A Question of Balance? Part III: Weighing Isn’t an Uncertain Process, Is It?

A new version of United States Pharmacopeia (USP) <41> Balances becomes effective in February 2026. This installment of “Questions of Quality” explores the impact of the changes this brings, along with developments in the Japanese and European Pharmacopoeias.

Observant readers of this column may ask the question, what happened to Parts I and II? Mature readers will know that they were published in 2004 and 2006 (1,2). These looked at the principles and practice of weighing on an analytical balance. In analytical procedures such as chromatography, the weighing of standards and samples are critical operations to ensure the validity and quality of a reportable result. Weighing is such a familiar laboratory operation, usually digitally displayed and captured, that we tend to intuitively believe that the numbers on our balances are error-free, or at least have such a small uncertainty that any errors will be negligible, which may be but is not always true.

What has happened in the intervening years from a regulatory and technical perspective? The answer is a lot, and the changes will be discussed in this article. There is also supplementary material with more details for readers at the end of this article.

United States Pharmacopeia(USP) <41> Balance has been updated and is accompanied by USP <1251>, a best practice informational chapter on weighing on an analytical balance. While this column will examine the regulatory aspects of analytical balances, the principles described here apply to all readers.

The purpose of this article is to:

1. Look at the uncertainties generated by the load cell in terms of their source, and one best practice to avoid errors in weighing by differences caused by buoyancy and drift.
2. Review the various regulatory requirements contained in the three main Pharmacopeias.

The Analytical Balance

Modern electronic balances rely on a load cell. As force is applied to the load cell or transducer by calibrated masses, the response is measured. The responses from the load cell are transformed by a mass calibration function supported by digital averaging, smoothing, and rounding firmware, which leads to the displayed mass on the balance.

To ensure data integrity, the minimum requirement is to attach a printer that can record the whole weighing process (for example, balance check, vessel weight, tare, sample weight, and vessel reweigh) along with the date and time and who did the weighing, as shown in Figure 1. A much better option is to automate the process by connecting the balance to an informatics application, such as an instrument data system (IDS) or laboratory information management system (LIMS) for all data and metadata capture.

In addition, modern balances have accounts for users and administrators that should be set up so that work can be traced to named individuals. It is important that administrator accounts that avoid conflicts of interest are established for managing users and adjusting the balance time and date when necessary.

Certain Uncertainties

The sources of uncertainties in the metrology of weighing coming from the balances are well known and form the basis for instrument qualification and calibration activities during routine preventative maintenance. Some include:

1. Readability of the display, u_Dis
2. Repeatability of measurement, u_Rep
3. Deviation from linearity over the operational range of the balance, u_Lin
4. Eccentricity, u_Ecc

The maximum values for these four uncertainties are usually specified by the balance manufacturer. Uncertainties 2 to 4 are expressed as standard deviations.

Therefore, we can estimate the overall intrinsic performance measurement error of the balance by combining them in an error budget in accordance with the measurement uncertainty principles (3).

If we assume some typical manufacturer values (4), u_Dis 0.0001g, u_Rep 0.0001g, u_Lin 0.0002g, and u_Ecc 0.0003g, using equation 1, we get a value for u_balance of 0.000387g

u_balance represents the performance capability expected of the balance itself. It does not consider other factors because we need to consider two other aspects in the dynamic weighing process, namely buoyancy corrections and drift. The former is difficult to assess routinely and requires considerable and complex calculations. See references 4, 5, and 6 for more technical details. However, there are approximate equations that will suffice for most practical purposes. There is also a simple method to compensate for drift (7). A NIST technical note 1900 (8) makes use of this method and avoids buoyancy corrections, providing data for an example calculation using Monte Carlo simulation, which is available as an electronic file associated with this article.

Best Practice “Weighing by Difference” Method

Buoyancy can introduce a negative systematic error during weighing. The balance itself can introduce both positive and negative bias owing to drift in the electronics of the load cell.

Weighing by difference is well used within the laboratory but is susceptible to both buoyancy and drift effects.

In general, it involves placing a container on the balance pan, noting the reading, and then adding a substance to it. The final balance reading is noted and the difference between the two readings is taken to be the amount of material in the container. This approach is fine for relatively low accuracy requirements but is not ideal for more demanding measurements.

This type of measurement is susceptible to problems with temporal balance drift (which is generally more of a problem with modern electronic analytical balances than was the case with mechanical balances)(7)

Davison et al. (7) proposed a simple four-step weighing-by-difference procedure, whereby, in addition to using a single sample container, they weighed a second reference container before and after the sample container and the container filled with the sample. Their method also compensates for buoyancy effects.

1. The reference container is placed on the tared balance pan and the reading is recorded (w₁).
2. The empty container (to be loaded with the test substance) is placed on the balance and the reading is recorded (w₂).
3. The container is filled with the test substance and the balance reading is recorded (w₃).
4. The reference container is put back on the pan and the reading is recorded (w₄).

Using this method, the weight of the sample in air is calculated, compensating for drift:

This method was modified and extended by Possolo (8) in a little-known NIST technical note in 2015 on measurement uncertainty in example E1.

He described adding a mass in the reference container to approximate half the expected value of the sample weight as the drift and buoyancy compensation. The idea is that any calibration error is minimized at the centroid of the operational range. He concluded that:

Since the containers are weighed with tightly fitting lids on, and assuming that they all displace essentially the same volume of air and that the density of air remained essentially constant in the course of the weighings, there is no need for buoyancy corrections.

Neither the weight of the reference container nor of the mass itself are required to be known because they are only used to find the difference to estimate the drift.

Requirements for Weighing in the Regulated Environment

The major source of these requirements is to be found predominantly in the pharmacopeias of Europe (European Pharmacopoeia), the USA (United States Pharmacopeia), and Japan (Japanese Pharmacopoeia). Fortunately, whilst not officially harmonized, there is not a large amount of difference between them. The European Pharmacopoeia has only one General Chapter, the USP has two, and the JP XVIII has four.

The contents of these General Chapters are summarized in Table 1.

USP <41> and <1251>

In the USP, there are two chapters on the topic, one for the mandatory performance requirements, USP <41> Balances, and one on guidance for best practice, USP <1251> Weighing on an Analytical Balance. USP <41> defines only repeatability (for the minimum weight calculation) and accuracy requirements, as shown in Figure 2. It does not reference external sources other than the American Society for Testing and Material E617 (ASTM) and International Organization of Legal Metrology (OIML) R111 for traceable masses. Figure 1 shows the relationships between <41> and <1251>.

The key point for USP <41> is clarification of the minimum weight requirements.

The repeatability measurement, s, establishes the minimum weight that may be weighed on the balance in conformance with the 0.10% limit. The minimum weight, which shall not include the weight of the tare vessel, is described by the following equation: m_min = 2000s

Note that the minimum weight is a calculated value based on the performance of the balance at the time of the repeatability measurement and is not constant over time. In contrast, the smallest net weight is a user-defined requirement, usually does not change over time, and shall not be smaller than the minimum weight.

Further clarification of the minimum weight requirement is given in <1251>

The minimum weight describes the lower limit of the balance below which the weighing tolerance is not met. While the concept of minimum weight applies to all balances, the equation above is only applicable to analytical balances because their performance at the lower end of the measurement range is determined by the repeatability s.

Factors that can influence repeatability while the balance is in use include:

• The performance of the balance and thus the minimum weight can vary over time because of changing environmental conditions.
• Different operators may weigh differently on the balance—i.e., the minimum weight determined by different operators may be different.
• The standard deviation of a finite number of replicate weighings is only an estimation of the true standard deviation, which is unknown.
• The determination of the minimum weight with a test weight may not be completely representative of the weighing application.
• The tare vessel also may influence minimum weight because of the interaction of the environment with the surface of the tare vessel.

It is therefore important to include minimum weight in ongoing and regular performance checks.

Japanese Pharmacopoeia (JP)

The JP has four chapters concerning balances. <G1-6-182> Concept of Weighing, <G1-7-182> Calibration, Inspection, and Weight of a weighing instrument (balance), and <G1-8-182> Installation Environment and Basic Handling Method of a Balance, and Precautions for Weighing together with 9.62 Measuring Instruments, Appliances. There is not such a strict demarcation between the chapters as in the USP.

However, <G1-6-182> says:

Measuring instrument, Appliances <9.62> in General Tests of the JP 18, it is required that balances and weights in the JP 18 shall be calibrated, ensuring traceability to the international system of units (SI).

Based on the results obtained from the requirements for repeatability shown in the section on balances and weights of the general test Measuring instruments, Appliance <9.62> of the JP 18, the minimum weight of the balance at that time is estimated.

and

Evaluation of accuracy (trueness) in a normal environment includes the three errors of sensitivity, linearity, and eccentricity, and the acceptance criterion, 0.10%, according to the error propagation rule (square root value of the sum of squares) satisfies the following equation.

which is compatible with USP and minimum weight requirements. In <G1-7-182>, it lists and defines the components required, which are presented in Table 2.

Uniquely, the JP 18 (2021) references old ISO source documents of the VIM and GUM (9,10), as well as an ISPE ISO based publication in 2009 (11).

European Pharmacopoeia

The European Pharmacopoeia has, since 2022, had a single General Chapter 2.1.7 Balances for Analytical Purposes covering both performance standards and best practices, and is compatible with the USP and JP. See Table 1 for more details.

A white paper from Mettler Toledo in 2021 discusses 2.1.7 in some detail (12).

Summary

Much has changed in the past 21 years since our original article (1), but weighing remains at the heart of an analytical laboratory’s operations. Based upon our auditing experiences over that period, it is regrettable that the knowledge concerning such a critical activity as weighing and sources of error has not improved much, if at all.

The apparent constancy of a digital display has some laboratory personnel convinced of the mantra that weighing is an exact science.

Hopefully, the updated pharmacopoeial General Chapters will facilitate improvements in laboratory knowledge and best practices.

References

(1) Burgess, C.; McDowall, R. D. A Question of Balance? Part 1: Principles. LCGC Europe 2004, 1 (7), 390–395.

(2) Burgess, C.; McDowall, R. D. A Question of Balance? Part 2: Putting Principles into Practice. LCGC Europe 2006, 3 (9), 152–159.

(3) Burgess, C. The Basics of Measurement Uncertainty in Pharmaceutical Analysis. Pharm. Technol. 2013, 37 (9), 62–64, 73.

(4) Eurachem/CITAC Guide: Quantifying Uncertainty in Analytical Measurement, 3rd ed.; Ellison, S. L. R.; Williams, A., Eds.; Eurachem: 2012; ISBN 978-0-948926-30 https://www.eurachem.org/index.php/ publications/guides/quam (accessed 2026-01-14).

(5) UKAS. LAB 14: Guidance on the Calibration of Weighing Machines Used in Testing and Calibration Laboratories, 7th ed.; United Kingdom Accreditation Service, 2022. https://www.ukas.com/wp-content/uploads/schedule_uploads/759162/LAB-14-Guidance-on-the-calibration-of-weighing-machines.pdf (accessed 2026-01-14).

(6) EURAMET. Calibration Guide No. 18: Guidelines on the Calibration of Non-Automatic Weighing Instruments, Version 4.0; EURAMET, 2015.
https://www.euramet.org/Media/docs/Publications/calguides/I-CAL-GUI-018_Calibration_Guide_No._18_web.pdf (accessed 2026-01-14).

(7) Davidson, S.; Perkin, M.; Buckley, M. The Measurement of Mass and Weight; NPL Measurement Good Practice Guide No. 71; National Physical Laboratory, 2004. https://www.npl.co.uk/special-pages/guides/gpg71_mass.aspx (accessed 2026-01-14).

(8) Possolo, A. Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results: Example E1, Weighing; NIST Technical Note 1900; National Institute of Standards and Technology, 2015.

(9) ISO/IEC. International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM); ISO/IEC Guide 99; International Organization for Standardization, 2007.

(10) ISO/IEC. Uncertainty of Measurement—Part 3: Guide to the Expression of Uncertainty in Measurement (GUM:1995); ISO/IEC Guide 98-3; International Organization for Standardization, 2008.

(11) Reichmuth, A.; Fritsch, K. Good Weighing Practices in the Pharmaceutical Industry – Risk-Based Qualification and Life Cycle Management of Weighing Systems. Pharm. Eng. 2009, 29 (6), 46–58.

(12) Mettler-Toledo White Paper. Weighing According to Ph. Eur.: Ensuring Compliance with Chapter 2.1.7; Mettler-Toledo, 2021.

Supplementary Material

In Part I of this series (1), the effect of buoyancy was discussed. In this supplement to Part III, we look at this in more detail and review a weighing by difference method that compensates for both buoyancy and drift. The result is confirmed using a Monte Carlo Simulation approach (2).

Buoyancy corrections can become important when the density of the object being weighed is very different from the density of the masses employed for the calibration function. For example, this is true when accurate masses of water are needed for the calibration of volumetric glassware.

When a sample is weighed on a balance, the observed weight is less than the true weight (in vacuo) owing to buoyance effects. In its simplest form, there are three factors involved: the density of air, ρ_Air, the density of the calibration masses, ρ_Mass, and the density of the sample, ρ_Sam, being weighed. Of these, the density of air ρ_Air is the most difficult to determine. However, it has a generally accepted value under normal laboratory conditions of 0.0012g mL. Equation 1.1 gives the relationship required to calculate the weight in vacuo.

Let us suppose that we need to calibrate a 100 mL volumetric flask. We weigh the flask empty and then fill it to the mark with water at 20 °C, which has a known density of 0.99705 g/mL and reweigh and find a value of w_air of 99.814 g. Dividing w_airby0.99705 yields a volume of 100.109 mL. Looking good you say, but is this the correct value? Well, the balance was calibrated using traceable austenitic stainless-steel masses, which have a density of 8.00g cm³.

Employing equation 1.2, we can find the weight in vacuo compensated for buoyancy effects:

which yields a volume of 100.215 mL. This is a small but significant systematic error of 0.11% in terms of calibrating the flask.

Monte Carlo Simulation of the Weighing by Difference Method

The Davison-Possolo method, as described in the main article (see references 3 and 4).

To simulate the sample weight and its uncertainty, it is necessary to have an estimate of the weight variability of the containers used. Sample containers like these are manufactured in large quantities, and the standard deviation for a batch will be well characterized. For this example, let us assume it is 0.005 g.

We can calculate the weighing uncertainty of the container by combining the uncertainty of the balance 0.000387 (see equation 2 in main article) with standard deviation of the container weight, 0.005, as per equation 1.3.

The uncertainty of weighing is the same for all the containers, irrespective of weight or content, as is illustrated in Figure S1.

The uncertainty of the difference in the reference weighings is given by:

The factor of two is required as there are two weighings.

Let us assume that we need to estimate the weight of a sample of low-density powder of 50 g. The mass in the reference container is selected as 25 g. This is approximately half the weight of the sample needed. We perform the four-step Davison weighing process (3) and record the values for w1 to w4 of 28.428 g, 3.436 g, 53.768 g, and 28.476 g, respectively. The difference between w1 and w4 is the drift and is equal to -0.048 g.

Therefore, the weight of the powder is given by:

We now calculate the uncertainty for w_pdr based upon the u_cont and u_Diff

The factor of two is again required as there are two weighings of the sample container.

However, the use of these error budget equations does not readily convey the dynamic nature of the weighing process. Running a Monte Carlo Simulation (5) on the three parameters, w3, w2, and (w1-w4) together with their standard deviations from equations 1.3 and 1.4 clearly does and visually demonstrates the spread of the weight data at k=2 (95.45% confidence) and k=3 (99.73% confidence) from n equal to 500,000 iterations.

This is shown in Figure S2. Note particularly that it is not necessary to calculate equation 1.6, but the u_wpdr value found is the same within rounding error.

References

(1) Burgess, C.; McDowall, R. D. A Question of Balance? Part 2: Putting Principles into Practice. LCGC Europe 2006, 3 (9), 152–159.
(2) Burgess, C. The Basics of Measurement Uncertainty in Pharmaceutical Analysis. Pharm. Technol. 2013, 37 (9), 62–64, 73.
(3) Davidson, S.; Perkin, M.; Buckley, M. The Measurement of Mass and Weight; NPL Measurement Good Practice Guide No. 71; National Physical Laboratory, 2004. https://www.npl.co.uk/special-pages/guides/gpg71_mass.aspx (accessed 2026-01-14).
(4) Possolo, A. Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results: Example E1, Weighing; NIST Technical Note 1900; National Institute of Standards and Technology, 2015.
(5) Minitab Workspace 1.3.0.0; 2024.