= Measurement reproducibility
<Measurement reproducibility> is <measurement precision> under specified <reproducibility conditions of measurement>. These include changed locations, operators and <measuring systems>; the <measurement procedures> may also differ. The comparison must state what changed and what remained common.<International vocabulary of metrology by JCGM>[VIM3 entries 2.24–2.25, definitions and entry 2.24 notes 1–2]{ref} This is a metrological use of reproducibility, concerned with agreement of measured values under declared conditions.
An original comparison sends a stable reference block to three laboratories. Each laboratory uses its own <differential length comparator> and operator, measures the same specified <measurand>, and reports three length errors relative to the common reference, in <micrometres>:
$$
\begin{array}{c|rrr|r}
\text{laboratory}&x_1&x_2&x_3&\text{mean error}\\
A&-1&0&1&0\\
B&3&4&5&4\\
C&-3&-2&-1&-2
\end{array}
$$
Each laboratory has <sample standard deviation> $1\,\mathrm{\mu m}$, yet their <arithmetic means> differ by as much as $6\,\mathrm{\mu m}$. Relative to the grand mean $2/3\,\mathrm{\mu m}$, the total squared deviation separates into within-laboratory and between-laboratory terms:
$$
\sum_{i=1}^{3}\sum_{j=1}^{3}(x_{ij}-\bar x)^2
=\sum_{i,j}(x_{ij}-\bar x_i)^2+3\sum_{i=1}^{3}(\bar x_i-\bar x)^2
=6+56=62\,\mathrm{\mu m^2}.
$$
Consequently the nine chosen values have pooled <sample standard deviation> $\sqrt{62/8}\approx2.78\,\mathrm{\mu m}$. This is a descriptive statistic for these observations, not a universal estimator prescribed by the definition of <measurement reproducibility>. Reporting only the three within-laboratory spreads would conceal the disagreement.
The physical comparison must also monitor transport damage, reference stability and temperature corrections. A changing reference block could imitate a laboratory difference. <Measurement reproducibility> alone does not establish <measurement trueness>: all laboratories could share the same erroneous reference value or correction. The changed apparatus and conditions belong alongside the reported statistic.
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