Материал: Крючков Фундаменталс оф Нуцлеар Материалс Пхысицал Протецтион 2011

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	2		N - n		σˆ	2
	2		N - n		σˆ
σ		(x ) =		×			, where
σ		(x ) =		×			, where
			N		n

For the total values we have:

X = N × x; σ ( X ) = N

∑(xi - x )2

σˆ 2 =	i=1		.	(4.61)

		n -1

×σ (4.62) (x).

Here and hereinafter: n is the sample size, N is the universe size, xi is the measurement result for the i–th sample element, x is the mean value of the characteristic of interest for the sample (and for the universe), and X is the total value of the characteristic of interest for the universe.

Cluster sampling. Single-stage sampling. The following formula is used to determine the cluster sample size from variables:

n =	m
	∑Ni,
	i=1		(4.63)
		t 2 × D2 × M
m =			,
		(M -1)ε 2 + t 2 × D2

where D = σ clust is the variance coefficient for the characteristic of

Хclust

interest between clusters (found from expert data or earlier measurement

∑M (X i - X clust )

results); σ clust2 =	i=1		is the intercluster (estimated) variance; М

		М

is the number of clusters in the universe; X i = ∑xij is the total value of the

j=1

characteristic in the i–th cluster; interest (variable) for the

xij is the quantity of the characteristic of j–th container in the i–th cluster;

X clust = ∑X i / M is the true mean value of the characteristic of interest for

i=1

206

		-
all clusters; ε =	x	-	X	is the permissible difference between the mean


		X
		X

(true) value of the characteristic in the universe and the mean sample value; and t is the quantity of the standardized deviation for a normal distribution of probabilities (determined by the probability that the relative difference between the sample estimate and the set value does not exceed ε).

The mean sample value is found by the following formulas:

M × ∑X i

i=1

x =

(4.64)

m × N

M - m

(x )

×σ clust ,

(4.65)

-1

N m

∑(X i -

clust )2

∑X i

M -1

i=1

where σclust

, and X clust

is the mean

m -1

total value of the characteristic in the cluster.

For the total values we have:

X = N ×

;

(4.66)

σ ( X )

= N

×σ (

Here: m is the sample size; M is the number of clusters in the set; X is the estimated total value of the characteristic of interest in the set; xij is the

quantity of the characteristic of interest (variable) for the j–th container in

the i–th cluster; Ni is the number of containers in the cluster i; N = ∑N i is

i=1

the universe size (total number of containers); xi is the measurement result for the i–th sample element, x is the average value of the characteristic of interest for the sample (and for the universe).

Stratified sampling. The mean sample value is determined as follows:

	L	Ni	ni
	= ∑		∑xij ,	(4.67)
x
x		Nni
	i=1	Nni	j=1

207

- n

s2 (

) = ∑

(4.68)

i=1 N

where

(xij

j )2

∑

si2 =

j=1

–

(4.69)

-1

is the variance inside the stratum, and

∑xij

i =

i=1

–

(4.70)

is the average value of the characteristic of interest in the i–th stratum. For the total values we have:

number of strata in the set; X is the estimated total value of the characteristic of interest in the set; xij is the quantity of the characteristic of interest (variable) for the j–th container in the i–th stratum; Ni is the number

of containers in the stratum i, N = ∑N i is the universe size (total number

i=1

of containers); x is the mean value of the characteristic of interest for the sample (and for the universe).

The cluster sample size determined from variables as required to ensure the specified accuracy level has been estimated from the following formula:

				L	2	)
				ε 2 = t 2 ∑Ni2	σ i (Ni - ni	)	,	(4.72)
				ε 2 = t 2 ∑Ni2	ni (Ni -1)		,	(4.72)
				i=1	ni (Ni -1)
		-
where ε =	x	-	X	is the permissible difference between the mean (true)


		X
		X

value of the characteristic in the universe and the mean value of the sample; t is the quantity of the standardized deviation from a normal distribution of probabilities (determined by the probability that the relative difference

208

between the sample estimate and the set value does not exceed ε); and

∑Ni (xij - X i )2

σ i2 =	j=1		is the variance (variation) of the characteristic of

		Ni

interest inside the i–th stratum.

A note should be made that Х and si are not known. Several sets of ni exist that satisfy the condition (4.72). One may determine ni from the requirement of minimizing the variance of the mean sampled and the total value of the characteristic in the sample:

n= ∑ni ,

i=1

where

n =


	N i ×σ i
ni = n	N i ×σ i	,
ni = n	L	,
	L
	∑N i ×σ i
	i =1


t	2 × ∑ N i		×σ i		×	∑N i ×σ i
	L	2	2			L
	i=1 N i -1					i=1			,
				L		2		2	,
				L		2		2
	ε 2 ( X ) + t2 ×			∑			N i	×σ i
	ε 2 ( X ) + t2 ×			∑
				i=1 N i -1

and the respective estimates are substituted for X and si.

4.5. Control and assurance of measurement quality

(4.73)

(4.74)

By quality control we will mean systematic actions one undertakes to ensure adequate operations of a structure, a system or a system component.

The quality of measurements is ensured by two activities: quality control and quality assessment.

1.Quality control includes procedures and actions developed and used to support the needed quality of measurement.

2.Quality assessment includes procedures and actions one undertakes to make sure that the quality control system operates properly.

NM measurement control (MC) is a component of any quality assurance program. MC is a system of procedures to track down and assess sources of errors. MC includes monitoring of instrument operations using standards and estimation of the error instability in particular measurements (this affects the variance of the value obtained and the ID calculation result).

209

Objectives of measurement quality control:

∙acquisition of quantitative data on uncertainties of measurements;

∙support of measurement process invariability;

∙detection and elimination of unusual occurrences.

Functional elements of NM measurement control systems:

∙use of standards;

∙qualification of measurement techniques;

∙interlaboratory comparisons of measurement data;

∙preparation of control charts;

∙calibrations;

∙other experiments to estimate measurement data uncertainties (auxiliary measurements).

Let us define more accurately some of the concepts.

Effect – a factor which does not represent a measurable q uantity and influences measurement results. MC suggests detection and description of effects inherent in the measurement system in use. So an error (the difference between the true and the measured values) is viewed as the result of cumulative effects.

Working standard – a material, a device or an instrument with the va lue known with respect to state standards or metrology systems.

A standard should be representative with respect to anything that affects measurements. It is normally suggested that the standard and the material to be measured have similar dimensions, shapes and chemical compositions. The hierarchy of standards is shown in Fig. 4.4.

ГосударственныйNational/internationalМеждународныйstandardэталон

SecondaryВторичные эталоныstandards

WorkingРабочие эталоныstandards

Fig. 4.4. Hierarchy of standards

A low-level standard is qualified by multiple measurements of its value against a higher-level standard. The mean arithmetic value of the measurement result will determine the value of the low-level standard. The uncertainty of the low-level standard value is obtained by summing up the

210

Источник: https://studfile.net/preview/16708779/

σ ( X ) = N

Here: ni is the number of containers selected from the i–th stratum;






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