# Uncertainty of measurement: conventional mass of a weight

This article will discuss the calibration of a 1 kg weight of class $Latex formula$ by comparison with a 1 kg weight of class $Latex formula$ (weight classes are described in OIML R 111-1), using a balance of resolution 0.1 mg. In particular, it will focus on the estimation of uncertainty of measurement (UoM) for this calibration, for which the result is intended to be reported as a conventional mass (as discussed in OIML D 28).

The 80-page document R 111-1 describes many characteristics of weights (for example, construction) that are of limited relevance to the calibration laboratory: we will use mostly the sections on Maximum Permissible Errors (section 5, p 11-12), Density (sections 10, p 17-18 and B.7.9.3, p 58) and, especially, Calibration (Annex C, p 61-70). The 12-page document D 28, “Conventional value of the result of weighing in air”, presents a convenient summary of the information most relevant to calibration.

The conventional mass of a body is equal to the mass $Latex formula$ of a standard weight that balances this body under “conventional” conditions, namely, ambient temperature $Latex formula$ = 20 °C, air density $Latex formula$ and standard weight density $Latex formula$ [D 28 section 4, p 5]. The conditions have been chosen such that mass, $Latex formula$, and conventional mass, $Latex formula$, of a weight do not differ “much” [D 28 section 0, p 4]. However, we will see, when we try to achieve the required calibration uncertainty for a 1 kg $Latex formula$ weight, which is $Latex formula$ [R 111-1 section 5.2 and Table 1, p 11-12], or, equivalently, a relative standard uncertainty of $Latex formula$, that typical deviations of weight density from the conventional $Latex formula$ can cause significant differences between $Latex formula$ and $Latex formula$.

Note: The subscript “c” indicates “conventional”, “r” refers to the reference weight (in other words, the measurement standard) and “t” refers to the test weight (or Unit Under Test).

The relation between conventional mass and mass is given by $Latex formula$ [D 28 equation 1, p 6].

In our example, both the reference weight and the test weight are made of stainless steel. The mass of the reference weight is chosen to be $Latex formula$, with the largest calibration uncertainty allowed by R 111-1 for class $Latex formula$ weights. Relevant data are presented in the table below.

 Quantity Uncertainty Reference $Latex formula$ $Latex formula$ $Latex formula$ [R 111-1 Table B.7, p 58] $Latex formula$ $Latex formula$ [R 111-1 Table 1, p 12] $Latex formula$ $Latex formula$ [R 111-1 Table B.7, p 58] Air pressure $Latex formula$ $Latex formula$ Humidity $Latex formula$ $Latex formula$ Air temperature $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ (see below for calculation) [R 111-1 equation (E.3-1), p 76]

Notes:

1.The conventional mass $Latex formula$ is $Latex formula$ lower than $Latex formula$: this difference is large (compared to the $Latex formula$ relative uncertainty goal), although the assumed weight density is not far from ideal.

2. As the densities of the reference and test weights are not known, they are “assumed” according to R 111-1 Table B.7 (for stainless steel). Although they are assumed to be the same in most of the data analysis below, the possible difference between the densities must be taken into account in the uncertainty analysis. In particular, during calculation of the uncertainty in the air buoyancy correction $Latex formula$, the sensitivity coefficient of $Latex formula$ will be calculated using the worst-case scenario: as $Latex formula$ for 1 kg $Latex formula$ weights and $Latex formula$ for 1 kg $Latex formula$ weights [R 111-1 Table 5, p 17], the sensitivity coefficient $Latex formula$ will be used. (The calculation is performed below.)

3. The air pressure $Latex formula$, and, consequently, air density $Latex formula$, are unusually low, because the measurements were performed at an altitude of 2400 m. The even-more-approximate formula for air density, $Latex formula$ [R 111-1 equation (E.3-2), p 76], yields a similar air density, $Latex formula$.

The uncertainty in air density is calculated as follows: $Latex formula$ $Latex formula$. Sensitivity coefficients come from R 111-1 section C.6.3.6 (p 68), with $Latex formula$ and $Latex formula$ being converted from $Latex formula$ to $Latex formula$ and from fractional relative humidity to %rh, respectively. (Both change by a factor of 100, though in opposite directions.) It is clear that air pressure has the largest effect on air density, followed by temperature, with the effect of relative humidity being negligible.

The conventional masses of test and reference weights are related as follows: $Latex formula$ [D 28 equation (9), p 8],

where $Latex formula$ is the measured difference in apparent masses.

The air buoyancy correction $Latex formula$ [D 28 equation (10), p 8] is applied, because the air density differs from $Latex formula$ by more than 10 % [R 111-1 section 10.2.1, p 18].

To determine the uncertainty $Latex formula$, we’ll need the sensitivity coefficients $Latex formula$, $Latex formula$ and $Latex formula$.

Then, $Latex formula$, where $Latex formula$ (the uncertainty of the weighing process and the balance, combined) and $Latex formula$, in the notation of R 111-1 section C.6, p 66-70.

Expressed as a relative uncertainty, $Latex formula$.

From the table above, $Latex formula$.

From equation (C.6.3-1) of R 111-1 (p 67), $Latex formula$ $Latex formula$ $Latex formula$,

where $Latex formula$ is the air density during the last calibration of the reference weight.

Using $Latex formula$, $Latex formula$ $Latex formula$ + $Latex formula$. So, $Latex formula$, with the terms in $Latex formula$ and $Latex formula$ dominating over the term in $Latex formula$.

The term $Latex formula$ contains the uncertainty of the weighing process, $Latex formula$, which is an ESDM calculated as 0.01 mg, the sensitivity of the balance, $Latex formula$, and the display resolution of the balance, $Latex formula$. (Eccentric loading of the balance is included in $Latex formula$, and magnetism is negligible, according to p 69 of R 111-1.) In total, let’s suppose that $Latex formula$, so that $Latex formula$.

Finally, $Latex formula$. So, we have achieved our goal of a relative standard uncertainty smaller than $Latex formula$.

(Contact the author at lmc-solutions.co.za.)