Measurement Uncertainty in Force Measurement


This article presents information about the general ways of evaluating uncertainty in force measurement applications. All the information is extracted from related documents provided by standard bodies — European Association of National Metrology Institutes(EURAMET), National Institute of Standards and Technology (NIST), American Society for Testing and Materials (ASTM), and the Joint Committee on Guides in Metrology (JCGM)and from published journals. Therefore, the information is reliable to work with and the details are explained intuitively so as to enable its proper understanding.

The concept of uncertainty is actually kind of new in the history of measurement science. According to [1], as of 1977, there was no international consensus on the expression of measurement uncertainty. This then made the CIPM request the BIPM to address this issue in collaboration with some national standard laboratories. A guide [1], was then developed that became applicable to a large spectrum of measurements – including force measurements.

The main body content is structured to contain: the definition of force measurement, the explanation of measurement uncertainty, the misconceptions between error and uncertainty, the importance of determining force measurement uncertainty, the general procedures for estimating measurement uncertainty, the procedures for estimating force measurement uncertainty, other sources of uncertainty, and force measurement calibration and hierarchy.

What Is Force Measurement?

According to [2], measurement is the process of experimentally obtaining one or more quantity values that can reasonably be attributed to a quantity. In metrology, measurement is done directly or indirectly. A direct form of measurement, for example, is obtaining the length of an object using a ruler; an indirect form, for example, is force measurement. Force measurement using a strain gauge transducer is not direct. It involves the transformation of the load into the deformation of the elastic element and then the transformation of the deforming stress into a strain that is then measured as the change in electrical resistance; this then modifies the output voltage of a Wheatstone bridge.

Force measurements involve the use of mathematical models that describe the relationship between the input and the output. An example is in describing the voltage output of the Wheatstone bridge; the input values can be the change in resistance of the strain gauge, the arm resistors, and the excitation voltage.

Now, from the datasheet of a strain gauge load cell, it should be noticed that the specification values are written as a single value with a ± sign. For example, the full-scale output (FSO) can be2.2mV/V ±0.25%; expanding this further gives 2.2mV/V±5.5μV/V hence, the actual value of the FSO probably lies within the range ( .2mV/V+5.5μV/V = 2.2055mV/V) and ( 2.2mV/V-5.5μV/V = 2.1945mV/V).

Furthermore, multiplying the 0.25% by 2 gives 0.5%; subtracting 0.5% from 100% gives 99.5%. Hence, the interpretation of the FSO output means: the probability that the FSO lies in this range/interval is 99.5%. Therefore, it can be said that the level of confidence for this value is 99.5%

Gradually, this becomes a problem involving statistics, where the set of possible values can be described by using probability distribution functions. With respect to a normal distribution, using the example of the FSO value above, the central value is 2.2mV/V, while the standard deviation from this value is 5.5uV/V. This standard deviation is taken as the measure of uncertainty.

What Therefore, Is The Meaning Of Uncertainty?

According to [2], it is defined as a non-negative parameter characterizing the dispersion of the quantity value being attributed to a measurand. From this definition, notice the word “dispersion”, in statistics, there is a type of descriptive statistics that involves the measure of dispersion; the mathematical tools used for this measure of dispersion include? Yes, you guessed right, standard deviation! There are others such as variance and range. This, therefore, emphasizes why the value 5.5μV/V is a standard deviation showing the dispersion of the central value 2.2mV/V around the range.

Intuitively, uncertainty can be defined as the doubt about the correctness of the value obtained from the force measurement. It is the quantitative indication of the quality of the measured force. This quantitative value is the value of a standard deviation.

Misconceptions: Measurement Error Vs. Measurement Uncertainty

Measurement error is the difference between the observed value of a quantity and its true value. This is, therefore, a theoretical concept that cannot be truly known; because it is impossible to determine the true value of a quantity.

Therefore, it can be concluded that measurement error is different from uncertainty because, an error is theoretical and single-valued while uncertainty is a range of values that are actually quantifiable and is included in the specification sheet of a force measuring instrument. Also, when all the sources of errors have been identified and appropriate adjustments made, there is still uncertainty about the reliability of the measured value.

Importance of Determining Force Measurement Uncertainty

The following are the importance of evaluating the uncertainty in force measurements [3]

  1. The uncertainty value of a force measurement provides a basis for the comparison of the results obtained with those obtained in other laboratories or by national standards.
  2. It helps to properly interpret the results obtained under different conditions. For example, calibration is performed under laboratory conditions, the application of the force transducer is/maybe under totally different conditions; hence, there will be differences in the results. This conditions can be grouped into geometrical, mechanical, temporal, electrical, and environmental effects [4]. These differences are easily accounted for by the expression of the uncertainty of each result.
  3. The results of the evaluation of uncertainty of measurement can also serve as a statement of compliance if requested by a customer or legally.
  4. It is a means of determining the capability of the force measurement in providing accurate measurement results.
  5. Specifically, the uncertainty components that form the combined uncertainty value can help to pinpoint the measurement variables to be improved.
  6. The methods of force measurement employed by the laboratory can be ascertained and even fine-tuned through the understanding of the principles of measurement uncertainty in combination with practical experiences.
  7. The principles of evaluating force measurement uncertainty can help in maintaining and improving product quality and quality assurance.

General Procedures of Estimating Measurement Uncertainty

In order to avoid confusion, this section explains the general steps to be followed in evaluating a measurement uncertainty in accordance with the GUM standard as in [1]. With that, the steps for a force measurement uncertainty will not be confusing.

1. Modeling the Measurement:

This involves the development of a mathematical model that defines the relationship between the measurand/output Y and the input quantities X_i. This is represented by the simple expressions below, where y is the output estimate, and the input estimates are x_1,x_2,…,x_N

At times, these input quantities can actually be the arithmetic mean value of the individual n observations of each input quantity, that is . Mathematically, the mean for each input quantity becomes

Therefore the estimate y of the measurand Y becomes a function of the average  of each, expressed as

2. Evaluating The Standard Uncertainty Of Each Input Quantity:

There are two methods of evaluating the standard uncertainty denoted as . These two methods are

Type A Evaluation: This method involves performing a series of repeated observations/measurements of the input quantities. It involves: calculating the mean of each input quantity as in step 1, calculating their experimental standard deviation, and finally calculating the standard deviation of the mean.  The standard deviation of the mean is what the standard uncertainty becomes. The illustration below should explain this better.

Type B Evaluation: This involves the determination of the standard uncertainty by means others than those employed by Type A. This involves the use of existing information that gives the uncertainty sources and their values. They can be gotten from the calibration certificate, authoritatively published quantity values, certified reference materials and handbooks, personal experiences and general knowledge of the instrument, etc. This method is used when there are difficulties in performing repeated measurements as they can be costly and time-consuming. It relies on the use of the triangle and rectangular distribution. On the other hand, Type A is associated with a normal or Gaussian distribution. The figure 1 below shows these distributions and their functions.

Figure 1. Probability Distribution
Figure 1. Probability Distribution

It could, therefore, be seen, that whatever method is used, the end result is a value of the standard deviation.

3. Determine The Combined Standard Uncertainty u_c (y).

The most commonly used method for this is the GUM’s law of propagation of uncertainty (LPU). LPU involves the expansion of the mathematical model in a Taylor series and simplifying based on only the first order terms.

The combined standard uncertainty is simply the appropriate combination of the standard uncertainties obtained for each input quantities in step 2. The right expression for the combined standard uncertainty depends on if the input quantities are independent or interdependent/correlated. If they are independent, the expression is

If they are interdependent, the expression below is added to the right-hand side of the one above

Where the partial derivatives are called the sensitivity coefficients and  is the standard uncertainty of the i-th input quantity. The unsquared value of the product of the partial derivatives with the standard uncertainty is called the uncertainty component.

4. Determining The Expanded Uncertainty:

The expanded uncertainty value defines an interval about the measurement result. It ensures the result encompasses a large fraction of the distributed values that could be reasonably attributed to the measurand. It is denoted by U and it is obtained as the product of the combined standard uncertainty and a coverage factor K, expressed as

U=k.u_c (y).

Therefore, the measurement result can be expressed conveniently as Y=y ±U. That is, (y-U)≤Y ≥(y+U). Recall the example of the FSO taken to be as 2.2 mV/V±0.05μV/V .

The value of the coverage factor K depends on the confidence level and the effective degree of freedom — it can be derived using the Welch-Satterthwaite formula.

Procedures for Estimating Force Measurements

These procedures follow the general steps listed in the previous section, but it is borrowed from NIST’s exact method for force calibration [5].

1.The polynomial equation modeling the force transducer response is given below

Where R is the response, F is the applied force and the Ai is the coefficient calculated by applying the least-square fit method to the dataset.

2. Evaluating The Standard Uncertainty: According to NIST, the sources of uncertainty considered to be attributable to the transducer force measurement are the applied force (the input quantity), the calibration of the indicating instrumentation, and the fit of the measured data to the model equation.

The Uncertainty In Applied Force: This is denoted by  and the mathematical model to obtain this for a deadweight is expressed below

Where F is the applied force in Newton’s, m is the deadweight in Kg, g is the acceleration due to gravity in m/s^2, ρ_a is the atmospheric density of the air, and ρ_w is the density of the weight material.

Therefore, to obtain the standard uncertainty of the applied force, it is simply the combination of the standard uncertainty of the input quantities in its model equation. That is, the standard uncertainty associated with the mass u_fa, the acceleration due to gravity u_fb, and the density u_fc. Hence,

The Uncertainty of The Voltage Ratio Instrumentation:

Assuming that the response is the voltage output of the force transducer, then the indicating instrument is a multimeter. The multimeter is actually like a data acquisition device that has an analog-to-digital converter, a digital signal processor, and a display; or it could be an analog multimeter.

The various standard uncertainty sources for this includes: the uncertainty associated with the multimeter’s linearity and resolution, denoted as u_vb, that affects its least square fit to a model data; the calibration factor of the multimeter, denoted as u_va, the calibration factor is the ratio of the voltage indicated by the multimeter to that of a reference voltage; the uncertainty associated with the results of the primary calibration of the multimeter using a primary transfer standard such as a precision load cell simulator, it is denoted as u_vc
In total, the standard uncertainty u_v associated with the instrument is a combination of its sources

In total, the standard uncertainty u_v associated with the instrument is a combination of its sources

2. The Uncertainty Due To the Deviation of the Observed Data from the Fitted Curve: This uncertainty is denoted as u_r and it is calculated as the deviation of the values of the measured data from those of the fitted model data. This is expressed below as

Where d_j are the differences between the measured response R_j and those calculated using the model equation in step 1, n is the number of individual measurements in the calibration dataset, and m is the order of the polynomial plus one

3. Evaluating The Combined Standard Uncertainty: The combined uncertainty is therefore expressed as

4. Evaluating The Expanded Uncertainty: The expanded uncertainty U is the product of the combined uncertainty Uc and the Coverage factor K. The value of K is obtained at a confidence level defined as 95%, making K equal to U=k.u_c = 2.u_c

Other Sources of Uncertainty

The following sources could be taken into account for a practical process of calibration or force measurement.

  • Creep of the transducer
  • Reproducibility and repeatability
  • Temperature effect on the zero balance and the span
  • Alignment of the application point
  • Stability and performance of the data acquisition instrument
  • Effects of different time-loading profiles.
  • Effects of the type of loading; this can be static, quasi-static, or dynamic loading

Force Measurement Calibration and Calibration Hierarchy

This is a process in which a comparison is performed between the results obtained from the measurements and that values from a reference standard.

The calibration hierarchy is the sequence of calibration from a reference to the final force measurement/generating system. This hierarchy in descending order, is as follows: the national force standard machines, the force calibration machines, the force measuring instrument, and the force generating machine [6]. The calibration hierarchy is such that the uncertainty at each level is based on two components

  • The uncertainty component due to equipment used in calibrating the instrument
  • The uncertainty component due to the calibration of the instrument itself and from other sources.


This article discussed important points that relate to force uncertainty measurement. In case you find any of the terms used here difficult to understand, please do check out the definition in the vocabulary information in [2].


[1] Joint Committee on Guides in Metrology (JCGM), “Evaluation of Measurement Data – Guide to the Expression of Uncertainty in Measurement,” JCGM, 2008.
[2] Joint Committee for Guides in Metrology (JCGM), “JCGM 200:2008 International vocabulary of metrology – Basic and general concepts and associated terms (VIM),” Joint Committee for Guides in Metrology, 2008.
[3] A. G. Piyal Aravinna, “Basic Concepts of Measurement Uncertainty,” 2018.
[4] Dirk Röske, Jussi Ala-Hiiro, Andy Knott, Nieves Medina, Petr Kaspar, Mikołaj Woźniak, “Tools for uncertainty calculations in force measurement,” ACTA IMEKO, vol. 6, pp. 59-63, 2017.
[5] Thoma W. Bartel, “Uncertainty in NIST Force Measurements,” Journal of Research of the National Institute of Standards and Technology, vol. 110, no. 6, pp. 589-603, 2005.
[6] European Association of National Metrology Institutes (EURAMET), “Uncertainty of Force Measurements,” 2011.
[7] Jailton Carreteiro Damasceno and Paulo R.G. Couto, “Methods for Evaluation of Measurement Uncertainty,” IntechOpen, 2018.