Understanding Measurement Uncertainty in Weighing

A measurement of any kind is affected by the errors and uncertainties that exist in that measurement process. During calibration, the performance of an instrument is assessed and its limitations originating from errors and uncertainties are made evident; this is the essential foundation for achieving accurate measurement results.

What is calibration?
Calibration is more than just comparing a “standard” number with a displayed value on the balance. There is a calibration process that relates to the combination of uncertainties achieved when a balance weigh cell is deviated from its rest position. Calibration establishes the relationship between the displayed value and a standard or accepted true value. The values must fall within an assigned measurement uncertainty range (1, 2).

A calibration is meaningless without the measurement uncertainty for that specific instrument. Only once the instrument uncertainty has been established, and providing that the number is within the required process tolerance, can it be classed as an accurate calibration.
Measurement uncertainty is derived from known scientific methodology. It is not something that can be decided by a company or committee. There is an internationally correct and recognized way of calculating measurement uncertainty, described in the Guide to the Expression of Uncertainty of Measurement (GUM) (3).

It is crucial that the weights used in routine testing are certified as traceable and that the certification is current. If the weight is out of certification, it is no longer guaranteed to give the maximum permissible error (MPE) that is expected.

Why is measurement uncertainty important?
A modern electronic weigh cell works differently than a traditional mechanical balance. In an electronic weigh cell, the relationship between what is displayed on the screen and the errors involved is not linear. The current approach to metrology, however, is often based on history and familiarity, rather than a scientific study of the process. Traditional methods of balance testing need to be updated and replaced with scientifically meaningful tests, which include an estimation of measurement uncertainty. A knowledge of measurement uncertainty can be used to develop a streamlined, robust, and meaningful testing regime that meets scientific requirements but doesn’t take up several hours every week.

What contributes to total measurement uncertainty?
The total measurement uncertainty is shown in Figure 1by the black line (U_tot). The significant factors that contribute to measurement uncertainty across the weighing range (4) are described as follows.

Repeatability. Repeatability, also known as precision, is the largest single contributor to the total measurement uncertainty. At the lower end of the weighing range, the total measurement uncertainty detected is almost entirely due to the repeatability, shown by the yellow line (U_RP) in Figure 1. The magnitude of the repeatability uncertainty completely obscures the chance to measure any other components in this range.

Figure 1: Individual uncertainty contributions to relative weighing uncertainty of an analytical balance with an expansion factor of k=2. U_tot is the total measurement uncertainty; U_RP is repeatability; U_EC is eccentricity; U_NL is nonlinearity; and U_SE is sensitivity. Repeatability is the dominant error at lower sample mass, and sensitivity and eccentricity are the dominant errors at higher sample mass.

Eccentricity. The eccentricity or cornerload error is the error associated with not placing the weight in the center of the weigh pan. The eccentricity is only significant, and therefore detectable, at the higher end of the weighing range, as shown by the green line (U_EC) in Figure 1.

Nonlinearity. Nonlinearity is the error due to the nonlinear behavior of the balance upon increasing the load on the weighing pan. It is not a dominant influence at any point on the weighing range. At the low end of the weighing range, the nonlinearity is superimposed by repeatability uncertainty. At the high end, it is dominated by sensitivity. In fact, at no point on the weighing range does it account for more than 0.1% of the total measurement uncertainty that can be detected. Nonlinearity is significant for the manufacturer because it provides information about the mechanical operation of the balance. For people who have worked for a long time in the pharmaceutical industry, where it may be normal to carry out linearity checks daily, it may come as a surprise to learn that these tests are not as significant as previously thought.

Sensitivity. Sensitivity (i.e., systematic deviation) is the key parameter (besides repeatability) that should be assessed periodically. At the higher end of the weighing range, the sensitivity and eccentricity become the dominant contributors to the overall measurement uncertainty. The high end of the weighing range is thus the ideal place to carry out these routine tests. Figure 1 clearly shows that the sensitivity component of uncertainty (pink line, U_SE) is a completely horizontal line, which means it can be safely assumed that if the sensitivity measurement is performed at the high end of the weighing range, the result represents the whole of the nominal capacity of the balance, providing that the weigh cell is working correctly. Incidentally, if the weigh cell is not working correctly, a test carried out at the high end of the weighing range (as near to 100% of the nominal capacity of the balance) on a daily or regular basis will usually indicate any problems with the weigh cell immediately.

How to measure uncertainty
A manufacturer or service provider will test the four significant components of measurement uncertainty when performing a calibration. These measurements are reflected on a calibration certificate. It is also expected that the balance user or internal metrology department will measure some of these contributors during routine balance testing operations.

Good Weighing Practice (GWP), Mettler Toledo’s science-based standard for lifecycle management of weighing instruments, provides a strategy for reducing measurement errors and ensuring accurate weighing results. Based primarily on the user’s weighing requirements and prevailing weighing risks, GWP guides users on how to optimize routine testing procedures and how to avoid unnecessary or erroneous testing (4). GWP specifies which of the following tests should be carried out during routine testing.

Repeatability (i.e., precision) test. Perform (typically) 10 replicate weighings under the same conditions, using the same methodology, to give a statistically significant weighing sequence, according to the GUM (3). Calculate the standard deviation (s), which is the repeatability of the instrument.

Eccentricity test (i.e., cornerload error). Place a weight in the center of the weigh pan and zero the balance. Move the weight out to the corners of the weigh pan and take the readings. With a round weighing pan, the protocol is no different. The aim is to compare the difference between the extremities of the weigh pan to the center. Depending on the weighing process, this test can be omitted by the user.

Nonlinearity (i.e., error of indication) test. Nonlinearity is not very significant at any point in the weighing range, so it is perfectly valid to omit this from the user routine testing regime. This test can be performed by the external service technician during the annual calibration or preventive maintenance visit. It is acceptable to perform a nonlinearity test only one or two times per year, even for a heavily used microbalance or analytical balance. Although it doesn’t need to be measured daily or frequently by the user, it does need to be included in the measurement uncertainty calculation, as per the GUM (3).

Sensitivity test. This test is sometimes referred to as the highest “error of indication” test point. The aim is to measure the error of indication by placing a single weight on the balance and comparing it with a standard value. For accuracy, this test must be done at or close to the top of the nominal working range of the balance. Below that, increasingly lower uncertainty due to sensitivity can actually be detected. At the very low end, with very small weights, there is almost no sensitivity uncertainty detectable at all. At this point, the repeatability (i.e., precision of the balance) becomes the completely dominant factor.

Although there are other contributors to uncertainty, such as external influences, the advantage of modern electronic weigh cell technology is that most of those uncertainties are included in the repeatability of the balance.

Combining uncertainty components
Once the four uncertainty components have been determined during the full calibration by the manufacturer or service provider, the correct way to calculate the total measurement uncertainty (i.e., “given uncertainty budget”) for any balance or scale, according to metrology guidelines, is a two-step calculation. First combine the individual components to calculate the so-called standard uncertainty (u) for the instrument by using a root-sum-square method, as shown in the simplified model in Equation 1:

where u_RP, u_EC, u_NL,  and u_SE are the uncertainties due, respectively, to repeatability, eccentricity, nonlinearity, and sensitivity.

Next, expand the uncertainty using standard scientific practices. For an expansion factor of k=2, which can be quite frequently applied, multiply the equation by 2 to give a 95.5% confidence level in the result (3).

Which test weights should be used?
According to GWP, only two weights are required for a user to perform regular routine testing of their balance (4). The larger weight at 100% or close to the nominal capacity of the balance is used to perform the sensitivity test. The smaller weight (5% of the nominal weighing capacity) is used to perform the repeatability test.

Linearity testing does require more than two weights. If a service provider performs the linearity testing as part of the annual calibration or preventative maintenance visit, however, then there is no scientific requirement for the user to check linearity more frequently.

Summary
According to the laws of metrology, there is no traceable calibration without a statement of measurement uncertainty. Traditional balance testing should be replaced with scientifically meaningful test points. Testing at multiple points on the weighing range every day can take a lot of time, effort, and resources. In addition, the test points may not even be relevant if the metrology behind the testing is not considered. An understanding of the basic principles of balance and scale properties, such as measurement uncertainty, enables the user to achieve a qualified weighing process.

References
1. Mettler Toledo, “Calibration: What is it?”, white paper no. 30260955 (May 2015),  accessed Feb. 29, 2016.
2. K. Fritsch and I. Ciesniewski, NCSLI Measure: J. Meas. Sci. 10 (3) 56-65 (September 2015).
3. BIPM, Evaluation of measurement data-Guide to the expression of uncertainty in measurement, JCGM 100:2008, GUM 1995 with minor correction, first ed. (Sèvres, France, September 2008).
4. K. Fritsch, NCSLI Measure: J. Meas. Sci. 8 (3) 60-69 (September 2013). 

About the authors
Ian Ciesniewski, PhD, is technical director, and Joanne Ratcliff, PhD, is communications project manager, both at laboratory Weighing, Mettler Toledo.

Article DetailsPharmaceutical Technology
Vol. 40, No. 4
Pages: 78–82

Citation:
When referring to this article, please cite it as I. Ciesniewski and J. Ratcliff, "Understanding Measurement Uncertainty in Weighing," Pharmaceutical Technology 40 (4) 2016.