= Proficiency-test z-prime score
{title2=$z_i'$}
A \b[proficiency-test z-prime score] modifies a <proficiency-test z-score> to include uncertainty of the <assigned value in proficiency testing>:
$$
z_i'=\frac{x_i-x_{\mathrm{pt}}}{\sqrt{\sigma_{\mathrm{pt}}^2+u^2(x_{\mathrm{pt}})}}.
$$
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.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.4(b), printed pp. 17–18; §7.2.6, printed p. 18]{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 $z_i'=2.4$. 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 $r=u(x_{\mathrm{pt}})/\sigma_{\mathrm{pt}}$. Then
$$
z_i'=\frac{z_i}{\sqrt{1+r^2}}.
$$
When $r=0.3$, 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.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.6, uncertainty criterion; §7.2.4, cautions on z-prime interpretation]{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.
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