**4.3 Illustrative example 1 - Simulation result**

A very important question can be addressed with regard to accuracy: How the measuring system's accuracy depends on the number of measurements for the prescribed measuring system's reliability and measuring system's dependability? We consider the following prescribed values for , :

a. 0.3, 0.5

10 Modern Metrology Concerns

where the sample mean value <sup>a</sup> and sample standard deviation a s are estimated from *n*

n 1 n 1 k z

kz 1

<sup>1</sup> <sup>2</sup>

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

1 1

x x x xx x x 3

2

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

´2 2 2 2 3 4 2 1

 <sup>2</sup> <sup>1</sup> ´2 <sup>2</sup> x x <sup>1</sup> z2 1

21 1 z 2 z 1 z 7z 6 z 14 z 32 3 9 2 405

<sup>2</sup> <sup>2</sup> <sup>x</sup>

 

u ln 1 x

z 5 z 10

2n 12n 

2

1 2

x x

n n 1

<sup>x</sup> x x <sup>2</sup>

1

n 1 

2.30753 0.27061 u

1 0.99229 u 0.04481 u

<sup>1</sup> <sup>2</sup> <sup>2</sup> (1 ) <sup>n</sup> k z

 

<sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> (1 )

a a a,y a a k s (1 ) (26)

(27)

(28)

(29)

(30)

(31)

(32)

PP ks

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

Ghosh [6] defines the next approximation form:

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

or a much simpler approximation form from [3]:

2 The approximation error is not greater than 0.003

<sup>1</sup> 9 z 256 z 433 z

5 3 <sup>2</sup> xxx

z u

We start by the task description

samples according to (16).

Bowker [5] defines:

and for <sup>2</sup>

4860 2

b. 0.05, 0.99

From (26) the measuring system's accuracy is given k sa . For finding the parameter *k* the equations (30), (31) and (32) were used. The Fig. 1 and Fig. 2 show the dependence of the parameter *k* on the number of measurements *n* for cases a) and b) respectively.
