PT formula sheet

The scores, robust statistics and acceptance criteria used in proficiency testing, on one page. From EqasOnline Academy, academy.eqasonline.com.

z score

z = (x − x_pt) / σ_pt

|z| ≤ 2.0 satisfactory · 2.0 < |z| < 3.0 questionable · |z| ≥ 3.0 unsatisfactory

z′ score

z′ = (x − x_pt) / √(σ_pt² + u²(x_pt))

Use when u(x_pt) > 0.3 σ_pt. Same limits as z.

ζ (zeta) score

ζ = (x − x_pt) / √(u²(x) + u²(x_pt))

Standard uncertainties. Same limits as z.

E_n number

E_n = (x − x_pt) / √(U²(x) + U²(x_pt))

Expanded uncertainties (usually k = 2). |E_n| ≤ 1.0 satisfactory.

Percent difference

D% = 100 × (x − x_pt) / x_pt

Judged against the limit in the scheme plan.

Expanded and standard uncertainty

U = k × u · u = U / k

Read k from the certificate or report; k = 2 is the usual value.

Algorithm A (robust mean and SD)

start: x* = median, s* = 1.483 × MAD

Pull values outside x* ± 1.5 s* to the limits; x* = mean, s* = 1.134 × SD of the adjusted values; repeat until stable.

Robust spread

MADe = 1.483 × MAD · nIQR = 0.7413 × IQR

Both estimate the SD of normally distributed data while ignoring outliers.

Uncertainty of a consensus value

u(x_pt) = 1.25 × s* / √p

p results. Negligible when u(x_pt) ≤ 0.3 σ_pt.

Homogeneity

s_s ≤ 0.3 σ_pt

g ≥ 10 items in duplicate; s_w = √(Σw² / 2g); s_s = √(max(0, s_x² − s_w²/2))

Stability

|ȳ₁ − ȳ₂| ≤ 0.3 σ_pt

Items measured after the round (or after transport) against the start.

σ_pt from a precision experiment

σ_pt = √(s_R² − s_r²(1 − 1/m))

m replicates per participant; s_R reproducibility, s_r repeatability SD.

Your scheme's plan decides which score, limits and rounding apply. Learn the methods with worked examples in the free course, or try them in the free tools.