= Intermediate measurement precision
<Intermediate measurement precision> is <measurement precision> under declared <intermediate precision conditions of measurement>. The <measurement procedure> and location remain the same, measurements extend over a longer period, and specified changes can include operators, <measuring systems> or <measurement calibration>. It lies between a tightly restricted <measurement repeatability> study and a comparison across locations for <measurement reproducibility>.<International vocabulary of metrology by JCGM>[VIM3 entries 2.22–2.23, definitions and entry 2.22 notes 1–2]{ref}
For a concrete study, one laboratory could measure the same stable reference block each working day with a <differential length comparator>, taking four complete placements per day. Keep the procedure fixed, record temperature and operator, and retain the day labels. The labels let the study distinguish variation between days from rapid reading-to-reading variation.
An original <measurement model> makes that distinction explicit:
$$
X_{dj}=\mu+D_d+\varepsilon_{dj},\qquad
\operatorname{Var}(D_d)=\tau^2,\qquad
\operatorname{Var}(\varepsilon_{dj})=\sigma^2.
$$
Here $D_d$ is a shared daily offset, while $\varepsilon_{dj}$ varies between placements. Assume zero-mean <independent random variables>, independent also between days. A single measurement then has <standard deviation> $\sqrt{\tau^2+\sigma^2}$. For stipulated $\tau=0.30\,\mathrm{\mu m}$ and $\sigma=0.10\,\mathrm{\mu m}$, this is $0.316\,\mathrm{\mu m}$, even though within-day scatter reflects only $0.10\,\mathrm{\mu m}$.
For four readings from one day, the daily <arithmetic mean> has
$$
\operatorname{Var}(\bar X_d)=\tau^2+\frac{\sigma^2}{4},\qquad
\operatorname{SD}(\bar X_d)=\sqrt{0.0925}\,\mathrm{\mu m}\approx0.304\,\mathrm{\mu m}.
$$
Averaging four such means from independent days reduces that modelled <standard deviation> to $0.152\,\mathrm{\mu m}$. Four same-day readings do not average away their shared daily offset. The numbers illustrate the model, not measured comparator performance. A real study must estimate its components from enough days and investigate trends: deterministic <instrumental drift> is not automatically an independent daily offset.
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