En number calculator for proficiency testing

How the calculation works

1. 01En = (x_lab − x_ref) / √(U_lab² + U_ref²), the performance statistic ISO 13528 gives for comparisons in which every participant reports its own expanded uncertainty — the normal case in calibration. |En| ≤ 1 is the published satisfactory criterion.
2. 02ISO/IEC 17043 is the standard the proficiency testing provider itself is accredited against, and it is the provider's report that decides the outcome. This recomputes the figure so you can check it and see how much margin there was.
3. 03Both expanded uncertainties are divided by their own stated coverage factor and re-expressed at k = 2 before being combined, because En is only meaningful when the two cover the same probability.
4. 04z = (x − X)/σ̂ is reported only when you supply σ̂, the standard deviation for proficiency assessment, from the scheme report. ISO 13528 clause 9.4 describes |z| = 2 as a warning signal and |z| = 3 as an action signal.
5. 05z′ = (x − X)/√(σ̂² + u(X)²), from ISO 13528 clause 9.5, is reported alongside it, taking the standard uncertainty of the assigned value into account.

Limitations

• En is only as good as the uncertainty you claimed. An overstated budget buys a pass and an understated one manufactures a failure, so the tool reports the smallest uncertainty that would still have passed, and how much of the denominator your own figure supplied, next to the En itself.
• It states no criterion beyond the published |En| ≤ 1 and the ISO 13528 z-signals. What your scheme calls satisfactory, and what it expects after a signal, comes from the scheme and from your own quality system.
• It cannot work out σ̂ for you. The standard deviation for proficiency assessment is the scheme's property, derived from the participants or set by the provider, and nothing in one laboratory's own figures implies it — so without it there is no z-score.
• One comparison is one data point. A satisfactory En is evidence rather than a validated capability, and a single failure is a trigger to investigate rather than a verdict on the laboratory.
• The scheme's report is the record. Where this and the report disagree, the report is what an assessor reads, and the disagreement is itself the thing to ask the provider about.

Frequently asked questions

How do I calculate an En number?

Take the difference between your result and the assigned value, then divide it by the square root of the sum of the squares of the two expanded uncertainties: En = (x_lab − x_ref) / √(U_lab² + U_ref²). Both uncertainties go in at the same coverage probability, normally k = 2, so one quoted at k = 1 has to be doubled first. The sign tells you which way your bias runs and is worth keeping — a run of comparisons all biased the same way says something a set of absolute values hides.

What does an En number of less than 1 mean?

That the difference between your result and the assigned value is covered by the uncertainties the two of you claimed, which ISO 13528 treats as satisfactory. Note what it does not say: it does not say the difference is small, and it does not say you measure well. A result 2 °C from the assigned value with a claimed uncertainty of 3 °C passes comfortably, and the honest reading of that pass is that the laboratory said it measured badly and was right.

What should I do if my En number fails?

Treat it as evidence that the uncertainty on your certificates is too small for the bias observed, not as a request to re-measure the artefact. That distinction matters because it reaches backwards: results already issued to customers carried the same budget, so a failing En normally becomes a nonconforming work and impact assessment case under ISO/IEC 17025 clause 7.10, alongside the technical investigation. Re-measuring and getting a better answer the second time does not close it.

What is the difference between an En number and a z-score?

They test different things against different yardsticks. En measures your difference from the assigned value against the uncertainty you claimed for yourself, so it asks whether your stated uncertainty is honest. A z-score measures the same difference against σ̂, the spread the scheme expects of competent participants, so it asks whether you agree with your peers. They can disagree, and the disagreement is informative: a satisfactory En with a z-score raising an action signal points at a budget large enough to absorb a real bias, and the bias is then the finding.

Can a laboratory pass a proficiency test by claiming a larger uncertainty?

Arithmetically, yes, and that is the known weakness of En. Inflating U_lab enlarges the denominator and shrinks En, so a generous budget turns a failing result into a passing one. It is also self-defeating, because the same inflated uncertainty goes on every certificate you issue, widens every guard band your customers have to apply, and will be examined against your stated scope. This calculator reports the smallest uncertainty that would still have passed, and what share of the denominator your own figure supplied, so a pass of that kind is visible rather than hidden inside a single number.

Do both uncertainties have to be at the same coverage factor?

Yes, and this is a routine way to get En wrong. En combines two expanded uncertainties, so they have to cover the same probability before they are added in quadrature; mixing a k = 1 figure with a k = 2 figure inflates En by almost a factor of two and can fail a comparison that actually passed. Check the coverage factor on both the certificate and the scheme report, and enter each with its own k so the conversion is done once and visibly.