3. Using the GUM approach on uncertainty evaluation

The following main steps summarize the methodology presented by the GUM.

#### 3.1. Definition of the measurand and of input quantities

It must be clear to the analyst which quantity will be the final object of the measurement in question. This quantity is known as the measurand. In addition, it is important to identify all the variables that directly or indirectly influence the measurand. These variables are known as the input quantities. As an example, Eq. (1) shows a measurand y as a function of three different input quantities: x1, x2, and x<sup>3</sup>:

$$y = f(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) \tag{1}$$

#### 3.2. Modeling the measurement process

competence and the ability to properly operate their management systems, and so they are

In addition, the use of uncertainty evaluation methods as a tool for technical management of measurement processes is extremely important to reduce, for example, the large number of losses that occurs in the industry, which can be highly significant in relation to the gross domestic product (GDP) of some countries. One of the probable causes of the waste can be attributed to instruments whose accuracy is inadequate to the tolerance of a certain measure-

In order to harmonize the uncertainty evaluation process for every laboratory, the Bureau International des Poids et Mesures (BIPM) published in 1980 the Recommendation INC-1 [2] on how to express uncertainty in measurement. This document was further developed and originated the "Guide to the Expression of Uncertainty in Measurement"—GUM in 1993, which was revised in 1995 and lastly in 2008. This document provides complete guidance and references on how to treat common situations on metrology and how to deal with uncertainties in metrology. Currently, it is published by International Organization for Standardization (ISO) as the ISO/IEC Guide 98-3 "Uncertainty of measurement—Part 3: Guide to the expression of uncertainty in measurement" (GUM), and by the Joint Committee for Guides in Metrology (JCGM) as the JCGM 100:2008 guide [3]. The JCGM was established by BIPM to maintain and further develop the GUM. They are in fact currently producing a series of documents and supplements to accompany the GUM, four of which are already published

Evaluation of uncertainty, as presented by the JCGM 100:2008, is based on the law of propagation of uncertainties (LPU). This methodology has been successfully applied for several years worldwide for a range of different measurement systems and is currently the most used procedure for uncertainty evaluation in metrology. However, since its twentieth anniversary in 2013, JCGM decided to revise it again [8–10]. In this new revision, uncertainty terms and concepts [11] will be aligned with the current International Vocabulary of Metrology (VIM) [12] and with the new GUM supplements [5, 6]. Aspects such as a new Bayesian approach, the redefinition of coverage intervals and the elimination of the Welch-Satterthwaite formula to evaluate the effective degrees of freedom will be covered [9]. In late 2014, a first draft of the newly revised version of the GUM was circulated among National Metrology Institutes. Remarkable changes were made that could affect the way laboratories deal with the measurement uncertainty results. This revision is still being discussed, and some information about it

In the field of analytical chemistry, there is also another document worth mentioning that is the "Quantifying Uncertainty in Analytical Measurement" guide [13], produced by a joint

required to evaluate the uncertainty for their measurement results.

In this chapter, detailed steps for uncertainty evaluation are given.

2. Main references for uncertainty evaluation

ment process.

10 Metrology

[4–7].

has also been released elsewhere [10].

In this step, the measurement procedure should be modeled in order to have a functional relationship expressing the measurand as a result of all the input quantities. The measurand y in Eq. (1) could be modeled, for example, as in Eq. (2)

$$y = \frac{\mathbf{x}\_1 \mathbf{x}\_2}{\mathbf{x}\_3^2} \tag{2}$$

The modeling step is critical for the uncertainty evaluation process as it defines how the input quantities impact the measurand. The better the model is defined, the better its representation of reality will be, including all the sources that impact the measurand on the uncertainty evaluation. The modeling process can be easily visualized by using a cause-effect diagram (Figure 1).

Example: To illustrate these steps, let us consider a measurement model for a torque test. Torque is a quantity that represents the tendency of a force to rotate an object about an axis. It can be mathematically expressed as the product of a force and the lever-arm distance. In metrology, a practical way to measure it is by loading a known mass to the end of a horizontal arm while keeping the other end fixed (Figure 2).

Note: This example is also presented, with a few adaptations, in other publications by the same authors [15].

Figure 1. A cause-effect diagram representing the model from Eq. (2).

Figure 2. A conceptual illustration of the experimental setup for a measurement of torque (T), where F is the applied force, m is the mass of the load, g is the local gravity acceleration, and L is the length of the arm.

A simple model that describes this experiment can be expressed as follows:

$$T = \text{mgL} \tag{3}$$

ux ¼ sð Þ¼ x

this input source is considered to be normal or Gaussian.

distributions: the uniform and the triangular distributions.

such an interval is given by Eq. (6):

would be evaluated as <sup>u</sup><sup>θ</sup> <sup>¼</sup> <sup>4</sup><sup>=</sup> ffiffiffiffiffi

where x is the mean value of the repeated measurements, s xð Þ<sup>i</sup> is its standard deviation, and sð Þx is the standard deviation of the mean. As such, the statistical distribution associated with

Note: This evaluation is not consistent with the GUM supplement 1 [5], where repeated indications are treated as Student's t-distributions to account for the lack of degrees of freedom or a low number of indications. In this way, the new proposal for the draft GUM is to consider repeated indications as t-distributions, just like in supplement 1. Therefore, its uncertainty would be evaluated as in Eq. (5). This equation takes the degrees of freedom for the indications (n � 1) into account, raising the uncertainty for a low number of indications. This correction would then be in accordance with the approach suggested by the other GUM supplements for this type of uncertainty

> ux <sup>¼</sup> <sup>n</sup> � <sup>1</sup> n � 3 � �<sup>1</sup>=<sup>2</sup>

It is important to note that the evaluation of uncertainties of Type B input sources must be based on careful analysis of observations or in an accurate scientific judgment, using all available information about the measurement procedure. This uncertainty type is generally used when repeated experiments would not be possible, not available, or would be too costly or time-consuming. In this case, the GUM also suggests the use of two more types of statistical

The uniform distribution should be used when only a range of values are available, that is, an interval with the minimum and maximum values, and no detailed information about the probability of values within this interval is available. The standard uncertainty associated with

ux <sup>¼</sup> <sup>b</sup> � <sup>a</sup> ffiffiffiffiffi

where b is the maximum and a is the minimum values for the range. For example, if the only information about the room temperature of a laboratory is known to be 20 ð Þ � 2 �C, then b � a ¼ 22 � 18 ¼ 4�C and the standard uncertainty associated with the room temperature

The triangular distribution can be used when there is a strong evidence that the most probable value lies in the middle of a given interval, but still without knowing exactly how this probability behave within the interval. In chemistry, for example, the uncertainty associated with the volume of a measuring flask could be evaluated by a triangular distribution. The

> ux <sup>¼</sup> <sup>a</sup> ffiffiffi 6

<sup>12</sup> <sup>p</sup> �<sup>C</sup> <sup>¼</sup> <sup>1</sup>:15�C.

standard uncertainty for a triangular distribution is given by Eq. (7):

where a is the semi-interval for the total range of the triangular distribution.

s xð Þ<sup>i</sup> ffiffiffi

> s xð Þ<sup>i</sup> ffiffiffi

<sup>n</sup> <sup>p</sup> (4)

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873 13

<sup>n</sup> <sup>p</sup> (5)

<sup>12</sup> <sup>p</sup> (6)

p (7)

where T is the torque (N.m), m is the mass of the applied load (kg), g is the local gravity acceleration (m/s<sup>2</sup> ), and L is the total length of the arm (m). Thus, m, g, and L are the input quantities for this model. This example will be further discussed in the subsections ahead.

#### 3.3. Evaluating the uncertainties of the input quantities

This step is also of great importance. Here, uncertainties for all the input quantities are individually evaluated. The GUM classifies uncertainty sources as being of two main types: Type A, which usually originates from some statistical analysis, such as the standard deviation obtained in a repeatability study; and Type B, which is determined from any other source of information, such as a calibration certificate or deduced from personal experience.

Type A uncertainties from repeatability studies are evaluated by the GUM as the standard deviation of the mean obtained from the repeated measurements. For example, if a set of n indications xi about a quantity x are available, the uncertainty ux due to repeatability of the measurements can be expressed by sð Þx as follows in Eq. (4):

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873 13

$$\mu\_{\mathfrak{x}} = \mathfrak{s}(\overline{\mathfrak{x}}) = \frac{\mathfrak{s}(\mathfrak{x}\_i)}{\sqrt{n}} \tag{4}$$

where x is the mean value of the repeated measurements, s xð Þ<sup>i</sup> is its standard deviation, and sð Þx is the standard deviation of the mean. As such, the statistical distribution associated with this input source is considered to be normal or Gaussian.

Note: This evaluation is not consistent with the GUM supplement 1 [5], where repeated indications are treated as Student's t-distributions to account for the lack of degrees of freedom or a low number of indications. In this way, the new proposal for the draft GUM is to consider repeated indications as t-distributions, just like in supplement 1. Therefore, its uncertainty would be evaluated as in Eq. (5). This equation takes the degrees of freedom for the indications (n � 1) into account, raising the uncertainty for a low number of indications. This correction would then be in accordance with the approach suggested by the other GUM supplements for this type of uncertainty

$$
\mu\_{\mathbf{x}} = \left(\frac{n-1}{n-3}\right)^{1/2} \frac{\mathbf{s}(\mathbf{x}\_i)}{\sqrt{n}}\tag{5}
$$

It is important to note that the evaluation of uncertainties of Type B input sources must be based on careful analysis of observations or in an accurate scientific judgment, using all available information about the measurement procedure. This uncertainty type is generally used when repeated experiments would not be possible, not available, or would be too costly or time-consuming. In this case, the GUM also suggests the use of two more types of statistical distributions: the uniform and the triangular distributions.

The uniform distribution should be used when only a range of values are available, that is, an interval with the minimum and maximum values, and no detailed information about the probability of values within this interval is available. The standard uncertainty associated with such an interval is given by Eq. (6):

A simple model that describes this experiment can be expressed as follows:

force, m is the mass of the load, g is the local gravity acceleration, and L is the length of the arm.

3.3. Evaluating the uncertainties of the input quantities

Figure 1. A cause-effect diagram representing the model from Eq. (2).

measurements can be expressed by sð Þx as follows in Eq. (4):

acceleration (m/s<sup>2</sup>

12 Metrology

where T is the torque (N.m), m is the mass of the applied load (kg), g is the local gravity

Figure 2. A conceptual illustration of the experimental setup for a measurement of torque (T), where F is the applied

This step is also of great importance. Here, uncertainties for all the input quantities are individually evaluated. The GUM classifies uncertainty sources as being of two main types: Type A, which usually originates from some statistical analysis, such as the standard deviation obtained in a repeatability study; and Type B, which is determined from any other source of

Type A uncertainties from repeatability studies are evaluated by the GUM as the standard deviation of the mean obtained from the repeated measurements. For example, if a set of n indications xi about a quantity x are available, the uncertainty ux due to repeatability of the

information, such as a calibration certificate or deduced from personal experience.

quantities for this model. This example will be further discussed in the subsections ahead.

), and L is the total length of the arm (m). Thus, m, g, and L are the input

T ¼ mgL (3)

$$
\mu\_{\mathfrak{x}} = \frac{b - a}{\sqrt{12}} \tag{6}
$$

where b is the maximum and a is the minimum values for the range. For example, if the only information about the room temperature of a laboratory is known to be 20 ð Þ � 2 �C, then b � a ¼ 22 � 18 ¼ 4�C and the standard uncertainty associated with the room temperature would be evaluated as <sup>u</sup><sup>θ</sup> <sup>¼</sup> <sup>4</sup><sup>=</sup> ffiffiffiffiffi <sup>12</sup> <sup>p</sup> �<sup>C</sup> <sup>¼</sup> <sup>1</sup>:15�C.

The triangular distribution can be used when there is a strong evidence that the most probable value lies in the middle of a given interval, but still without knowing exactly how this probability behave within the interval. In chemistry, for example, the uncertainty associated with the volume of a measuring flask could be evaluated by a triangular distribution. The standard uncertainty for a triangular distribution is given by Eq. (7):

$$
\mu\_{\mathfrak{x}} = \frac{a}{\sqrt{\mathfrak{G}}} \tag{7}
$$

where a is the semi-interval for the total range of the triangular distribution.

Another common Type B source of uncertainty is due to calibration certificates, related to a standard or to a calibrated instrument. In this case, the standard uncertainty to be used is normally obtained by dividing the expanded uncertainty U by the coverage factor k, both provided by the calibration certificate (Eq. (8))

$$
\mu\_{\text{x}} = \frac{\mathcal{U}}{k} \tag{8}
$$

3.4. Propagation of uncertainties

3.4.1. The law of propagation of uncertainties

valid for a range of simplistic models.

u2 <sup>y</sup> <sup>¼</sup> <sup>X</sup><sup>N</sup> i¼1

simplified as

the authors).

The GUM uncertainty approach is based on the law of propagation of uncertainties (LPU). This methodology encompasses a set of approximations to simplify the calculations and is

According to the LPU, the propagation of uncertainties is accomplished by expanding the measurand model in a Taylor series and simplifying the expression by considering only the first-order terms. This approximation is viable as uncertainties are very small numbers compared with the values of their corresponding quantities. In this way, the treatment of a model where the measurand y is expressed as a function of N variables x1, …, xN (Eq. (9)) leads to the

> X<sup>N</sup>�<sup>1</sup> i¼1

X<sup>N</sup> j¼iþ1

∂y ∂xi � �<sup>2</sup>

u2

where uy is the combined standard uncertainty for the measurand y and uxi is the uncertainty for the ith input quantity. The second term of Eq. (10) is due to the correlation between the input quantities. If there is no supposed correlation between them, Eq. (10) can be further

The partial derivatives of Eq. (11) are known as sensitivity coefficients and describe how the output estimate y varies with changes in the values of the input estimates x1, x2,…, xN. It also

Another important observation regarding the sensitivity coefficient occurs when the mathematical model that defines the measurand does not contemplate a given quantity, known as influence quantity. In this case, the determination of the sensitivity coefficient of the measurand in relation to the input quantity must be done experimentally. For example, biodiesel is susceptible to oxidation when exposed to air, and this oxidation process affects fuel quality. The oxidation time is determined by measuring the conductivity of an oil sample when inflated with air at a given flow rate. There are a number of influence quantities that impact the oxidation time of biodiesel such as temperature, air flow, conductivity, sample mass, and so on. In this case, the sensitivity coefficients for oxidation time with respect to each of these influence quantities are determined from an interpolation function obtained with experimental data. For example, Figure 3 presents the table and its resulting graph, which shows the model of the function that relates the oxidation time to the temperature of a biofuel sample (case study of

∂y ∂xi � � ∂y

y ¼ f xð Þ <sup>1</sup>;…; xN (9)

� �COV xi; xj

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873 15

xi (11)

� � (10)

∂xj

general expression for the propagation of uncertainties shown in Eq. (10)

u2 xi þ 2

> u2 <sup>y</sup> <sup>¼</sup> <sup>X</sup><sup>N</sup> i¼1

∂y ∂xi � �<sup>2</sup>

converts the units of the inputs to the unit of the measurand.

Several good examples on how to treat some of the most common uncertainty sources can be found on the GUM [3], the EURACHEM/CITAC guide [13], and elsewhere [16].

Example: Returning to the example of torque measurement and considering the model defined in Eq. (3), the following sources of uncertainty are considered:

Mass (m). The mass m was repeatedly measured 10 times in a calibrated balance. The average mass was 35.7653 kg, with a standard deviation of 0.3 g. This source of uncertainty is purely statistical and is classified as being of Type A according to the GUM. The standard uncertainty (umR ) that applies in this case is obtained by Eq. (4), that is, umR <sup>¼</sup> <sup>0</sup>:3 g<sup>=</sup> ffiffiffiffiffi <sup>10</sup> <sup>p</sup> <sup>¼</sup> <sup>9</sup>:<sup>49</sup> � <sup>10</sup>�<sup>5</sup> kg.

In addition, the balance used for the measurement has a certificate stating an expanded uncertainty for this range of mass of Um = 0.1 g, with a coverage factor k = 2 and a coverage probability of 95%. The uncertainty of the mass due to the calibration of the balance constitutes another source of uncertainty involving the same input quantity (mass). In this case, the standard uncertainty (umC ) is calculated by using Eq. (8), that is, umC ¼ Um=k ¼ 0:1 g=2 ¼ 0:00005 kg.

Local gravity acceleration (g). The value for the local gravity acceleration is stated in a certificate of measurement as 9.80665 m/s<sup>2</sup> , as well as its expanded uncertainty of Ug = 0.00002 m/s<sup>2</sup> , for k = 2 and p = 95%. Again, Eq. (8) is used to calculate the standard uncertainty (ug), that is, ug <sup>¼</sup> Ug=<sup>k</sup> <sup>¼</sup> <sup>0</sup>:00002 m=s<sup>2</sup> � �=<sup>2</sup> <sup>¼</sup> <sup>0</sup>:00001 m/s2 .

Length of the arm (L). Let us suppose that in this hypothetical case, the arm used in the experiment has no certificate of calibration, indicating its length value and uncertainty, and that the only measuring method available for the arm's length is by the use of a ruler with a minimum division of 1 mm. The use of the ruler leads then to a measurement value of 2000.0 mm for the length of the arm. However, in this situation, very poor information about the measurement uncertainty of the arm's length is available. As the minimum division of the ruler is 1 mm, one can assume that the reading can be done with a maximum accuracy of up to 0.5 mm, which can be thought as an interval of �0.5 mm as limits for the measurement. As no information of probabilities within this interval is available, the assumption of a uniform distribution is the best option, on which there is equal probability for the values within the whole interval. Thus, Eq. (6) is used to determine the standard uncertainty (uL), that is, uL <sup>¼</sup> ð Þ <sup>2000</sup>:<sup>5</sup> � <sup>1999</sup>:<sup>5</sup> mm<sup>=</sup> ffiffiffiffiffi <sup>12</sup> <sup>p</sup> <sup>¼</sup> <sup>0</sup>:000289 m.

In practice, one can imagine several more sources of uncertainty for this experiment, like, for example, the thermal dilatation of the arm as the room temperature changes. However, the objective here is not to exhaust all the possibilities, but instead to provide basic notions of how to implement the methodology of the GUM on a simple model.

#### 3.4. Propagation of uncertainties

Another common Type B source of uncertainty is due to calibration certificates, related to a standard or to a calibrated instrument. In this case, the standard uncertainty to be used is normally obtained by dividing the expanded uncertainty U by the coverage factor k, both

ux <sup>¼</sup> <sup>U</sup>

Several good examples on how to treat some of the most common uncertainty sources can be

Example: Returning to the example of torque measurement and considering the model defined

Mass (m). The mass m was repeatedly measured 10 times in a calibrated balance. The average mass was 35.7653 kg, with a standard deviation of 0.3 g. This source of uncertainty is purely statistical and is classified as being of Type A according to the GUM. The standard uncertainty

In addition, the balance used for the measurement has a certificate stating an expanded uncertainty for this range of mass of Um = 0.1 g, with a coverage factor k = 2 and a coverage probability of 95%. The uncertainty of the mass due to the calibration of the balance constitutes another source of uncertainty involving the same input quantity (mass). In this case, the standard uncertainty (umC ) is calculated by using Eq. (8), that is, umC ¼ Um=k ¼ 0:1 g=2 ¼

Local gravity acceleration (g). The value for the local gravity acceleration is stated in a

Length of the arm (L). Let us suppose that in this hypothetical case, the arm used in the experiment has no certificate of calibration, indicating its length value and uncertainty, and that the only measuring method available for the arm's length is by the use of a ruler with a minimum division of 1 mm. The use of the ruler leads then to a measurement value of 2000.0 mm for the length of the arm. However, in this situation, very poor information about the measurement uncertainty of the arm's length is available. As the minimum division of the ruler is 1 mm, one can assume that the reading can be done with a maximum accuracy of up to 0.5 mm, which can be thought as an interval of �0.5 mm as limits for the measurement. As no information of probabilities within this interval is available, the assumption of a uniform distribution is the best option, on which there is equal probability for the values within the whole interval. Thus, Eq. (6) is used to

In practice, one can imagine several more sources of uncertainty for this experiment, like, for example, the thermal dilatation of the arm as the room temperature changes. However, the objective here is not to exhaust all the possibilities, but instead to provide basic notions of how

determine the standard uncertainty (uL), that is, uL <sup>¼</sup> ð Þ <sup>2000</sup>:<sup>5</sup> � <sup>1999</sup>:<sup>5</sup> mm<sup>=</sup> ffiffiffiffiffi

to implement the methodology of the GUM on a simple model.

, for k = 2 and p = 95%. Again, Eq. (8) is used to calculate the standard uncertainty

.

found on the GUM [3], the EURACHEM/CITAC guide [13], and elsewhere [16].

(umR ) that applies in this case is obtained by Eq. (4), that is, umR <sup>¼</sup> <sup>0</sup>:3 g<sup>=</sup> ffiffiffiffiffi

in Eq. (3), the following sources of uncertainty are considered:

<sup>k</sup> (8)

10

, as well as its expanded uncertainty of Ug =

12

<sup>p</sup> <sup>¼</sup> <sup>0</sup>:000289 m.

<sup>p</sup> <sup>¼</sup> <sup>9</sup>:<sup>49</sup> � <sup>10</sup>�<sup>5</sup> kg.

provided by the calibration certificate (Eq. (8))

certificate of measurement as 9.80665 m/s<sup>2</sup>

(ug), that is, ug <sup>¼</sup> Ug=<sup>k</sup> <sup>¼</sup> <sup>0</sup>:00002 m=s<sup>2</sup> � �=<sup>2</sup> <sup>¼</sup> <sup>0</sup>:00001 m/s2

0:00005 kg.

14 Metrology

0.00002 m/s<sup>2</sup>

#### 3.4.1. The law of propagation of uncertainties

The GUM uncertainty approach is based on the law of propagation of uncertainties (LPU). This methodology encompasses a set of approximations to simplify the calculations and is valid for a range of simplistic models.

According to the LPU, the propagation of uncertainties is accomplished by expanding the measurand model in a Taylor series and simplifying the expression by considering only the first-order terms. This approximation is viable as uncertainties are very small numbers compared with the values of their corresponding quantities. In this way, the treatment of a model where the measurand y is expressed as a function of N variables x1, …, xN (Eq. (9)) leads to the general expression for the propagation of uncertainties shown in Eq. (10)

$$y = f(\mathbf{x}\_1, \dots, \mathbf{x}\_N) \tag{9}$$

$$u\_y^2 = \sum\_{i=1}^N \left(\frac{\partial y}{\partial \mathbf{x}\_i}\right)^2 u\_{x\_i}^2 + 2\sum\_{i=1}^{N-1} \sum\_{j=i+1}^N \left(\frac{\partial y}{\partial \mathbf{x}\_i}\right) \left(\frac{\partial y}{\partial \mathbf{x}\_j}\right) \text{COV}(\mathbf{x}\_i, \mathbf{x}\_j) \tag{10}$$

where uy is the combined standard uncertainty for the measurand y and uxi is the uncertainty for the ith input quantity. The second term of Eq. (10) is due to the correlation between the input quantities. If there is no supposed correlation between them, Eq. (10) can be further simplified as

$$
\mu\_y^2 = \sum\_{i=1}^N \left(\frac{\partial y}{\partial \mathbf{x}\_i}\right)^2 \mu\_{x\_i}^2 \tag{11}
$$

The partial derivatives of Eq. (11) are known as sensitivity coefficients and describe how the output estimate y varies with changes in the values of the input estimates x1, x2,…, xN. It also converts the units of the inputs to the unit of the measurand.

Another important observation regarding the sensitivity coefficient occurs when the mathematical model that defines the measurand does not contemplate a given quantity, known as influence quantity. In this case, the determination of the sensitivity coefficient of the measurand in relation to the input quantity must be done experimentally. For example, biodiesel is susceptible to oxidation when exposed to air, and this oxidation process affects fuel quality. The oxidation time is determined by measuring the conductivity of an oil sample when inflated with air at a given flow rate. There are a number of influence quantities that impact the oxidation time of biodiesel such as temperature, air flow, conductivity, sample mass, and so on. In this case, the sensitivity coefficients for oxidation time with respect to each of these influence quantities are determined from an interpolation function obtained with experimental data. For example, Figure 3 presents the table and its resulting graph, which shows the model of the function that relates the oxidation time to the temperature of a biofuel sample (case study of the authors).

ruler used in the measurement. This analysis shows to the analyst that, to reduce the final uncertainty and improve the measurement system, a calibrated ruler, with a better uncertainty, should be used. This represents the importance of the GUM as a management tool to the

The Kragten method is an approximation that facilitates the calculation of the combined uncertainty using finite differences in place of the derivatives [13]. This approximation is valid when the uncertainties of the inputs are relatively small compared to the respective values of the input quantities, generating discrepancies in the final result in relation to the LPU that

Assuming a measurand y, which is calculated from the input quantities x1, x<sup>2</sup> and x<sup>3</sup> according to the mathematical model of Eq. (2), the uncertainties ux<sup>1</sup> , ux<sup>2</sup> and ux<sup>3</sup> for the input quantities are evaluated normally, according to methodologies already explained previously. From there, the calculations of the measurand are performed individually for each input magnitude (yx<sup>1</sup>

> yx<sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup> <sup>þ</sup> ux<sup>1</sup> ð Þx<sup>2</sup> x2 3

> yx<sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> ux<sup>2</sup> ð Þ x2 3

The value of the measurand y varies for yxi due to the addition of the uncertainty uxi to the value of its respective input quantity. Thus, the uncertainty component of each input source in

> uyð Þ¼ x<sup>1</sup> yx<sup>1</sup> � y � � �

> uyð Þ¼ x<sup>2</sup> yx<sup>2</sup> � y � � �

> uyð Þ¼ x<sup>3</sup> yx<sup>3</sup> � y � � �

> > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>u</sup><sup>2</sup> <sup>y</sup>ð Þ xi

� � �

� �

� �

� � � �

the unit of the measurand y is defined by the difference yxi � y

Thus, the combined standard uncertainty of y is finally evaluated as

or by Eq. (20), if there are correlated uncertainties

uy ¼

r

yx<sup>3</sup> <sup>¼</sup> <sup>x</sup>1x<sup>2</sup>

) so that each time their respective values are added with their uncertainties, as

<sup>x</sup><sup>3</sup> <sup>þ</sup> ux<sup>3</sup> ð Þ<sup>2</sup> (15)

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873

�, according to Eqs. (16)–(18)

� (16)

� (17)

� (18)

,

17

(13)

(14)

(19)

measurement process.

3.4.2. The Kragten method

yx<sup>2</sup> and yx<sup>3</sup>

shown in Eqs. (13)–(15)

occur in decimals that can be ignored.

Figure 3. A table and a graph representing the variation of the oxidation time of a biofuel sample as a function of temperature.

Example: On returning to the torque measurement example, assuming that all the input quantities are independent, the combined standard uncertainty for the torque is calculated by using the LPU (Eq. (11)). The final expression is then

$$u\_T = \sqrt{\left(\frac{\partial T}{\partial m}\right)^2 u\_{m\_k}^2 + \left(\frac{\partial T}{\partial m}\right)^2 u\_{m\_c}^2 + \left(\frac{\partial T}{\partial g}\right)^2 u\_g^2 + \left(\frac{\partial T}{\partial L}\right)^2 u\_L^2} = 0.096 \text{ N m} \tag{12}$$

It is important to note that the terms (not squared) of Eq. (12), that is, each sensitivity coefficient multiplied by its corresponding uncertainty, are known as uncertainty components. These components can be compared to each other as they are in the same units of the measurand. Figure 4 shows the comparison between the uncertainty components for the torque measurement model.

As can be noted, the dominant uncertainty component is due to the uncertainty associated with the measurement of the arm length, which was taken as the resolution of the non-calibrated

Figure 4. Uncertainty component balance for the input quantities in the torque measurement model.

ruler used in the measurement. This analysis shows to the analyst that, to reduce the final uncertainty and improve the measurement system, a calibrated ruler, with a better uncertainty, should be used. This represents the importance of the GUM as a management tool to the measurement process.

#### 3.4.2. The Kragten method

Example: On returning to the torque measurement example, assuming that all the input quantities are independent, the combined standard uncertainty for the torque is calculated by

Figure 3. A table and a graph representing the variation of the oxidation time of a biofuel sample as a function of

It is important to note that the terms (not squared) of Eq. (12), that is, each sensitivity coefficient multiplied by its corresponding uncertainty, are known as uncertainty components. These components can be compared to each other as they are in the same units of the measurand. Figure 4 shows the comparison between the uncertainty components for the

As can be noted, the dominant uncertainty component is due to the uncertainty associated with the measurement of the arm length, which was taken as the resolution of the non-calibrated

∂T ∂g � �<sup>2</sup>

u2 <sup>g</sup> þ

∂T ∂L � �<sup>2</sup>

u2 L ¼ 0:096 N m (12)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2 mC þ

∂T ∂m � �<sup>2</sup>

Figure 4. Uncertainty component balance for the input quantities in the torque measurement model.

using the LPU (Eq. (11)). The final expression is then

u2 mR þ

∂T ∂m � �<sup>2</sup>

s

uT ¼

temperature.

16 Metrology

torque measurement model.

The Kragten method is an approximation that facilitates the calculation of the combined uncertainty using finite differences in place of the derivatives [13]. This approximation is valid when the uncertainties of the inputs are relatively small compared to the respective values of the input quantities, generating discrepancies in the final result in relation to the LPU that occur in decimals that can be ignored.

Assuming a measurand y, which is calculated from the input quantities x1, x<sup>2</sup> and x<sup>3</sup> according to the mathematical model of Eq. (2), the uncertainties ux<sup>1</sup> , ux<sup>2</sup> and ux<sup>3</sup> for the input quantities are evaluated normally, according to methodologies already explained previously. From there, the calculations of the measurand are performed individually for each input magnitude (yx<sup>1</sup> , yx<sup>2</sup> and yx<sup>3</sup> ) so that each time their respective values are added with their uncertainties, as shown in Eqs. (13)–(15)

$$y\_{\mathbf{x}\_1} = \frac{(\mathbf{x}\_1 + \mathbf{u}\_{\mathbf{x}\_1})\mathbf{x}\_2}{\mathbf{x}\_3^2} \tag{13}$$

$$y\_{x\_2} = \frac{\mathbf{x}\_1(\mathbf{x\_2} + \mathbf{u\_{x\_2}})}{\mathbf{x\_3^2}} \tag{14}$$

$$y\_{\mathbf{x}\_3} = \frac{\mathbf{x}\_1 \mathbf{x}\_2}{\left(\mathbf{x}\_3 + \mathbf{u}\_{\mathbf{x}\_3}\right)^2} \tag{15}$$

The value of the measurand y varies for yxi due to the addition of the uncertainty uxi to the value of its respective input quantity. Thus, the uncertainty component of each input source in the unit of the measurand y is defined by the difference yxi � y � � � � � �, according to Eqs. (16)–(18)

$$
\mu\_y(\mathbf{x}\_1) = \left| y\_{x\_1} - y \right| \tag{16}
$$

$$\mu\_{\mathcal{Y}}(\mathbf{x}\_2) = \left| y\_{x\_2} - y \right| \tag{17}$$

$$
\mu\_y(\mathbf{x}\_3) = \left| y\_{x\_3} - y \right| \tag{18}
$$

Thus, the combined standard uncertainty of y is finally evaluated as

$$
\mu\_{\mathcal{Y}} = \sqrt{\sum\_{i=1}^{N} \mu\_{\mathcal{Y}}^2(\mathbf{x}\_i)} \tag{19}
$$

or by Eq. (20), if there are correlated uncertainties

$$u\_y = \sqrt{\sum\_{i=1}^{N} u\_y^2(\mathbf{x}\_i) + 2\sum\_{i=1}^{N-1} \sum\_{j=i+1}^{N} u\_y(\mathbf{x}\_i) u\_y(\mathbf{x}\_j) r(\mathbf{x}\_i, \mathbf{x}\_j)} \tag{20}$$

where r xi; xj � � is the correlation coefficient between xi and xj.

#### 3.5. Evaluation of the expanded uncertainty

The result provided by Eqs. (10) and (11) corresponds to an interval that contains only one standard deviation (or approx. 68.2% of the measurements for a normal distribution). In order to have a better coverage probability for the result, the GUM approach expands this interval by assuming that the measurand follows the behavior of a Student's t-distribution. An effective degrees of freedom veff for the t-distribution can be obtained by using the Welch-Satterthwaite formula (Eq. (21))

$$\nu\_{\rm eff} = \frac{u\_y^4}{\sum\_{i=1}^N \frac{\left(\frac{\partial y}{\partial x\_i}\right)^4 u\_{x\_i}^4}{\nu\_{x\_i}}} \tag{21}$$

Using t-distribution tables, the coverage factor for this value of υeff and p = 95% is k = 1.96. Therefore, the expanded uncertainty is calculated as U ¼ kuT ¼ 1:96 � 0:096 ¼ 0:2 N m, and the measurement result is expressed as 668.0 � 0.2 N m. The GUM recommends that the final

One of the most valuable tools for the metrologist is the calibration curve. It is widely used in measurement systems on which one cannot directly obtain the property value to be measured of an object. Instead, a response from the system is measured. In this way, a calibration curve is used to correlate the response from the system with well-known property values, usually

With a calibration curve in hands, the property value for a new unknown sample can be directly determined by using the equation for the fitted curve, which is usually adjusted by a linear regression. However, the calibration curve contains errors due to the lack of fit for the experimental data, causing an uncertainty source to arise. Thus, when evaluating the uncertainty for a predicted property value of xo corresponding to a new observation yo (for a new unknown sample, for example), the LPU with correlation terms is applied on the linear regression model in the form of Eq. (24). Eq. (25) represents the model for a predicted value yo

<sup>x</sup><sup>0</sup> <sup>¼</sup> yo � <sup>a</sup>

where a and b are, respectively, the intercept and the slope parameters of the linear regression.

Figure 5. An example of a linear calibration curve for atomic absorption spectroscopy: the absorption signals (instrument

responses) are plotted against the concentrations for a specific analyte.

<sup>b</sup> (24)

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873 19

yo ¼ a þ bx<sup>0</sup> (25)

uncertainty should be expressed with one or at most two significant digits.

corresponding to a new observed value xo, in the case of the inverse process

4. Calibration curve and correlated uncertainties

calibration standards (see Figure 5).

where νxi is the degrees of freedom for the ith input quantity.

The effective degrees of freedom is used to obtain a coverage factor k that depends also of a chosen coverage probability p, which is often 95%. The expanded uncertainty Uy is then evaluated by multiplying the combined standard uncertainty by the coverage factor k that finally expands it to a coverage interval delimited by a t-distribution with a coverage probability p (Eq. (22))

$$
\Delta I\_y = k u\_y \tag{22}
$$

Note: The draft for the new GUM proposal suggests that the final coverage interval cannot be reliably determined if only an expectation y and a standard deviation uy are known, mainly if the final distribution deviates significantly from a normal or a t-distribution. Thus, they propose distribution-free coverage intervals in the form of y � Up, with Up ¼ kpuy: (a) if no information is known about the final distribution, then a coverage interval for the measurand Y for coverage probability of at least p is determined using kp ¼ 1=ð Þ 1 � p 1=2 . If p ¼ 0:95, a coverage interval of y � 4:47uy is evaluated. (b) If it is known that the distribution is unimodal and symmetric about y, then kp ¼ 2= 3 1ð Þ � p <sup>1</sup>=<sup>2</sup> h i and the coverage interval <sup>y</sup> � <sup>2</sup>:98uy would correspond to a coverage probability of at least p ¼ 0:95.

Example: The effective degrees of freedom for the torque measurement example is calculated using Eq. (21). As the number of degrees of freedom for Type B uncertainties is considered infinite, only Type A uncertainties are accounted. In this case,

$$\nu\_{\rm eff} = \frac{\mu\_T^4}{\left(\frac{\rm \vartheta T}{\rm \nu \pi\_R}\right)^4 \nu\_{m\_R}^4} = 6.5 \times 10^7 \tag{23}$$

Using t-distribution tables, the coverage factor for this value of υeff and p = 95% is k = 1.96. Therefore, the expanded uncertainty is calculated as U ¼ kuT ¼ 1:96 � 0:096 ¼ 0:2 N m, and the measurement result is expressed as 668.0 � 0.2 N m. The GUM recommends that the final uncertainty should be expressed with one or at most two significant digits.
