**1. Introduction**

In this chapter, the emphasis is on the analysis of performance parameters, assigned to different system components, functions or processes, together with the methodology of their statistical assessment based on limited measured samples, fulfilling the condition of normal distribution. The practical examples of GNSS (Global Navigation Satellite Systems) applications assessment and certification are presented.

### **2. Definition of performance parameters**

The methodology for the definition and measurement of following individual system parameters is being developed within the frame of complex system's assessment. The basic performance parameters can be defined as follows:

 **Accuracy** is the degree of conformance between system true parameters and its measured values that can be defined as the probability

$$\mathbf{P}\left(\left|\mathbf{p}\_{\mathrm{i}}-\mathbf{p}\_{\mathrm{m},\mathrm{i}}\right|\leq\varepsilon\_{1}\right)\geq\gamma\_{1}\tag{1}$$

that the difference between the required system parameter pi and the measured parameter p will not exceed the value m,i <sup>1</sup> on probability level 1 where this definition is applicable for all N system parameters p ,p ,..,p 12 N .

 **Reliability** is the ability to perform a required function (process) under given conditions for a given time interval that can be defined as the probability

$$\mathbf{P}\left(\left|\vec{\mathbf{v}}\_{\rm t} - \vec{\mathbf{v}}\_{\rm m,t}\right| \leq \varepsilon\_2\right) \geq \gamma\_2, \mathbf{t} \in \{0, \mathbf{T}\}\tag{2}$$

that the difference between required system functions (processes) represented by parameters vt and the vector of measured parameters vm,t will not exceed the value <sup>2</sup> on probability level 2 in each time interval *t* from the interval 0,T .

 **Availability** is the ability to perform required functions (processes) at the initialization (triggering) of the intended operation that can be defined as the probability

$$\mathbf{P}\left(\left|\mathbf{q}\_{i}-\mathbf{q}\_{\mathrm{m},i}\right|\leq\varepsilon\_{3}\right)\geq\chi\_{3}\tag{3}$$

that the difference between the required rate1 of successful performing of the function *i* (process *i*) qi and the measured q will not exceed the value m,i <sup>3</sup> at the probability level <sup>3</sup> .

 **Continuity** is the ability to perform required functions (processes) without nonscheduled interruption during the intended operation that can be defined as the probability

$$P\left(\left|\mathbf{r}\_{i} - \mathbf{r}\_{m,i}\right| \le \varepsilon\_{4}\right) \ge \gamma\_{4} \tag{4}$$

that the difference between the required rate of successful performing of the function *i* (process *i*) without interruption ir and the measured m,i r will not exceed the value 4 at the probability level 4 .

 **Integrity** is the ability to provide timely and valid alerts to the user, when a system must not be used for the intended operation, that can be defined as the probability

$$P\left(\left|\mathbf{S}\_{i} - \mathbf{S}\_{m,i}\right| \leq \mathbf{c}\_{5}\right) \geq \mathbf{y}\_{5} \tag{5}$$

that the difference between the required rate of successful performing of the alert limit (AL) *i* not later than predefined time to alert (TTA) i S and the measured S will not m,i exceed the value 5 on the probability level 5 .

 **Safety** can also be covered among the performance parameters, but the risk analysis and the risk classification must be done beforehand with a knowledge of the system environment and potential risk, and then the safety can be defined as the probability

$$\mathbf{P}\left(\left|\mathbf{W}\_{\mathrm{i}} - \mathbf{W}\_{\mathrm{m},\mathrm{i}}\right| \leq \varepsilon\_{6}\right) \geq \gamma\_{6} \tag{6}$$

that the difference between the required rate of *i* risk situations Wi and the measured ones W will not exceed the value m,i <sup>6</sup> on the probability level 6 .

A substantial part of the system parameters analysis is represented by a decomposition of system parameters into individual sub-systems of the telematic chain. One part of the analysis is the establishment of requirements on individual functions and information linkage so that the whole telematic chain can comply with the above defined system parameters.

The completed decomposition of system parameters will enable the development of a methodology for a follow-up analysis of telematic chains according to various criteria (optimisation of the information transfer between a mobile unit and a processing centre, maximum use of the existing information and telecommunication infrastructure, etc.).

The following communication performance parameters quantify the quality of telecommunication service [16]:

<sup>1</sup> i m,i <sup>Q</sup> <sup>q</sup> <sup>Q</sup> where Qi is the number of successful experiments (successful performing of the function

*i*, successful performing of the process *i*) and Q is the number of all experiments (both successful and unsuccessful).


 **Continuity** is the ability to perform required functions (processes) without nonscheduled interruption during the intended operation that can be defined as the

<sup>3</sup> .

parameters.

<sup>1</sup> i m,i

unsuccessful).

telecommunication service [16]:

probability

the probability level 4 .

exceed the value 5 on the probability level 5 .

that the difference between the required rate1 of successful performing of the function *i* (process *i*) qi and the measured q will not exceed the value m,i <sup>3</sup> at the probability level

that the difference between the required rate of successful performing of the function *i* (process *i*) without interruption ir and the measured m,i r will not exceed the value 4 at

 **Integrity** is the ability to provide timely and valid alerts to the user, when a system must not be used for the intended operation, that can be defined as the probability

PS S i m,i 5 5 (5)

 **Safety** can also be covered among the performance parameters, but the risk analysis and the risk classification must be done beforehand with a knowledge of the system environment and potential risk, and then the safety can be defined as the probability

that the difference between the required rate of successful performing of the alert limit (AL) *i* not later than predefined time to alert (TTA) i S and the measured S will not m,i

that the difference between the required rate of *i* risk situations Wi and the measured

A substantial part of the system parameters analysis is represented by a decomposition of system parameters into individual sub-systems of the telematic chain. One part of the analysis is the establishment of requirements on individual functions and information linkage so that the whole telematic chain can comply with the above defined system

The completed decomposition of system parameters will enable the development of a methodology for a follow-up analysis of telematic chains according to various criteria (optimisation of the information transfer between a mobile unit and a processing centre, maximum use of the existing information and telecommunication infrastructure, etc.).

The following communication performance parameters quantify the quality of

<sup>Q</sup> <sup>q</sup> <sup>Q</sup> where Qi is the number of successful experiments (successful performing of the function *i*, successful performing of the process *i*) and Q is the number of all experiments (both successful and

ones W will not exceed the value m,i <sup>6</sup> on the probability level 6 .

Pr r i m,i 4 4 (4)

PW W i m,i 6 6 (6)

Performance indicators described for communications applications must be transformed into telematic performance indicators structure, and vice versa. Such transformation allows for a system synthesis.

Transformation matrix construction is dependent on detailed communication solution and its integration into telematic system. Probability of each phenomena appearance in the context of other processes is not deeply evaluated in the introductory period. Each telematic element is consequently evaluated in several steps, based on a detailed analysis of the particular telematic and communications configuration and its appearance probability in the context of the whole system performance. This approach represents a subsequent iterative process, managed with the goal of reaching the stage where all minor indicators (relations) are eliminated, and the major indicators are identified under the condition that relevant telematic performance indicators are kept within a given tolerance range.

#### **3. Quality of measured performance parameters**

In this chapter, unified approach applicable for all above mentioned performance parameters [18] will be introduced.

 **Absolute measuring error (** <sup>a</sup> **)** is the difference between a measured value and the real value or the accepted reference

$$
\boldsymbol{\mu}\_{\rm a} = \mathbf{x}\_{\rm d} - \mathbf{x}\_{s} \tag{7}
$$

<sup>d</sup> x - measured dynamic value


$$
\mu\_r = \frac{\mathbf{x}\_d - \mathbf{x}\_s}{\mathbf{x}\_s} \tag{8}
$$

 **Accuracy (δ) of a measuring system** is the range around the real value in which the actual measured value must lie. The measurement system is said to have accuracy δ if:

$$
\mathbf{x}\_s - \delta \le \mathbf{x}\_d \le \mathbf{x}\_s + \delta \tag{9}
$$

or straightforwardly:

$$-\delta \le \mu\_{\mathfrak{a}} \le +\delta \tag{10}$$

Accuracy is often expressed as a relative value in ± %.

 **Reliability (1-α) of a measuring system** is the minimal probability of a chance that a measuring error <sup>a</sup> lies within the accuracy interval , :

$$\mathbf{P}\left(1-\alpha\right)\leq\mathbf{P}\left(\left|\mu\_{\mathbf{a}}\right|\leq\delta\right)\tag{11}$$

where P(.) means the probability value.

 **Error probability (α) of a measuring system** is the probability that a measured value lies further from the actual value than the accuracy:

$$\alpha \ge \mathbf{P}\left( \left| \mu\_{\mathbf{a}} \right| > \delta \right) \tag{12}$$

The reliability of measuring system is often controlled by the end-user of the measurement system while error probability is generally assessed by the International Organization for Legal Metrology (OIML).

 **Dependability (β) of an acceptance test** is the probability that - on the basis of the sample - a correct judgment is given on the accuracy and reliability of the tested system:

$$\Pr(\mathfrak{a} \le \mathbf{P}(-\delta < \mu\_{\mathfrak{a}} < \delta)) \ge \beta \tag{13}$$

The desired dependability determines the size of the sample; the higher the sample, the higher the dependability of the judgment.

## **4. Estimation of performance parameters**

#### **4.1 Tests of normality**

With regard to [12] normal distribution will be expected, because using different kinds of statistics, such as order statistics (distribution independent) for small sample sizes, typical for performance parameters, the result may be fairly imprecise. Testing normality is important in the performance parameters procedure, because in analyses containing a lot of data this data is required to be at least approximately normally distributed. Furthermore, the confidence limits assessment requires the assumption of normality. Several kinds of normality tests are available, such as [1]:


All the above mentioned tests for normality are based on the *empirical distribution function* (EDF) and are often referred to as *EDF tests*. The empirical distribution function is defined for a set of *n* independent observations X ,X ,...,X 12 n with a common distribution function F(x). Under the null hypothesis, F(x) is the normal distribution. Denote the observations ordered from the smallest to the largest as X ,X ,..,X . The empirical distribution (1) (2) (n) function, F (x) <sup>n</sup> , is defined as

$$\begin{aligned} \mathbf{F\_0(x)} &= 0, & \mathbf{x} < \mathbf{X\_{(1)}}\\ \mathbf{F\_i(x)} &= \frac{\mathbf{i}}{\mathbf{n}}, & \mathbf{X\_{(i)}} \le \mathbf{x} < \mathbf{X\_{(i+1)}}, & \mathbf{i} = \mathbf{1}, \dots, \mathbf{n-1} \\ \mathbf{F\_n(x)} &= \mathbf{1}, & \mathbf{X\_{(n)}} \le \mathbf{x} \end{aligned} \tag{14}$$

**Reliability (1-α) of a measuring system** is the minimal probability of a chance that a

**Error probability (α) of a measuring system** is the probability that a measured value

The reliability of measuring system is often controlled by the end-user of the measurement system while error probability is generally assessed by the International

The desired dependability determines the size of the sample; the higher the sample, the

With regard to [12] normal distribution will be expected, because using different kinds of statistics, such as order statistics (distribution independent) for small sample sizes, typical for performance parameters, the result may be fairly imprecise. Testing normality is important in the performance parameters procedure, because in analyses containing a lot of data this data is required to be at least approximately normally distributed. Furthermore, the confidence limits assessment requires the assumption of normality. Several kinds of

All the above mentioned tests for normality are based on the *empirical distribution function* (EDF) and are often referred to as *EDF tests*. The empirical distribution function is defined for a set of *n* independent observations X ,X ,...,X 12 n with a common distribution function F(x). Under the null hypothesis, F(x) is the normal distribution. Denote the observations ordered from the smallest to the largest as X ,X ,..,X . The empirical distribution (1) (2) (n)

<sup>i</sup> F (x) , X x X , i 1,....,n-1

(14)

0 (1)

F (x) 0, x X

n (n)

n F (x) 1, X x

i (i) (i 1)

 **Dependability (β) of an acceptance test** is the probability that - on the basis of the sample - a correct judgment is given on the accuracy and reliability of the tested system:

<sup>a</sup> (1 ) P (11)

P <sup>a</sup> (12)

P( P( )) <sup>a</sup> (13)

measuring error <sup>a</sup> lies within the accuracy interval , :

where P(.) means the probability value.

Organization for Legal Metrology (OIML).

higher the dependability of the judgment.

**4. Estimation of performance parameters** 

normality tests are available, such as [1]:

Kolmogorov-Smirnov test

function, F (x) <sup>n</sup> , is defined as

Pearson test (Chi-Square Goodness-of-Fit Test)

Anderson-Darling and Cramer-von Mises test

**4.1 Tests of normality** 

lies further from the actual value than the accuracy:

Note that F (x) n is a step function that takes a step of height *1/n* at each observation. This function estimates the distribution function F(x). At any value x, F (x) <sup>n</sup> is the proportion of observations less than or equal to x, while F(x) is the probability of an observation less than or equal to x. *EDF statistics* measure the discrepancy between F (x) <sup>n</sup> and F(x).

In the following part the *Pearson test* (Chi-Square Goodness-of-Fit Test) will be introduced as a practical example of EDF tests. The chi-square goodness-of-fit statistic <sup>2</sup> q for a fitted parametric distribution is computed as follows:

$$\chi^2\_{\mathbf{q}} = \sum\_{i=1}^{L} \frac{\left(\mathbf{m}\_i - \mathbf{n} \cdot \mathbf{p}\_i\right)^2}{\mathbf{n} \cdot \mathbf{p}\_i} \tag{15}$$

where L is the number of histogram intervals, mi is the observed percentage in *i*-th histogram interval, *n* is the number of observations, pi is the probability of *i*-th histogram interval computed by means of theoretical distribution. The degree of freedom for the chisquare test <sup>2</sup> is equal to L-r-1, where r is parameters number of theoretical distribution (in case of normal distribution r=2).

#### **4.2 Estimation of measuring system's accuracy, reliability and dependability**

Let us assume we have a normally distributed set of *n* measurements of performance parameters a,1 a,2 a,n , ,..., (absolute error between prescribed and measured parameters as defined in (7)).

If the mean value or a standard deviation is not known we can estimate both the mean value a and standard deviation a s from the measured data as follows:

$$\begin{aligned} \overline{\mu}\_{\text{a}} &= \frac{1}{n} \sum\_{i=1}^{n} \mu\_{\text{a},i} \\ \mathbf{s}\_{\text{a}} &= \sqrt{\frac{1}{n-1} \sum\_{i=1}^{n} \left(\mu\_{\text{a},i} - \overline{\mu}\_{\text{a}}\right)^{2}} \end{aligned} \tag{16}$$

Let *n* be non-negative integer, , are given real numbers 0,1 and let a,1 a,2 a,n a,y , ,..., , be *n+1* independent identically distributed random variables.

*Tolerance limits* L L , ,..., a,1 a,2 a,n and U U , ,..., a,1 a,2 a,n are defined as values so that the probability is equal to that the limits include at least a proportion 1 of the population. It means that such limits L and U satisfy:

$$P\left\{\mathbf{P}\left(\mathbf{L} < \mu\_{\mathbf{a},\mathbf{y}} < \mathbf{U}\right) \ge 1 - \alpha\right\} = \mathfrak{B} \tag{17}$$

A *confidence interval* covers population parameters with a stated confidence. The *tolerance interval* covers a fixed proportion of the population with a stated confidence. *Confidence limits* are limits within which we expect a given population parameter, such as the mean, to lie. *Statistical tolerance limits* are limits which we expect a stated proportion of the population to lie within.

For the purpose of this chapter we will present only results derived under the following assumptions:


Under the above given assumptions, the condition (17) can be rewritten as follows:

$$\mathbf{P}\left\{\Phi\left(\frac{\mathbf{U}-\mu\_0}{\sigma\_0}\right)-\Phi\left(\frac{\mathbf{L}-\mu\_0}{\sigma\_0}\right)\geq\mathbf{1}-\alpha\right\}=\mathfrak{P}\tag{18}$$

where is the distribution function of the normal distribution with mean zero and standard deviation equal to one:

$$\Phi(\mathbf{u}) = \frac{1}{\sqrt{2 \cdot \pi}} \int\_{-\phi}^{\mathbf{u}} \mathbf{e}^{-\frac{1}{2}\mathbf{t}^2} \,\mathrm{d}\mathbf{t} \tag{19}$$

The solution of the problem to construct tolerance limits depend on the level of knowledge of the normal distribution, i.e., on the level of knowledge of mean deviation <sup>a</sup> and standard deviation a s .

In the following part the *accuracy, reliability and dependability of the measuring system* will be mathematically derived for a known mean value and standard deviation, for a known mean value and unknown standard deviation, and for both an unknown mean value and standard deviation.

#### **Known mean value and standard deviation**

We can start with the equation [3]:

$$\mathbb{P}\left\{\mathbb{P}\left[\boldsymbol{\mu}\_{0} - \mathbf{z}\_{\left(1-\alpha\right\rangle\_{2}}\right] \cdot \sigma\_{0} \le \mu\_{\mathrm{a},\mathrm{y}} \le \mu\_{0} + \mathrm{z}\_{\left(1-\alpha\right\rangle\_{2}} \cdot \sigma\_{0}\right] \ge \left(1-\alpha\right)\right\} = 1\tag{20}$$

where a,y is the measured value, 0 0 , are known mean value and standard deviation and (1 ) <sup>2</sup> z is a percentile of normal distribution (e.g. for 0.05 we can find in statistical table 0.975 z 1.96 ) .

Based on (20) we can decide that measuring system's accuracy 0 (1 ) <sup>2</sup> z is guarantied with measuring system's reliability 1 . Because the mean value and standard deviation are known, the measuring system's dependability is equal to 1 .

#### **Known standard deviation and unknown mean value**

8 Modern Metrology Concerns

For the purpose of this chapter we will present only results derived under the following

a,1 a,2 a,n a,y , ,..., , are *n+1* independent normally distributed random variables

 The tolerance limits are restricted to the simple form a a k s and a a k s , where *k* is a so called *tolerance factor*, <sup>a</sup> and a s are *sample mean and sample standard deviations*,

> 0 0 0 0

where is the distribution function of the normal distribution with mean zero and

 <sup>u</sup> <sup>1</sup> <sup>2</sup> <sup>t</sup> <sup>2</sup> 1 u e dt 2

The solution of the problem to construct tolerance limits depend on the level of knowledge of the normal distribution, i.e., on the level of knowledge of mean deviation <sup>a</sup> and standard

In the following part the *accuracy, reliability and dependability of the measuring system* will be mathematically derived for a known mean value and standard deviation, for a known mean value and unknown standard deviation, and for both an unknown mean value and standard

> 0 0a,y0 0 (1 ) (1 ) 2 2 PP z z (1 ) 1

Based on (20) we can decide that measuring system's accuracy 0 (1 ) <sup>2</sup>

are known, the measuring system's dependability is equal to 1 .

where a,y is the measured value, 0 0 , are known mean value and standard deviation and

z is a percentile of normal distribution (e.g. for 0.05 we can find in statistical table

with measuring system's reliability 1 . Because the mean value and standard deviation

(20)

U L P 1 (18)

Under the above given assumptions, the condition (17) can be rewritten as follows:

random sample of size *n+1* from the normal distribution with mean 0 and variance

<sup>2</sup> (equivalently a,1 a,2 a,n a,y , ,..., , is a

(19)

z is guarantied

with the same mean 0 and variance 0

respectively, given by (16).

standard deviation equal to one:

**Known mean value and standard deviation** 

We can start with the equation [3]:

The symmetry about the mean or its estimation is required.

assumptions:

0 <sup>2</sup> ).

deviation a s .

deviation.

(1 ) <sup>2</sup>

0.975 z 1.96 ) .

Now we expect that the mean value is estimated according to (16). Then we can write the equation [3]:

$$\mathbb{P}\left\{\mathbb{P}\left[\overline{\mu}\_{\mathsf{a}} - \mathsf{k} \cdot \boldsymbol{\sigma}\_{0} \leq \mu\_{\mathsf{a},\mathsf{y}} \leq \overline{\mu}\_{\mathsf{a}} + \mathsf{k} \cdot \boldsymbol{\sigma}\_{0} \right] \geq \left(1 - \alpha\right)\right\} = \emptyset \tag{21}$$

where <sup>2</sup> <sup>0</sup> is the known variance and *k* is computed from the following equation:

$$\Phi\left(\frac{\mathbf{z}\_{\{1+\emptyset\}}}{\sqrt{\mathbf{n}}}+\mathbf{k}\right)-\Phi\left(\frac{\mathbf{z}\_{\{1+\emptyset\}}}{\sqrt{\mathbf{n}}}-\mathbf{k}\right)=\mathbf{1}-\alpha\tag{22}$$

where the function u was defined in (19) and sample <sup>a</sup> computed according to (16).

Based on the equation (22) we can say that for the predefined values of measuring system's reliability 1 and dependability and the number of measurements *n* the accuracy of measuring system will be

$$\delta \mathcal{S} = \left( \mathbf{z}\_{1 - \frac{\alpha}{2}} + \frac{\mathbf{1}}{\sqrt{\mathbf{n}}} \cdot \mathbf{z}\_{\frac{(1+\beta)}{2}} \right) \cdot \boldsymbol{\sigma}\_0 \tag{23}$$

#### **Known mean value and unknown standard deviation**

For a known mean value and unknown standard deviation we can write the equation:

$$\mathbf{P}\left[\mathbf{P}\left[\boldsymbol{\mu}\_{0} - \left(\mathbf{z}\_{\left(1-\frac{\mathbf{a}}{2}\right)} \cdot \left(\frac{\mathbf{n}}{\chi^{2}\_{\left(1-\emptyset\right)}\left(\mathbf{n}\right)}\right)^{\frac{1}{2}}\right) \cdot \mathbf{s}\_{\mathbf{a}} \leq \mu\_{\mathbf{a},\mathbf{y}} \leq \mu\_{0} + \left(\mathbf{z}\_{\left(1-\frac{\mathbf{a}}{2}\right)} \cdot \left(\frac{\mathbf{n}}{\chi^{2}\_{\left(1-\emptyset\right)}\left(\mathbf{n}\right)}\right)^{\frac{1}{2}}\right) \cdot \mathbf{s}\_{\mathbf{a}}\right] \geq \left(1-\alpha\right) = \boldsymbol{\beta} \quad \text{(24)}$$

where a s is estimated according to (16), <sup>2</sup> (1 )(n) means chi-quadrate distribution with *n* degree of freedom.

Based on the equation (24) we can say that for predefined values of measuring system's reliability 1 and dependability and the number of measurements *n* the accuracy of measuring system will be:

$$\delta = \left( \mathbf{z}\_{(1-\frac{\alpha}{2})} \cdot \left( \frac{\mathbf{n}}{\chi^2\_{(1-\beta)}(\mathbf{n})} \right)^{\frac{1}{2}} \right) \cdot \mathbf{s}\_{\mathbf{a}} \tag{25}$$

#### **Unknown mean value and standard deviation**

This variant is the most important in many practical cases, but the solution is theoretically very difficult. However, a lot of approximation forms exist based on which the practical simulation could be feasible.

We start by the task description

$$\mathbf{P}\left(\mathbf{P}\left[\overline{\mu}\_{\text{a}} - \mathbf{k} \cdot \mathbf{s}\_{\text{a}} \le \mu\_{\text{a},\text{y}} \le \overline{\mu}\_{\text{a}} + \mathbf{k} \cdot \mathbf{s}\_{\text{a}}\right] \ge \left(1 - \alpha\right)\right) = \mathbf{\beta} \tag{26}$$

where the sample mean value <sup>a</sup> and sample standard deviation a s are estimated from *n* samples according to (16).

Howe [4] defines a very simple approximation form for *k*:

$$\mathbf{k} \approx \left(\frac{\mathbf{n} + 1}{\mathbf{n}}\right)^{\frac{1}{2}} \cdot \mathbf{z}\_{\left(1 - q\frac{Q}{2}\right)} \cdot \left(\frac{\mathbf{n} - 1}{\chi^2\_{\left(1 - \beta\right)}\left(\mathbf{n} - 1\right)}\right)^{\frac{1}{2}}\tag{27}$$

Bowker [5] defines:

$$\mathbf{k} \approx \mathbf{z}\_{\left(1-\alpha\_{\beta}^{\prime}\right)} \cdot \left[ 1 + \frac{\mathbf{z}\_{\beta}}{\sqrt{2\mathbf{n}}} + \frac{\mathbf{5} \cdot \mathbf{z}\_{\beta}^{2} + 10}{12\mathbf{n}} \right] \tag{28}$$

Ghosh [6] defines the next approximation form:

$$\mathbf{k} \approx \mathbf{z}\_{\left(1-\alpha\frac{\alpha}{2}\right)} \cdot \left(\frac{\mathbf{n}}{\chi^2\_{\left(1-\beta\right)}\left(\mathbf{n}-1\right)}\right)^{\frac{1}{2}}\tag{29}$$

If we take the approximation forms for x z for x>0.5 [2]2:

$$\begin{aligned} \mathbf{z}\_{\mathbf{x}} &= \mathbf{u}\_{\mathbf{x}} - \frac{2.30753 + 0.27061 \cdot \mathbf{u}\_{\mathbf{x}}}{1 + 0.99229 \cdot \mathbf{u}\_{\mathbf{x}} + 0.04481 \cdot \mathbf{u}\_{\mathbf{x}}^2} \\ \mathbf{u}\_{\mathbf{x}} &= \left[ \ln \left( 1 - \mathbf{x} \right)^{-2} \right]^{\frac{1}{2}} \end{aligned} \tag{30}$$

and for <sup>2</sup> <sup>x</sup> ( ) [3]3 (the number of degree of freedom is usually n 1 ):

$$\begin{split} \chi\_{\mathbf{x}}^{\gamma}(\mathbf{y}) &= \mathbf{y} + \mathbf{z}\_{\mathbf{x}} \cdot \sqrt{2} \cdot \boldsymbol{\gamma}^{\frac{1}{2}} + \frac{2}{3} \cdot \left( \mathbf{z}\_{\mathbf{x}}^{2} - 1 \right) + \frac{1}{9\sqrt{2}} \Big( \mathbf{z}\_{\mathbf{x}}^{3} - 7 \cdot \mathbf{z}\_{\mathbf{x}} \Big) \cdot \boldsymbol{\gamma}^{-\frac{1}{2}} - \frac{1}{405} \Big( 6 \cdot \mathbf{z}\_{\mathbf{x}}^{4} + 14 \cdot \mathbf{z}\_{\mathbf{x}}^{2} - 32 \Big) \cdot \boldsymbol{\gamma}^{-1} + \\ &+ \frac{1}{4860\sqrt{2}} \cdot \left( 9 \cdot \mathbf{z}\_{\mathbf{x}}^{5} + 256 \cdot \mathbf{z}\_{\mathbf{x}}^{3} - 433 \cdot \mathbf{z}\_{\mathbf{x}} \Big) \cdot \boldsymbol{\gamma}^{-\frac{3}{2}} \Big) \end{split} \tag{31}$$

or a much simpler approximation form from [3]:

$$\chi\_{\infty}^{\gamma^2}(\gamma) = \frac{1}{2} \cdot \left[ \mathbf{z}\_{\infty} + (\mathbf{2} \cdot \gamma - 1)^{\frac{1}{2}\zeta} \right]^2 \tag{32}$$

<sup>2</sup> The approximation error is not greater than 0.003

<sup>3</sup> For x 0.01,0.99 and 20 the absolute error of approximation is not greater than 0.001

then the analytical equation for the estimation of measuring system's accuracy based on an estimated mean value and standard deviation of *n*-sample data with the predefined measuring system's reliability 1 and measuring system's dependability can be computed.
