Choice of sample size for estimating the average quality of a lot or process
Some standard content:
ICS 03. 120. 30
National Standard of the People's Republic of China
GB/T4891--2008
Replaces GB/T4891-1985
Choice of sample size for estimating the average quality of a lot or process
Issued on July 28, 2008
General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China Administration of Standardization of the People's Republic of China
Dual Code Effect
Implemented on January 1, 2009
1Fan Guo
2Normative References
3Terms, Definitions and Symbols·
3. 1 Terms and definitions
3.2 Symbols
General requirements
5 Formula for calculating sample size
6 Calculation of sample size when historical sample data is available 6.1 Use of formula (1)
6.2 Use of formula (2)·
6.3 Use of formula (3)
7 Calculation of sample size when historical sample data is not available 7.1 Use of formula (1)
7.2 Use of formula (2)·
Use of formula (3)
Cost considerations
Sample selection
GB/T 4891--2008
This standard replaces GB/T 4891-1985 "Method for selecting samples for estimating batch (or process) average quality". The main differences between this standard and GB/T4851-1985 are as follows: a) The standard text is redrafted according to the requirements of GB/T1.1-2000; b) Normative references are added: IS03534-1:2006: ISO3534-2:2006: In order to facilitate the application of the standard, relevant terms and definitions are added; e
uses \absolute error limit E(x一|)\ to replace the "precision E(|一)\ in the original standard; GB/T 4891—2008
Use the formula for calculating sample size under the general confidence level of 1-a to replace the special formula under the confidence level of 99.73% in the original standard:
Given that when is very small, after calculating the mountain using formula (3), if, replace the general formula in formula 3); f)
g) Give the expression for directly calculating S when using formula (1), and delete the corresponding Tables 2 and 3 in the original standard; h) Delete Figure 2 in the original standard.
This standard is proposed by the China National Institute of Standardization. This standard is issued by the whole The National Technical Committee for Standardization of Statistical Methods is responsible for this standard. The drafting units of this standard are: Ordnance Engineering College of the Chinese People's Liberation Army, China Institute of Standardization and Application, Institute of Mathematics and Systems Science of the Chinese Academy of Sciences, and Fuzhou Chunlun Tea Co., Ltd. The main drafters of this standard are: Zhang Yuzhu, Yu Zhenfan, Chen Min, Ding Wenxing, Chen Yuzhong, Feng Shiyong, and Fu Tianlong. The previous versions of the standards replaced by this standard are: GB/T4891-1985. 1 Scope
For estimating the average quality of a batch (or process)
Method for selecting the sample size
GB/T 4891--2008
This standard specifies the method for selecting sample size for estimating the average quality of a batch (or process) under simple random sampling for a given confidence level and error limit.
This standard is applicable to the estimation of the mean of a characteristic of a batch of products or processes. 2 Normative references
The provisions of the following documents are incorporated by reference in this standard. For dated references, all subsequent amendments (excluding errata) or revisions are not applicable to this standard. However, parties to agreements based on this standard are encouraged to study whether these documents can be used. The latest version of the document shall apply. For any undated referenced document, the latest version shall apply to this standard. GB/T 10111?2008 Generation of random numbers and their application in sampling inspection of product quality 1 ISO 3531-1 2006 Statistical vocabulary. And symbols Part 1: General statistical terms and terms used in probability ISO 3534-2 2006 Statistical vocabulary. And symbols Part 2: Applied statistics 3 Terms, definitions and symbols
3.1 Terms and definitions
The following terms and definitions apply to this standard.
Simple random sampling simple random sampling From a population containing N sampling units, n units are selected without replacement. If the probability of any π units being selected is equal, that is, equal to 1, this sampling method is called simple random sampling. NOTE 1 Simple random sampling can be carried out by taking a sample of N sampling units as follows: the first sample unit is randomly selected from all N sampling units in the population, the second sample unit is randomly selected from the remaining N-1 sampling units, and so on. NOTE 2 A sample obtained by simple random sampling is called a simple random sample. [IS0 3534-1.2006, 1.3, 4]
Absolute limit of error limit of error
The degree of agreement between independent test results under specified conditions. NOTE 1 Absolute limit of error depends on random errors and is independent of the true value or other agreed value of the measurand. [ISO 3534 1:2006, 4. 11]
Lot
is a sampled, pooled portion of a population with the same substantial conditions. NOTE 2 The purpose of sampling can be to determine the acceptability of a lot or to estimate the mean of an individual characteristic. [ISO 3534-2:2006,1. 2. 1]
processprocess
A set of interrelated or interacting activities that transform inputs into outputs. 1
GB/T 4891—2008
Note 1: The inputs to one process are often the inputs to other processes. Note 2: Organizations usually plan processes and operate them under controlled conditions in order to add value. Note 3: Processes that cannot be easily or economically verified as to whether the resulting product is conforming are usually called "special processes". [IS0 9000 :2005,3. 4. 1]
samplesample
A group of (one or more) individuals (or sampling units) selected from a population according to a certain procedure. Note 1: Each individual in a sample is sometimes also called a sample. Note 2: If the sample is selected according to a random procedure, the sample can be regarded as a group of random variables, each of which is a sample component. [ISO 3534-1:2006,3.5]
sample sizesample size
the number of individuals (or sampling units) included in the sample: [ISO 3534 1:2006,3.7]
standard deviationstandard deviation
the positive square root of the variance.
ISO 3534-1:2006,3,19
coeficient of yariationratio of the standard deviation to the absolute value of the average.
150 3534-1 : 2006 ,2. 207
3.2 Symbol
X represents the sample mean of the random variable X of the individual characteristic value under examination
the mean of the batch (or process), or the expected prior estimate (estimate based on past experience or data) E absolute error limit of the individual characteristic value X under examination in the batch (or process). The maximum allowable value of μe=
Relative error limit
NBatch
nSample size
pBatch (or process) defective rate
Prior estimate of force (estimated value based on past experience or data) Sample defective rate
. Standard deviation of the batch (or process), or the standard deviation of the individual observations in the batch (or process). Prior estimate
SSample standard deviation
5Average value of sample standard deviation (same sample size)CV=.g
Batch (or process) coefficient of variation
CV. Prior estimate of CV
General requirements
Sample coefficient of variation
GB/T 4891—2008
4.1 Based on the previous observation data of individual characteristic values, determine the estimated value of the standard deviation of the characteristic value, or determine the dispersion range and distribution shape of the characteristic value.
4.2 When estimating the defective rate of a batch (or process), the characteristic value of the individual is 0 or 1, 0 means that the individual is a qualified product, and 1 means that the individual is a defective product. At this time, the shape of the distribution and the standard deviation depend only on the defective rate P of the batch (or process). It can be estimated by pre-sampling or past experience.
4.3 When past information is not sufficient, the accuracy of the standard deviation estimate is insufficient, and the required absolute error limit is relatively small, the actual required sample size will be larger than the sample size obtained by the following formulas. 4.4 Before using the formula for calculating the sample size, the absolute error limit E or relative error limit e required for the batch average quality estimate and its corresponding confidence level 1-a must be specified.
5 Formula for calculating sample size
5.1 Given the absolute error limit, the sample size is calculated using the following formula: (-a/o
Using formula (1) to determine the sample size, the probability that the absolute error x is greater than E is a, where the coefficient 1-2 is the quantile of the standard normal distribution. Table 1 gives the corresponding relationship between commonly used confidence levels 1-a and the corresponding coefficient u1-a/2. Table 1 Correspondence between commonly used confidence levels 1-α and the corresponding coefficient u1 a/2 Confidence level 1-
Note: The data given in Table 1 are applicable to situations where the observed characteristic values follow a normal distribution or the sample size is relatively large. 5.2 Given the relative error limit, the sample size is calculated using formula 5.3 to estimate the batch defective rate, which should be Vp(1-p). At this time, formula (1) becomes p(1 When np)
is very small, after calculating the mountain n by formula (3), if np<5, the coefficient u1-α-2
should be used to replace 1-/2 in formula (3) to calculate the corrected sample size. If np~4, the coefficient u1-/2 in Table 1 should be replaced by ua/2+0.25. If np~1, the coefficient u1-ar in Table 1 should be replaced by ui-g/2 0.5. Example: To estimate the defective rate of a certain product, calculate the required sample size. When 1 α=0.9, F=0.002, the pre-estimated value is =0.007, then n=32.95 is calculated by formula (3), and npα0.23 is set. Then ±-a/2 +3
GB/T 4891—2008
-1.64+1.04=2.68, replace u1-a/z in formula (3) to calculate the corrected sample size n88. 5.4 is the average value of a finite batch. When the surface is not the average value of the process, the required sample size is less than the sample size determined by formula (1), (2) or (3). The formula for estimating the sample size screened by the average value of a finite batch is N
n, = (Nn)
, where n is the sample size determined by formula (1), (2) or (3) Determine the sample size. 6 Calculation of sample size when there is historical sample data 6.1 Use of formula (1)
If there is a batch of historical data with a sample size of n, the sample standard deviation S
is calculated using the following formula as the formula in formula (1).
(X--X)
If the historical data comes from the next batch, let the sample size of the batch be n, and calculate the sample standard deviation S of the batch according to the above formula, then use the following formula S-
to calculate the sample standard deviation as the formula in formula (1). . [2(n; 1)S,2
NZ(n; -1)
Example 1: When the specified value of F is 3.64×10*Pa, in order to find the average flexural strength of a batch of bricks, calculate the required sample cover. Based on the data of the previous two batches of bricks (the sample size of each batch is 1C0), the estimated value of the standard deviation of each batch is (15.64, 13.96 and 14.69)×10 Pa. The average value of these standard deviations is 14.76×104Pa. When the supervision level is 99%, the coefficient ±1- is 2.58 from Table 1. The following result is obtained from formula (1): n-(2.58X14.76)2
—(10.45)~110(blocks)
6.2 Use of formula (2)
a) If there is a batch of historical data with a sample size of n, the sample mean and sample standard deviation are calculated as follows: Xbzxz.net
Use number as C in formula (2).
b) If the historical data is from several batches, and. As the mean of the observed product characteristics changes, the mean difference is not large, and the CV change is not large, the sample size of the batch is \ According to the above formula, the sample mean of the i-th batch is calculated, and the sample standard deviation S, and then Zn
[≥(n,-1)S
Z(n; -1)
is used as the CV
in formula (2). Then use the same
Example 2 When the specified value of e is G. 05 or 5%, in order to estimate the average tensile strength of a certain product, calculate the required sample size. There is no sample data of previous products. From the sample data of three similar products in Table 2, the sample size when the confidence level is about 1-α is S9% is 4. (2.85X0.06692)2
-14.5*15
sample maximum code
average tensile strength
sample standard deviation S,, sample variance s,3
12, 02.144.48
14.60.213.16
18.07,326.52
22.96,527.16
25.38,644.14
GB/T 4891—2008
c) If. As the mean of the observed product characteristics changes and the mean difference is large (see Table 4), the mean value Ai and standard deviation S are calculated for several samples. If the difference between the (C)
values is not large, the average value of (CV)\ can be taken as CX
When the sample size is small, the following formula can be used to obtain CV. t
(cv)a, +(CV)har
n+n++ne
Where (CV)i=1,.…,) is the coefficient of variation of the nth sample, α; is a constant that depends on n, and its value is obtained from the noncentral distribution with noncentral parameters. n
Table 4 gives, according to the value of α, obey
Example 3: When the specified value of e is 0.10 or 10%, in order to estimate the average wear resistance of a product, calculate the required sample size, 1.0579
1, 0418
There is no previous sample data for the same product. The sample data of 6 similar products show that the range of wear resistance is relatively wide. However, the estimated value of the standard deviation is approximately proportional to the observed average value, as shown in Table 4. 5
GB/T4891-2008
samples and n
average wear resistance X
In formula (2), the average value of the observed coefficient of variation is used as the estimated value of the CV standard drug
, assuming that the level 1-a is 99.73%, then
3×0.152/
=(4.6)=21,222
In this example, due to the small sample size of each sample, it can also be calculated as follows: +.-
Wa+(va)
1.094 2(0.110.17+0.130.16+0.12+0.19/60
C =0.162(=16.2%)
n=(3×0.162)
In this example, if e=0.05 or 5%, the sample size is 85.6.3 Use of formula (3)
The calculation formula is:
Total number of defectives in all samples
Total number of individuals in all samples
Coefficient of variation
(%)
Example 4: When the specified value of E is 0.04, calculate the required sample size to estimate the defective rate of a batch of alloy steel strip bolts and nuts. Using the data of the first four batches given in Table 5, the prior estimate is given. Table 5
Sample size
Number of defectives
-(0.054) +(0, 9463 -287. 4228810.047
If the specified value of E is 0.01, the required sample size is 1600. Calculation of sample size when there is no historical sample data 7.1 Use of formula (1)
Defective product rate
Based on past experience,The maximum value b and the minimum value a of the observed characteristic are posted, and the distribution of the observed values is shown graphically. 6
GB/T 4891—2008
a) When the distribution form is unclear and the absolute error limit E is required to be strict, the uniform distribution can be used. Since the standard deviation of this distribution is relatively large, a relatively large sample is required. If the isosceles triangle distribution is used instead of other diagonal distributions and uniform distributions, the difference in the obtained standard deviation does not exceed 40%. b) Use the standard deviation estimated by the formula in Figure 1 as the standard deviation in formula (1). This method of prior estimation is often used. Equilateral distribution
Unilateral distribution
Standard deviation
Coefficient of variation
Figure 1 Several distribution forms and their mean, standard deviation and coefficient of variation Positive score
Example: Problem Example 1, when the specified value of F is 3.61×10\Pa, calculate the required sample size to estimate the average flexural strength of a batch of bricks. According to past experience, the scatter range of flexural strength values is about 87.27×10°Ps. These values are in the middle of this range, but they are not necessarily normally distributed. The triangular distribution shown in Figure 1 is the most suitable, and the prior estimate is e87.27
2=17.81×10*Pa
. The sample required in this case is larger than that in Example 1, which is caused by the lack of previous sample data. 7.2 Use of formula (2)
In formula (2), although Figure 1 can be used to estimate CV, it is not recommended. By analyzing actual data, it is generally believed that using CV is better than using. If so, the methods of 6.1 and 6.2 can be used. 7.3 Use of formula (3)
According to past experience, it is possible to roughly estimate the range in which the defective rate may fall. From the midpoint of the range, find the value of \-(1-force) and use it in formula (3). When the absolute error limit is required to be strict, it can be used within the range. The maximum value of is the endpoint value closest to 0.5 (0.5 should be taken when 0.5 is included). For example, the possible range of values is 0.05 to 0.1, so it should be taken as 0.1. The value of \ is the largest, so a=o.1x0.9=0.3
8 Cost considerations
8.1 After calculating the sample size required to meet the specified absolute error limit according to formula (1), (2) or (3), the next step is to calculate the cost of observing this sample. If the cost is too high, the required absolute error limit can be relaxed and the sample size can be reduced as appropriate to meet the requirements for the allowable cost. 8.2 When the allowable cost is specified, the sample size n can be determined from this, and then the possible density can be calculated using formula (1), (2) or (3). 8.2.1 In the case of 5.1, the estimate of the maximum allowable error E is given by the following formula: E-uigaa
GB/4891--2008
If the previous sample data can be used, when the standard deviations of the samples are not much different, F=wrs
Where S, is the standard deviation of the ith sample (i1,,). When the size of each sample is small, E
8.2.2 In the case of 5.2, the estimate of the relative error e is given by the following formula: e=m
If the previous sample data can be used, when the coefficients of variation of the samples are not much different, 3u:-(CV)
Where (CV) is the coefficient of variation of the ith sample (i-1,,). When the size of each sample is small, 241-2/2
(CV)rn)
8.2.3 In the case of 5.3, the estimated value of the maximum error E allowed is given by the following formula: E/p(-)
If previous sample data can be used. In the above formula, it is taken as the ratio of the total number of defective products in all samples to the total number of individuals in all samples (see 5.3).
8.3 Either the cost or the absolute error limit must be specified, otherwise the sample size cannot be determined. 9 Selection of samples
9. 1 In order to estimate the average quality of a batch (or process), samples should be randomly selected using the method specified in CB/T 10111. 9.2 This standard does not discuss the method of handling products and forming sampling units, but assumes that there is an appropriate method to form sampling units and then answers the question of how many sampling units to select.
GB/T 4891-2008
National Standard of the People's Republic of China
Method for selecting sample size for estimating batch (or process) average quality
GI/T 4891—2008
Published and distributed by China Standards Press
No. 16, Sanlihebei Street, Fuxingmenwai, Beijing
Postal Code: 100045
Website spc. nct. cn
Tel: 6852394668517518
Printed by China Standard Press Chaofudao Yincha Factory and distributed by Xinhua Bookstores in various places
Format 880×12301/16 Printing sheet 1 Word count 20,000 words November 2008, first edition: November 2008, first printing*
Book number: 155066·1-34361 Price 16.00 yuan If there is any printing error, the publishing center of this company will replace it. Copyright infringement will be investigated
Report telephone: (010) 68533533
800218In the above formula, the ratio of the total number of nonconforming products in all samples to the total number of individuals in all samples is taken (see 5.3).
8.3 Either the cost or the absolute error limit must be specified, otherwise the sample size cannot be determined. 9 Selection of samples
9.1 In order to estimate the average quality of a batch (or process), samples should be randomly selected using the method specified in CB/T 10111. 9.2 This standard does not discuss the method of handling products and forming sampling units, but assumes that there is an appropriate method to form sampling units and then answers the question of how many sampling units to select.
GB/T 4891-2008
National Standard of the People's Republic of China
Method for selecting sample size for estimating batch (or process) average quality
GI/T 4891—2008
Published and distributed by China Standards Press
No. 16, Sanlihebei Street, Fuxingmenwai, Beijing
Postal Code: 100045
Website spc. nct. cn
Tel: 6852394668517518
Printed by China Standard Press Chaofudao Yincha Factory and distributed by Xinhua Bookstores in various places
Format 880×12301/16 Printing sheet 1 Word count 20,000 words November 2008, first edition: November 2008, first printing*
Book number: 155066·1-34361 Price 16.00 yuan If there is any printing error, the publishing center of this company will replace it. Copyright infringement will be investigated
Report telephone: (010) 68533533
800218In the above formula, the ratio of the total number of nonconforming products in all samples to the total number of individuals in all samples is taken (see 5.3).
8.3 Either the cost or the absolute error limit must be specified, otherwise the sample size cannot be determined. 9 Selection of samples
9.1 In order to estimate the average quality of a batch (or process), samples should be randomly selected using the method specified in CB/T 10111. 9.2 This standard does not discuss the method of handling products and forming sampling units, but assumes that there is an appropriate method to form sampling units and then answers the question of how many sampling units to select.
GB/T 4891-2008
National Standard of the People's Republic of China
Method for selecting sample size for estimating batch (or process) average quality
GI/T 4891—2008
Published and distributed by China Standards Press
No. 16, Sanlihebei Street, Fuxingmenwai, Beijing
Postal Code: 100045
Website spc. nct. cn
Tel: 6852394668517518
Printed by China Standard Press Chaofudao Yincha Factory and distributed by Xinhua Bookstores in various places
Format 880×12301/16 Printing sheet 1 Word count 20,000 words November 2008, first edition: November 2008, first printing*
Book number: 155066·1-34361 Price 16.00 yuan If there is any printing error, the publishing center of this company will replace it. Copyright infringement will be investigated
Report telephone: (010) 68533533
800218
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