ISO 13528 z‑score calculator

Algorithm A is the robust statistics procedure in Annex C of ISO 13528. It estimates the centre and the spread of the participants’ results so that a few extreme values have little pull on either.

It starts from the median and the median absolute deviation. Then, pass after pass, any result further than 1.5 s* from x* is pulled back to that limit, and x* and s* are recalculated from the adjusted results. s* is their standard deviation times 1.134, which keeps it on the scale of an ordinary standard deviation. When neither figure moves any more, x* is the robust mean and s* the robust standard deviation.

Leave the assigned value and σpt blank and the calculator uses x* and s*.

Algorithm A needs enough results to settle, so, like the EqasOnline engine, the calculator uses it for 12 or more results. For 3 to 11 results it takes the median as the assigned value and MADe (1.483 × the median absolute deviation) as the spread, or nIQR when the MAD is zero. Below 3 results it gives no z-score.

A z-score says how many σpt a result lies from the assigned value: z = (x − x_pt) / σpt. ISO 13528 reads |z| ≤ 2 as satisfactory, 2 < |z| < 3 as questionable (a warning signal) and |z| ≥ 3 as unsatisfactory (an action signal). The calculator rounds z to two decimals and takes the band from the rounded value.

The assigned value is an estimate too. When it is the participants’ robust mean, its standard uncertainty is u(x_pt) = 1.25 × s* / √p for p results. ISO 13528 treats that as negligible when it is no more than 0.3 σpt, and z is then the score to use.

When it is larger, the standard points to z′, which adds the uncertainty of the assigned value to σpt: z′ = (x − x_pt) / √(σpt² + u(x_pt)²). The calculator makes this check and shows z′ beside z when it fails. An assigned value you enter yourself comes with no uncertainty here, so it gets neither the check nor z′.