 02The standard uncertainty of the mean is s/√n, which GUM clause 4.2.3 writes as u(q̄). The standard uncertainty of a single reading is s itself, and the tool asks which of the two your reported result is, because they differ by a factor of √n.
- 03Degrees of freedom are reported as ν = n − 1, which is the figure Welch–Satterthwaite needs in the budget to arrive at the effective degrees of freedom and from there the coverage factor.
- 04The coverage factor shown against that ν comes from GUM Table G.2 at p = 95.45 %. It is there to show what a short run costs, not to be applied to this one component on its own.
- 05How well s itself is known is reported as σ(s)/s ≈ 1/√(2ν) — the reason GUM clause 4.2.3 notes that the reliability of s deserves consideration when n is small.
Limitations
- It measures repeatability, which is the narrowest scatter there is: one operator, one setup, one short run, nothing touched in between. Variation between days, operators, re-fits and re-immersions is larger, usually dominates a real budget, and is a different experiment.
- It produces one line of a budget, not a measurement uncertainty. The reference standard's certificate, its drift since calibration, resolution, gradients and every other Type B contribution still have to be combined with it.
- It rejects nothing. No outlier rule is applied, because discarding a reading needs a criterion written down before the data was seen; the furthest reading is marked so you can see it and decide, and record the decision.
- It cannot fully separate independent repeats from a drift. A run where every reading moves the same way is flagged, but a partial trend still distorts s in ways only the person who took the readings can judge.
- A standard deviation of zero means the indication is coarser than the variation, not that the measurement is perfect. The tool says so rather than handing over a zero that would flatter a budget.
Frequently asked questions
- What is a Type A uncertainty?
- A Type A evaluation is an uncertainty worked out from the statistical analysis of a series of observations you actually made — in practice, the scatter of repeated readings. Everything else is Type B: a figure taken from a certificate, a specification, a data sheet, a handbook or a judgement. The distinction is about how you obtained the number, not about what sort of quantity it describes, which is why the same effect can be evaluated either way. Resolution is Type B because it comes from the display; repeatability is Type A because you measured it.
- Do I divide the standard deviation by the square root of n or not?
- It depends on what you report. If the result that leaves your laboratory is the mean of those n readings, the standard uncertainty is s/√n, because that describes how well the mean is known. If the result is a single reading — one indication taken in service, with the repeat run only there to characterise the scatter — then the standard uncertainty is s itself. The two differ by √n, so at n = 10 you are choosing between two answers that are three times apart, and this is the single most common Type A mistake. The tool makes you say which one it is before it shows an answer.
- Why is the standard deviation divided by n − 1 rather than n?
- Because the mean was estimated from the same readings. Using the sample mean in place of the true value of the measurand uses up one piece of information about where the distribution sits, so dividing the sum of squared residuals by n gives a standard deviation biased low — it understates the scatter, in the direction nobody notices. n − 1 corrects for that, and it is also the degrees of freedom the budget needs.
- How many repeat readings should I take?
- No standard sets a number, and a tool that gave you one would be inventing it. What can be said is the cost of a short run, which is arithmetic: s is itself uncertain by roughly 1/√(2ν), so four readings leave the standard deviation 41 % uncertain and ten leave it 24 %. On top of that, few degrees of freedom draw a larger coverage factor from GUM Table G.2 — 3.31 at ν = 3 against 2.28 at ν = 10 — so a short run is paid for twice. Ten is common practice in calibration for that reason rather than because any document requires it, and the right number for your work comes from your method and from how much of the budget repeatability turns out to carry.
- What degrees of freedom do I enter in the uncertainty budget?
- n − 1, for every Type A component. Leaving it blank is not neutral: a budget with no degrees of freedom entered treats the component as infinitely well known and returns k = 2, where ten readings actually earn 2.28 and four earn 2.87. The expanded uncertainty is then smaller than the data supports, which is exactly the direction that causes trouble on assessment. This tool reports ν alongside the value and includes it in the row it hands over.
- What does it mean if every reading is identical?
- Almost always that the indication is coarser than the variation, so the scatter is hidden rather than absent — a 0.1 °C display cannot show a 0.02 °C wobble. A Type A standard uncertainty of zero is not a defensible budget line. The resolution belongs in the budget as its own rectangular contribution, and the repeatability needs re-measuring with a finer indication, over a longer run, or at a point where the reading is not sitting squarely in the middle of a digit.
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Type A vs Type B uncertainty evaluation
The GUM splits components by how they were evaluated, not by what they are: Type A from statistical analysis of repeated observations, Type B from anything else. The split changes nothing about how they combine, and everything about their degrees of freedom.
Repeatability vs reproducibility
Repeatability holds everything constant except time; reproducibility changes operators, instruments, locations or days. The gap between them reveals hidden influences.
Building an uncertainty budget step by step
Write the measurement model, list every input, evaluate each as a standard uncertainty, apply sensitivity coefficients, combine in quadrature, expand with k. Eight steps, one table.
Effective degrees of freedom and when k = 2 is not enough
k = 2 assumes you know your uncertainty well. When a Type A component from a handful of readings dominates the budget, you do not, and the Welch-Satterthwaite calculation says by how much: three readings can push k to 4.53 and the expanded uncertainty to more than double.
Measurement uncertainty explained
What an uncertainty statement means, how it is built up, and why a coverage factor matters.
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