Proficiency-test z-score ($z_i=(x_i-x_{\mathrm{pt}})/{\sigma }_{\mathrm{pt}}$)

A 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: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.^[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, $|z|\le 2$ is satisfactory, $2<|z|<3$ raises a warning, and $|z|\ge 3$ 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.^[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 usesWith 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.^[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.

A proficiency-test z-prime score modifies a proficiency-test z-score to include uncertainty of the assigned value in proficiency testing:Here ${\sigma }_{\mathrm{pt}}>0$ is the standard deviation for proficiency assessment, and $u(x_{\mathrm{pt}})$ is the assigned value's standard uncertainty. The latter describes uncertainty in the comparator, not the participant's reported uncertainty.^[ref]

An original example gives $x_i-x_{\mathrm{pt}}=3.6$ units, ${\sigma }_{\mathrm{pt}}=1.2$ units and $u(x_{\mathrm{pt}})=0.9$ units. The ordinary score is 3; the modified denominator is $\sqrt{1.44+0.81}=1.5$ units and $z_i^{\prime }=2.4$. Under the guide's common interpretation, the first reaches the action boundary and the second lies in the warning interval. The change reflects benchmark uncertainty, not a change in the measured result.

The effect can be separated algebraically. Put $r=u(x_{\mathrm{pt}})/{\sigma }_{\mathrm{pt}}$. ThenWhen $r=0.3$, the magnitude is about 0.958 times the ordinary score. The guide treats a contribution below $0.3{\sigma }_{\mathrm{pt}}$ as negligible for ordinary score interpretation. Conversely, when $r$ becomes very large, the modified score tends toward zero for any fixed deviation. A weak benchmark can therefore make a discrepancy difficult to interpret; it cannot make a measurement intrinsically accurate.^[ref]

Do not replace the assessment scale with the laboratory's uncertainty and still call the result this score. An uncertainty-normalized compatibility calculation asks a different question. Likewise, if the assigned value uses the participant's own result, participant influence on a consensus assigned value creates dependence that must be understood before interpreting the score as an independent check.

The scheme should state its score and benchmark treatment in advance. When assigned-value uncertainty is too large, options include improving the reference assignment or explicitly reporting its limitations. Enlarging a denominator after seeing an inconvenient result does not supply new measurement evidence.