4. Calibration curve and correlated uncertainties

uy ¼

3.5. Evaluation of the expanded uncertainty

where r xi; xj

18 Metrology

ity p (Eq. (22))

X<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>u</sup><sup>2</sup>

<sup>y</sup>ð Þþ xi 2

� � is the correlation coefficient between xi and xj.

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � <sup>r</sup>

X<sup>N</sup>

<sup>j</sup>¼iþ<sup>1</sup> uyð Þ xi uy xj

� �r xi; xj

Uy ¼ kuy (22)

1=2

and the coverage interval y � 2:98uy would

<sup>¼</sup> <sup>6</sup>:<sup>5</sup> � 107 (23)

. If p ¼ 0:95, a

(20)

(21)

X<sup>N</sup>�<sup>1</sup> i¼1

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))

<sup>ν</sup>eff <sup>¼</sup> <sup>u</sup><sup>4</sup>

P<sup>N</sup> i¼1

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 probabil-

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

coverage interval of y � 4:47uy is evaluated. (b) If it is known that the distribution is unimodal

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

> T ∂T ∂mR � �<sup>4</sup> u4 mR νmR

<sup>1</sup>=<sup>2</sup> h i

Y for coverage probability of at least p is determined using kp ¼ 1=ð Þ 1 � p

<sup>ν</sup>eff <sup>¼</sup> <sup>u</sup><sup>4</sup>

and symmetric about y, then kp ¼ 2= 3 1ð Þ � p

correspond to a coverage probability of at least p ¼ 0:95.

infinite, only Type A uncertainties are accounted. In this case,

y

∂y ∂xi � �<sup>4</sup> u4 xi νxi

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 calibration standards (see Figure 5).

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

$$\alpha\_0 = \frac{y\_o - a}{b} \tag{24}$$

$$y\_o = a + b\mathbf{x}\_0\tag{25}$$

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.

The application of the LPU with the correlation term to Eqs. (24) and (25) leads to Eqs. (26) and (27), respectively, for both cases:

$$u\_{x\_o} = \sqrt{\left(\frac{\partial \mathbf{x}\_o}{\partial y\_o}\right)^2 u\_{y\_o}^2 + \left(\frac{\partial \mathbf{x}\_o}{\partial a}\right)^2 u\_a^2 + \left(\frac{\partial \mathbf{x}\_o}{\partial b}\right)^2 u\_b^2 + 2\left(\frac{\partial \mathbf{x}\_o}{\partial a}\right)\left(\frac{\partial \mathbf{x}\_o}{\partial b}\right)\mu\_a u\_b r\_{a,b}}\tag{26}$$

$$
\mu\_{\mathcal{Y}\_o} = \sqrt{\left(\frac{\partial y\_o}{\partial \mathbf{x}\_o}\right)^2 u\_{\mathbf{x}\_o}^2 + \left(\frac{\partial y\_o}{\partial a}\right)^2 u\_a^2 + \left(\frac{\partial y\_o}{\partial b}\right)^2 u\_b^2 + 2\left(\frac{\partial y\_o}{\partial a}\right)\left(\frac{\partial y\_o}{\partial b}\right)\mu\_a u\_b r\_{a,b}}\tag{27}
$$

For Eq. (26), uxo is the combined uncertainty for the predicted value xo and uyo is the uncertainty for the new observed response yo. For Eq. (27), uyo is the combined uncertainty for the predicted value yo and uxo is the uncertainty for the new observed response xo. In both cases, ua and ub are, respectively, the uncertainties for the intercept and the slope, and ra, <sup>b</sup> is the correlation coefficient between a and b. These last equations can also be found in the ISO/TS 28037 [17], concerning the use of straight-line calibration functions.

The uncertainties for a and b can be obtained by Eqs. (28) and (29), respectively, while the correlation coefficient ra, <sup>b</sup> is given by Eq. (30)

$$\mu\_a = S\_\epsilon \sqrt{\frac{\sum x\_i^2}{n\sum x\_i^2 - \left(\sum x\_i\right)^2}}\tag{28}$$

Using Eqs. (28)–(31), it is possible to calculate the values of Table 2 that shows the statistical

Considering that there is no uncertainty for the observed point xo = 22�C, that is, uxo = 0, the uncertainty of yo arising from the interpolation process of the point xo = 22�C can then be calculated by applying Eq. (27) and the data from Table 2, resulting in the following:

Another frequently used expression for the standard uncertainty of the predicted value uxo is

where Se is the residual standard deviation of the fitted line, m is the number of observations of yo, n is the number of points composing the calibration curve, and yo is the average value obtained from the observations of yo. In this expression, the uncertainty component due to the

> uyo <sup>¼</sup> Se ffiffiffiffi

However, Hibbert [19] suggests that if the standard deviation of the indications is known from

1 m þ 1 n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> yo � <sup>y</sup> � �<sup>2</sup> <sup>b</sup><sup>2</sup> <sup>P</sup>ð Þ xi � <sup>x</sup>

2 vuut (32)

¼ 0:021�C.

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873 21

<sup>m</sup><sup>p</sup> (33)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>∙</sup>0:<sup>00842</sup> <sup>þ</sup> <sup>2</sup>∙1∙22∙0:1943∙0:0084∙ð Þ �0:<sup>995</sup> <sup>q</sup>

uxo <sup>¼</sup> Se b

Data Value Unit

<sup>e</sup> 0.0024 �C2 ua 0.1943 �C

data for the thermometer calibration curve.

Table 1. Values of the calibration certificate of a thermometer.

Indication (xi) (�C) Reference value (yi) (�C)

20 20.3 21 21.3 22 22.2 23 23.1 24 24.2 25 25.1 27 27.0

<sup>∙</sup>0:1943<sup>2</sup> <sup>þ</sup> 222

observations of yo is evaluated by [19]

ub 0.0084 ra, <sup>b</sup> �0.995

consistent data, uyo can be better evaluated by

Table 2. Statistical data for the calibration curve of a thermometer.

uyo ¼

S2

12

given by Eq. (32) [13, 18]:

$$\mu\_b = \mathcal{S}\_e \sqrt{\frac{n}{n\sum x\_i^2 - \left(\sum x\_i\right)^2}}\tag{29}$$

$$r\_{a,b} = -\frac{\sum x\_i}{\sqrt{n\sum x\_i^2}}\tag{30}$$

where n is the number of points used to construct the curve, xi are the values for the independent variable of the linear equation for each yi , and S<sup>2</sup> <sup>e</sup> is the residual variance of the fitted curve, obtained by Eq. (31)

$$S\_{\varepsilon}^{2} = \frac{\sum \left( y\_{i} - \widehat{y}\_{i} \right)^{2}}{n - 2} \tag{31}$$

where <sup>b</sup>yi are the interpolated values in the fitted curve for each xi, that is, <sup>b</sup>yi <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bxi.

Example: This time, let us consider that the calibration certificate of a thermometer presents the results shown in Table 1.

For the data shown in Table 1, the calibration curve of the thermometer is expressed by <sup>b</sup>yo <sup>¼</sup> <sup>1</sup>:<sup>1484</sup> <sup>þ</sup> <sup>0</sup>:9578xo. For a temperature value indicated by the thermometer of xo = 22�C, applying the equation of the calibration curve yields a reference value of <sup>b</sup>yo = 22.22�C.


Table 1. Values of the calibration certificate of a thermometer.

The application of the LPU with the correlation term to Eqs. (24) and (25) leads to Eqs. (26) and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∂yo ∂b � �<sup>2</sup>

For Eq. (26), uxo is the combined uncertainty for the predicted value xo and uyo is the uncertainty for the new observed response yo. For Eq. (27), uyo is the combined uncertainty for the predicted value yo and uxo is the uncertainty for the new observed response xo. In both cases, ua and ub are, respectively, the uncertainties for the intercept and the slope, and ra, <sup>b</sup> is the correlation coefficient between a and b. These last equations can also be found in the ISO/TS

The uncertainties for a and b can be obtained by Eqs. (28) and (29), respectively, while the

n Px<sup>2</sup>

n Px<sup>2</sup>

s

s

ra, <sup>b</sup> ¼ �

S2 <sup>e</sup> ¼

where <sup>b</sup>yi are the interpolated values in the fitted curve for each xi, that is, <sup>b</sup>yi <sup>¼</sup> <sup>a</sup> <sup>þ</sup> bxi.

applying the equation of the calibration curve yields a reference value of <sup>b</sup>yo = 22.22�C.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>P</sup>x<sup>2</sup> i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n

Pxi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n Px<sup>2</sup> i

, and S<sup>2</sup>

where n is the number of points used to construct the curve, xi are the values for the indepen-

<sup>P</sup> yi � <sup>b</sup>yi � �<sup>2</sup>

Example: This time, let us consider that the calibration certificate of a thermometer presents

For the data shown in Table 1, the calibration curve of the thermometer is expressed by <sup>b</sup>yo <sup>¼</sup> <sup>1</sup>:<sup>1484</sup> <sup>þ</sup> <sup>0</sup>:9578xo. For a temperature value indicated by the thermometer of xo = 22�C,

<sup>i</sup> � <sup>P</sup>ð Þ xi

<sup>i</sup> � <sup>P</sup>ð Þ xi

2

2

u2

u2 <sup>b</sup> þ 2

<sup>b</sup> <sup>þ</sup> <sup>2</sup> <sup>∂</sup>xo ∂a � � ∂xo

> ∂yo ∂a � � ∂yo

∂b � �

∂b � �

<sup>q</sup> (30)

<sup>n</sup> � <sup>2</sup> (31)

<sup>e</sup> is the residual variance of the fitted

uaubra, <sup>b</sup>

uaubra, <sup>b</sup>

(26)

(27)

(28)

(29)

∂xo ∂b � �<sup>2</sup>

(27), respectively, for both cases:

20 Metrology

uxo ¼

uyo ¼

∂xo ∂yo � �<sup>2</sup>

∂yo ∂xo � �<sup>2</sup>

s

s

correlation coefficient ra, <sup>b</sup> is given by Eq. (30)

dent variable of the linear equation for each yi

curve, obtained by Eq. (31)

the results shown in Table 1.

u2 yo þ

u2 xo þ

28037 [17], concerning the use of straight-line calibration functions.

ua ¼ Se

ub ¼ Se

∂xo ∂a � �<sup>2</sup>

∂yo ∂a � �<sup>2</sup>

u2 <sup>a</sup> þ

u2 <sup>a</sup> þ

> Using Eqs. (28)–(31), it is possible to calculate the values of Table 2 that shows the statistical data for the thermometer calibration curve.

> Considering that there is no uncertainty for the observed point xo = 22�C, that is, uxo = 0, the uncertainty of yo arising from the interpolation process of the point xo = 22�C can then be calculated by applying Eq. (27) and the data from Table 2, resulting in the following: uyo ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 <sup>∙</sup>0:1943<sup>2</sup> <sup>þ</sup> 222 <sup>∙</sup>0:<sup>00842</sup> <sup>þ</sup> <sup>2</sup>∙1∙22∙0:1943∙0:0084∙ð Þ �0:<sup>995</sup> <sup>q</sup> ¼ 0:021�C.

> Another frequently used expression for the standard uncertainty of the predicted value uxo is given by Eq. (32) [13, 18]:

$$\mu\_{x\_o} = \frac{\mathcal{S}\_\varepsilon}{b} \sqrt{\frac{1}{m} + \frac{1}{n} + \frac{\left(\overline{y}\_o - \overline{y}\right)^2}{b^2 \sum \left(\chi\_i - \overline{\chi}\right)^2}}\tag{32}$$

where Se is the residual standard deviation of the fitted line, m is the number of observations of yo, n is the number of points composing the calibration curve, and yo is the average value obtained from the observations of yo. In this expression, the uncertainty component due to the observations of yo is evaluated by [19]

$$
\mu\_{\mathbb{Y}\_o} = \frac{\mathbb{S}\_e}{\sqrt{m}}\tag{33}
$$

However, Hibbert [19] suggests that if the standard deviation of the indications is known from consistent data, uyo can be better evaluated by


Table 2. Statistical data for the calibration curve of a thermometer.

$$
\mu\_{y\_o} = \frac{\mathcal{S}\_{y\_o}}{\sqrt{m}}\tag{34}
$$

uei ¼

values or systematic errors can also be reported.

6.1. Limitations of the GUM approach

same order of magnitude than its quantity.

Table 3. A typical format for the result of calibration of an instrument.

∂ei <sup>∂</sup>Vind � �<sup>2</sup>

6. Monte Carlo simulation applied to metrology

updated with procedures to use Monte Carlo simulations [13].

u2 VindRes þ

These standard uncertainties are obtained as described in Section 3.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2 VindRep þ

∂ei <sup>∂</sup>Vref � �<sup>2</sup>

u2

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

(37)

23

∂ei <sup>∂</sup>Vind � �<sup>2</sup>

where uVindRes , uVindRep , and uVref are, respectively, standard uncertainties due to resolution of the instrument, repeatability of indication values, and certificate of calibration of the reference.

The final calibration result can then be presented according to Table 3. In addition, correction

This section presents the limitations of the GUM and shows an alternative methodology based on the propagation of distributions that overcome those limitations. For further details, please refer to the authors' publication that addresses the use of the Monte Carlo methodology applied to uncertainty in measurement [15] or to the JCGM 101:2008 guide [5]. Also, in the field of analytical chemistry, the latest version of EURACHEM/CITAC guide (2012) was

As mentioned earlier, the approach to evaluate measurement uncertainties using the LPU as presented by the GUM is based on some approximations that are not valid for every measurement model [5, 20–22]. These approximations comprise (1) the linearization of the measurement model made by the truncation of the Taylor series, (2) the use of a t-distribution as the distribution for the measurand, and (3) the calculation of an effective degrees of freedom for the measurement model based on the Welch-Satterthwaite formula, which is still an unsolved problem [23]. Moreover, the GUM approach usually presents deviated results when one or more of the input uncertainties are relatively much larger than others, or when they have the

The limitations and approximations of the LPU are overcome when using a methodology that relies on the propagation of distributions. This methodology carries more information than the simple propagation of uncertainties and generally provides results closer to reality. It is

Range Indicated value Reference value Expanded uncertainty Coverage factor

Range 1 Vind<sup>1</sup> Vref<sup>1</sup> U<sup>1</sup> k<sup>1</sup> Range 2 Vind<sup>2</sup> Vref<sup>2</sup> U<sup>2</sup> k<sup>2</sup> …… … … … Range N VindN VrefN UN kN

Vref <sup>s</sup>

where Syo is the standard deviation of the observations of yo, and Eq. (32) is then expressed as Eq. (35) [18, 19]:

$$\mu\_{\mathbf{x}\_o} = \frac{1}{b} \sqrt{\frac{S\_{y\_o}^2}{m} + \frac{S\_e^2}{n} + \frac{S\_e^2 \left(\overline{y}\_o - \overline{y}\right)^2}{b^2 \sum \left(\mathbf{x}\_i - \overline{\mathbf{x}}\right)^2}}\tag{35}$$

### 5. Assessment of uncertainty in instrument calibration

The methodology presented in the GUM can also be used to evaluate the uncertainty in the calibration of a measuring instrument. Following the steps of the GUM, the measurand for the model used in the calibration must be defined by the quantity that evaluates the systematic error of an instrument over its entire measurement range. Thus, Eq. (36) can be generally used for the evaluation of uncertainty in a calibration process:

$$e = V\_{\rm ind} - V\_{\rm ref} \tag{36}$$

where e is the systematic error of the instrument for a fixed range, Vind is the value indicated by the instrument, and Vref is the reference value corresponding to the indicated value.

From Eq. (36), a basic cause-and-effect diagram can be assembled for the calibration uncertainty assessment of an instrument, as shown in Figure 6.

The sources of uncertainty in this case involve the repeatability of indicated values, the resolution of the instrument in calibration, and the certificate of calibration of the reference values. Thus, an evaluation of the uncertainty about the systematic error should be done for each nominal value of the instrument in calibration. The combined standard uncertainties uei for each calibrated nominal value are obtained by applying the LPU, as shown in Eq. (37)

Figure 6. A general cause-and-effect diagram for the calibration of an instrument.

$$u\_{c\_i} = \sqrt{\left(\frac{\partial \mathbf{e}\_i}{\partial V\_{ind}}\right)^2 u\_{V\_{ind\_{R\alpha}}}^2 + \left(\frac{\partial \mathbf{e}\_i}{\partial V\_{ind}}\right)^2 u\_{V\_{ind\_{Rp}}}^2 + \left(\frac{\partial \mathbf{e}\_i}{\partial V\_{ref}}\right)^2 u\_{V\_{nf}}^2} \tag{37}$$

where uVindRes , uVindRep , and uVref are, respectively, standard uncertainties due to resolution of the instrument, repeatability of indication values, and certificate of calibration of the reference. These standard uncertainties are obtained as described in Section 3.

The final calibration result can then be presented according to Table 3. In addition, correction values or systematic errors can also be reported.
