= Proficiency-test z-score
{title2=$z_i=(x_i-x_{\mathrm{pt}})/\sigma_{\mathrm{pt}}$}
A \b[proficiency-test z-score] expresses a laboratory result's deviation from the <assigned value in proficiency testing> in units of the scheme's <standard deviation for proficiency assessment>:
$$
z_i=\frac{x_i-x_{\mathrm{pt}}}{\sigma_{\mathrm{pt}}},\qquad \sigma_{\mathrm{pt}}>0.
$$
The score is dimensionless because numerator and denominator describe the same kind of quantity in the same units. Its sign gives the direction of the deviation. The denominator is an assessment scale, not automatically the participant's own <measurement uncertainty>.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.4(a), printed p. 16, z score]{ref}
For an original resistance example, take $x_{\mathrm{pt}}=100\,\Omega$, $\sigma_{\mathrm{pt}}=2\,\Omega$ and a reported value $105\,\Omega$. The score is $2.5$. Under the guide's commonly used interpretation, $|z|\le2$ is satisfactory, $2<|z|<3$ raises a warning, and $|z|\ge3$ raises an action signal. Thus this example is questionable; a result of $96\,\Omega$ gives $z=-2$ and lies exactly at the satisfactory boundary, whereas $106\,\Omega$ gives $z=3$. The scheme must state its interpretation.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.4, printed p. 17, score interpretation]{ref}
The word z does not by itself make this a <hypothesis test> with a universally fixed error probability. If scores actually followed a standard <normal distribution>, the stated thresholds would have the corresponding normal-tail probabilities. That distribution is an additional model assumption; the numerical scoring rule alone does not establish it.
Uncertainty in the benchmark can change the comparison. The <proficiency-test z-prime score> uses
$$
z_i'=\frac{x_i-x_{\mathrm{pt}}}{\sqrt{\sigma_{\mathrm{pt}}^2+u^2(x_{\mathrm{pt}})}}.
$$
With the same $5\,\Omega$ deviation and $\sigma_{\mathrm{pt}}=2\,\Omega$, an assigned-value standard uncertainty $1.5\,\Omega$ changes the denominator to $2.5\,\Omega$ and the modified score to 2. A smaller modified score may reflect a less certain benchmark, not improved laboratory measurement.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.4(b), printed pp. 17–18, modified score and caution]{ref}
A score should therefore travel with the assigned-value method, assessment scale and relevant uncertainty information. Comparing scores from schemes with different criteria can confuse a change in the benchmark with a change in performance.
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