**4. Evaluating and expressing uncertainty**

Accurate measurements and associated uncertainty propagation are the backbone of science and industry [20]. Measurements have been the cornerstone of the quantitative sciences since antiquity. However, concepts, terms, units and methods for expressing measurement results [21] and their uncertainties are still contested despite extensive and successful attempts at international consensus resulting in the International Vocabulary of Metrology (VIM) and Guide to the Expression of Uncertainty in Measurement (GUM) more than a decade ago [22–26]. The philosophy of measurement also continues to be a dynamic field of enquiry [27–30] rekindled since the early 2000s [31–34] when the Bureau International des Poids et Mesures (BIPM) began to engage in chemical measurements in addition to physical measurements.

The concept of uncertainty as a quantifiable attribute is relatively new in the history of measurement, although error and error analysis have long been a part of the practice of measurement science or metrology. It is now widely recognized that, when all of the known or suspected components of error have been evaluated and the appropriate corrections have been applied, there still remains an uncertainty about the correctness of the stated result, that is, a doubt about how well the result of the measurement represents the value of the quantity being measured. The uncertainty of the result of the measurement reflects the lack of exact knowledge of the value of the measurand. The result of a measurement after correction for recognized systematic effects is still only an estimate of the value of the measurand because of the uncertainty arising from random effects and from imperfect correction of the result for systematic effects.

The ideal method for evaluating and expressing the uncertainty of the result of a measurement should be applicable to all kinds of measurements and to all types of input data used in measurements. Also, the actual quantity used to express uncertainty should be directly derivable from the components that contribute to it. A measurement is a set of operations having the object of governing values of a particular quantity called the measurand. In general, the result of a measurement is only an estimate of the value of the measurand and thus is complete only when accompanied by a statement of the uncertainty of that estimate [35]. In general, a measurement has imperfections that give rise to an error in the measurement result. Traditionally, an error is viewed as having a random component and a systematic component. Random error presumably arises from unpredictable variations of influence quantities. It is not possible to compensate for the random error of a measurement, but increasing the number of observations can usually reduce it. A systematic error arises from a recognized effect of an influence quantity on a measurement; it can be quantified and a correction can be applied to compensate for the effect. According to GUM [35], it is assumed that the result of a measurement has been corrected for all recognized significant systematic effects and that every effort has been made to identify such effect. Uncertainty components are grouped into two categories based on their method of evaluation "A" and "B." Both types are based on probability distributions, and the uncertainty resulting from either type is quantified by variances or standard deviations. Type A standard uncertainty is calculated from series of repeated observations and is the square root of the statistically estimated variance (i.e., the estimated standard deviation). Type B standard uncertainty is also the square root of an estimated variance, but rather than being evaluated by repeated measurement, it is obtained from an assumed probability density function based on the degree of belief that an event will occur. This degree of belief is usually based on a pool of comparatively reliable information such as previous measurement data, experience, manufacturer's specifications, calibration certificates, and so on. Once all the uncertainty components, either Type A or Type B, have been estimated, they are used to calculate the combined standard uncertainty, which equals the square root of the combined variance obtained from all variance and covariance components using what is termed as the law of propagation of uncertainty. When reporting expanded uncertainty instead of combined standard uncertainty, the multiplying factor k should be stated as well as the approximate level of confidence associated with the interval covered by the expanded uncertainty.

The Joint Committee for Guides in Metrology (JCGM) provides authoritative guidance documents to address measurement needs and is currently developing an expanded Guide to the Expression of Uncertainty in Measurement (GUM) that will provide measurement uncertainty propagation methods for a range of applications. Therefore, a comprehensive set of new worked examples to support modern industrial and research practices and to promote the consistent evaluation of measurement uncertainties are needed for this document [36].

## **4.1. Type A evaluation**

Generally, goods and services are produced by a process that operates under a quality system. Nowadays, people are more conscious about quality more than before. In modern economy, calibration and testing activities play important roles in assuring the quality of goods, services and purchasing decisions. Currently, quality system registration seems to be a popular method of providing assurance of product quality, but it has become quite clear that, for testing and calibration activities, this is not good enough. The internationally recognized standard for the accreditation of laboratories is ISO 17025: General Requirements for the Competence of Testing and Calibration Laboratories [18]. Accreditation is verification that a laboratory has executed a featured system appropriate for its operations. It is verification of measurement uncertainty claims and of traceability to the International System of Units (SI). Accreditation facilitates trade and commerce by eradicating technical barriers to trade. The accreditation of calibration laboratories is particularly important through its impact on international commerce. A final benefit is that an accredited laboratory has been found to perform better in interlaboratory comparisons than unaccredited laboratories, providing additional

Accurate measurements and associated uncertainty propagation are the backbone of science and industry [20]. Measurements have been the cornerstone of the quantitative sciences since antiquity. However, concepts, terms, units and methods for expressing measurement results [21] and their uncertainties are still contested despite extensive and successful attempts at international consensus resulting in the International Vocabulary of Metrology (VIM) and Guide to the Expression of Uncertainty in Measurement (GUM) more than a decade ago [22–26]. The philosophy of measurement also continues to be a dynamic field of enquiry [27–30] rekindled since the early 2000s [31–34] when the Bureau International des Poids et Mesures (BIPM)

began to engage in chemical measurements in addition to physical measurements.

The concept of uncertainty as a quantifiable attribute is relatively new in the history of measurement, although error and error analysis have long been a part of the practice of measurement science or metrology. It is now widely recognized that, when all of the known or suspected components of error have been evaluated and the appropriate corrections have been applied, there still remains an uncertainty about the correctness of the stated result, that is, a doubt about how well the result of the measurement represents the value of the quantity being measured. The uncertainty of the result of the measurement reflects the lack of exact knowledge of the value of the measurand. The result of a measurement after correction for recognized systematic effects is still only an estimate of the value of the measurand because of the uncertainty arising from random effects and from imperfect correction of the result for systematic effects.

The ideal method for evaluating and expressing the uncertainty of the result of a measurement should be applicable to all kinds of measurements and to all types of input data used in measurements. Also, the actual quantity used to express uncertainty should be directly derivable from the components that contribute to it. A measurement is a set of operations having the object of governing values of a particular quantity called the measurand. In general, the result

assurance to users of accredited services [19].

84 Metrology

**4. Evaluating and expressing uncertainty**

In the simplest case (and fortunately the most usual one) of Type A evaluation, the input quantity Xi is treated as a random variable and is reasonably well approximated by the normal distribution [10]. The best estimate of the expected value of the random variable is denoted by xi and is obtained from the arithmetic mean of a series of *n* independent observations obtained under the same conditions of measurement:

$$\|\boldsymbol{x}\_{i}\| = \frac{1}{n} \sum\_{k=1}^{n} \boldsymbol{x}\_{i,k} \tag{7}$$

The individual observations *x<sup>i</sup>*,*<sup>k</sup>* differ in values because of random variations in the influence quantities or random effects. The experimental variance of the observations, which estimates the variance of the probability distribution of Xi , is given by:

$$\left(s^{\mathbb{Z}}\left(\mathbf{x}\_{i,k}\right) = \frac{1}{n-1} \sum\_{k=1}^{n} \left(\mathbf{x}\_{i,k} - \mathbf{x}\_{i}\right)^{\mathbb{Z}} \tag{8}$$

• **Long-term drift of the reference photometers between recalibrations:** Estimated maxi-

Optical Radiation Metrology and Uncertainty http://dx.doi.org/10.5772/intechopen.75205 87

• **Photometer temperature variation:** If the photometer has no temperature controller or

The components of a typical uncertainty budget for total luminous flux measurements are

**Uncertainty factor Type Relative standard uncertainty (%)**

mum drift of the reference photometer between calibrations.

• **Distance scale of the bench (0.5 mm in 3 m)**

• **Spectral mismatch correction**

• **Stray light**

shown in **Table 2** [9].

• **Lamp-current regulation (0.01%)**

• **Random noise (lamp drift, etc.)**

**Relative combined standard uncertainty Relative expanded uncertainty (k = 2)**

• **Alignment of the lamp distance (1 mm in 3 m)**

• **Lamp-current measurement uncertainty (0.01%)**

• **Repeatability of the test lamp (including alignment)**

Calibration of reference photometers **B** Long-term drift of the reference photometers between recalibrations **B** Photometer temperature variation **A** Distance scale of the bench (0.5 mm in 3 m) **B** Alignment of the lamp distance (1 mm in 3 m) **A** Spectral mismatch correction **B** Lamp-current regulation (0.01%) **A** Lamp-current measurement uncertainty (0.01%) **B** Stray light **B** Random noise (lamp drift, etc.) **A** Repeatability of the test lamp (including alignment) **A**

*4.3.2. The typical uncertainty budget for total luminous flux measurements*

**Table 1.** Typical uncertainty budget for luminous intensity calibrations (detector-based method).

temperature sensor, and the laboratory temperature is kept within

This estimate of variance and its positive square root, *S*(*x<sup>i</sup>*,*<sup>k</sup>* ) termed as the experimental standard deviation, characterize the variability of the observed values *x<sup>i</sup>*,*<sup>k</sup>* .

According to statistical theory, the best estimate for the variance of the mean xi is given by:

$$s^{\mathbb{Z}}(\mathbf{x}\_{i}) = \frac{s^{\mathbb{Z}}(\mathbf{x}\_{i,k})}{n} = \frac{1}{n(n-1)}\sum\_{k=1}^{n}(\mathbf{x}\_{i,k} - \mathbf{x}\_{i})^{\mathbb{Z}}\tag{9}$$

The Type A standard uncertainty for that component is then defined as the positive square root of this last quantity:

$$u(\mathbf{x}\_i) = \sqrt{\frac{1}{n(n-1)} \sum\_{k=1}^{n} (\mathbf{x}\_{i,k} - \mathbf{x}\_i)^2} \tag{10}$$

The number of observations *n* should be large enough to ensure that xi provides a reliable estimate of the expectation for Xi and that *u*<sup>2</sup> (*x<sup>i</sup>* ) provides a reliable estimate of the variance of the expectation for Xi . The number of degrees of freedom, defined as *vi* = *n* − 1 should always be given when Type A evaluations of uncertainty components are documented.

#### **4.2. Type B evaluation**

With Type B evaluation, an estimate xi of an input quantity Xi has not been obtained from repeated observations and the associated standard uncertainty is evaluated by scientific judgment based on all of the available information on the possible variability of X<sup>i</sup> [10]. This information may include previous measurement data, experience, manufacturer's specifications, calibration certificates, and so on. Type B evaluation calls for insight based on experience and general knowledge; it is, however, as reliable as Type A evaluation.

#### **4.3. The typical uncertainty budget for measurements**

*4.3.1. The components of a typical uncertainty budget for luminous intensity calibrations (detector-based method) [9]*

The following are the descriptions of the abovementioned uncertainty budget items [9] (**Table 1**):

• **Calibration of reference photometers:** The uncertainty of reference photometer is stated in the calibration report issued by the national laboratory or the calibration laboratory that conducted the calibration.


The individual observations *x<sup>i</sup>*,*<sup>k</sup>*

86 Metrology

root of this last quantity:

the expectation for Xi

**4.2. Type B evaluation**

*(detector-based method) [9]*

conducted the calibration.

estimate of the expectation for Xi

With Type B evaluation, an estimate xi

**4.3. The typical uncertainty budget for measurements**

the variance of the probability distribution of Xi

differ in values because of random variations in the influence

.

(*x<sup>i</sup>* ) provides a reliable estimate of the variance of

. The number of degrees of freedom, defined as *vi* = *n* − 1 should always

of an input quantity Xi

repeated observations and the associated standard uncertainty is evaluated by scientific

information may include previous measurement data, experience, manufacturer's specifications, calibration certificates, and so on. Type B evaluation calls for insight based on experi-

The following are the descriptions of the abovementioned uncertainty budget items [9] (**Table 1**): • **Calibration of reference photometers:** The uncertainty of reference photometer is stated in the calibration report issued by the national laboratory or the calibration laboratory that

(8)

(9)

(10)

[10]. This

is given by:

provides a reliable

has not been obtained from

quantities or random effects. The experimental variance of the observations, which estimates

This estimate of variance and its positive square root, *S*(*x<sup>i</sup>*,*<sup>k</sup>* ) termed as the experimental stan-

The Type A standard uncertainty for that component is then defined as the positive square

dard deviation, characterize the variability of the observed values *x<sup>i</sup>*,*<sup>k</sup>*

According to statistical theory, the best estimate for the variance of the mean xi

The number of observations *n* should be large enough to ensure that xi

and that *u*<sup>2</sup>

be given when Type A evaluations of uncertainty components are documented.

judgment based on all of the available information on the possible variability of X<sup>i</sup>

*4.3.1. The components of a typical uncertainty budget for luminous intensity calibrations* 

ence and general knowledge; it is, however, as reliable as Type A evaluation.

, is given by:


#### *4.3.2. The typical uncertainty budget for total luminous flux measurements*

The components of a typical uncertainty budget for total luminous flux measurements are shown in **Table 2** [9].


**Table 1.** Typical uncertainty budget for luminous intensity calibrations (detector-based method).

The following are the descriptions of the abovementioned uncertainty budget items:

• **Calibration of luminous-flux standard lamps:** The uncertainty of the luminous-flux standard lamps is stated in the calibration report issued by the national laboratory or the calibration laboratory that performed the calibration. This uncertainty normally includes the repeatability of the lamp.

• **Repeatability of test lamps:** Calculated as the standard deviation of the all measurements

Optical Radiation Metrology and Uncertainty http://dx.doi.org/10.5772/intechopen.75205 89

• **Spatial nonuniformity of the sphere response:** This uncertainty is associated with differences of the angular intensity distribution of the test lamps and the standard lamp.

• **Lamp electrical control [10]:** The uncertainty of less than 0.01% in the calibration of voltmeter and standard resistor used for measuring and setting the electrical operating conditions

In this chapter, we concentrate on the measurement of absolute amounts of optical radiation, which requires careful definition for the photometric and radiometric quantities such as total flux, intensity, illuminance, luminance, radiance, exitance and irradiance. Also, it was necessary to distinguish between the difference of the exitance and irradiance quantities in the physical meaning. The metrological traceability chain is the sequence of measurement standards and calibrations that were used to relate the measurement result to the reference. To produce accurate, reproducible and international acceptable results, the measurement of absolute amounts of optical radiation needs careful and detailed consideration of a broad range of physical concepts. A measurement has imperfections that give rise to an error in the measurement result. Therefore, measurement results should be expressed in terms of estimated value and an associated uncertainty. Actually, an error is viewed as having a random component and a systematic component. Random error presumably arises from unpredictable variations of influence quantities and is not possible to compensate for the random error of a measurement, but increasing the number of observations can usually reduce it. We provide an explanation to how to estimate and build the uncertainty budget of measurements for the most important quantities. The components of a typical uncertainty budgets for luminous intensity calibrations (detector-based method) and total luminous flux measurements are rep-

of the lamps may result in 0.08% uncertainty in the lamp output.

resented and explained in detail in this chapter.

**ω<sup>0</sup>** Steradian unit of solid angle

SI unit International system of units

EM electromagnetic radiation

for each lamp.

**5. Conclusion**

**Abbreviations**

**ω** solid angle

*L*<sup>e</sup> radiance

*I*

<sup>Φ</sup>*<sup>e</sup>* radiant flux

*<sup>e</sup>* radiant Intensity


$$SCF = \frac{\int\_{360}^{850nm} P\_{\boldsymbol{\epsilon}}^{\boldsymbol{r}}(\boldsymbol{\lambda}) \times V(\boldsymbol{\lambda}) \, d\boldsymbol{\lambda} \int\_{\boldsymbol{all-wavelength}} P\_{\boldsymbol{\epsilon}}^{\boldsymbol{S}}(\boldsymbol{\lambda}) \times R(\boldsymbol{\lambda}) \, d\boldsymbol{\lambda}}{\int\_{all-\,parallel \,} P\_{\boldsymbol{\epsilon}}^{\boldsymbol{T}}(\boldsymbol{\lambda}) \times R(\boldsymbol{\lambda}) \, d\boldsymbol{\lambda} \int\_{360}^{850nm} P\_{\boldsymbol{\epsilon}}^{\boldsymbol{S}}(\boldsymbol{\lambda}) \times V(\boldsymbol{\lambda}) \, d\boldsymbol{\lambda}} \tag{11}$$

where

is the relative spectral output of the test source;

is the relative spectral output of the standard source;

is the relative spectral responsivity of the photometer; and.

is the spectral luminous efficiency function that defines a photometric measurement.


$$\mu^2 = \sum\_{\text{varitude}} \frac{\partial SCF}{\partial \text{var} \, \text{a} \, \text{b} \, \text{e}^2} \times \mu^2 \, \text{(var} \, \text{a} \, \text{b} \, \text{be)} \tag{12}$$

**Table 2.** Typical uncertainty budget for luminous intensity calibration (source-based method) [9].

