Proficiency-test z-prime score ($z_i^{\prime }$)

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.