= Manometer height-to-pressure calibration
{title2=$d(\Delta p)/d(\Delta h)=\rho g$}
The ideal <measurement calibration> is a straight line through the equal-pressure zero. For the stipulated <water> <mass density> and <gravitational acceleration> above, its slope is $9.81\,\mathrm{Pa}$ per vertical millimetre. A lower <liquid> <mass density> gives a larger <length> change for the same <pressure>, improving the height-reading contribution to <measurement resolution> but requiring more vertical space for a given <pressure> range. The <pressure> scale depends on the actual <mass density>, which depends on <temperature>; $1000\,\mathrm{kg\,m^{-3}}$ here is a chosen model value, not an exact property of all <water> samples.
\Image[/manometer-height-pressure-resolution.png]
{title=Calculated <water>-column <pressure> scale and a two-meniscus rounding model}
{description=The left panel is the theoretical <measurement calibration> for $\rho=1000\,\mathrm{kg\,m^{-3}}$ and $g=9.81\,\mathrm{m\,s^{-2}}$. The right panel models two <liquid meniscus> heights read separately to the nearest millimetre: each rounding <measurement error> is assumed independently uniform between $-0.5$ and $0.5\,\mathrm{mm}$. Their difference gives a triangular <probability density function> for the inferred <pressure> error. The bounds and <standard uncertainty> describe this constructed model, not observed instrument data.}
Suppose each <liquid meniscus> height is rounded to the nearest scale division $q=1\,\mathrm{mm}$. Each height <measurement error> lies within $q/2$, so the error in their difference lies within $q$. The resulting <pressure> error is bounded by $\rho gq=9.81\,\mathrm{Pa}$. To quantify a <standard uncertainty>, make the additional model assumption that the two rounding errors are <independent random variables>, each having a <continuous uniform distribution> on $[-q/2,q/2]$. Each then has <variance> $q^2/12$; adding the two <variances> gives
$$
u(\Delta h)=\frac{q}{\sqrt6},\qquad
u(\Delta p)=\frac{\rho gq}{\sqrt6}=4.0049\,\mathrm{Pa}.
$$
Here $u$ denotes <standard uncertainty>. The $9.81\,\mathrm{Pa}$ worst-case bound and $4.0049\,\mathrm{Pa}$ <standard uncertainty> answer different questions. At $588.6\,\mathrm{Pa}$ they are respectively $1.67\%$ and $0.680\%$ of the reading. Shared reading bias or a different reading method can violate the assumed <independence>.
If the <mass density>, <gravitational acceleration> and height difference also have small independent <standard uncertainties>, a first-order estimate follows from the <partial derivatives> of $\Delta p=\rho g\Delta h$:
$$
u^2(\Delta p)\simeq(g\Delta h)^2u^2(\rho)+(\rho\Delta h)^2u^2(g)+(\rho g)^2u^2(\Delta h).
$$
This displays why a finer scale does not remove uncertainty in <mass density> or <gravitational acceleration>. A <pressure transducer> compared against the column needs its own <differential-pressure calibration> to establish an electrical response.
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