GB/T 3359-2009 Statistical processing and interpretation of data Determination of statistical tolerance intervals GB/T3359-2009 standard download decompression password: www.bzxz.net
This standard gives the procedure for establishing tolerance intervals, which contain at least a specified proportion of the population with a certain confidence level. This standard considers the determination of one-sided and two-sided statistical tolerance intervals, and two determination methods are given. The parametric method is applicable to the case where the characteristic under study follows a normal distribution, and the distribution-free method is applicable to the case where only the distribution is known to be continuous.

IS 03.120.30
National Standard of the People's Republic of China
GB/T 3359--2009/IS0 16269-6:2005 Generation of GB/T33591982
Statistical interpretation of data-Determination of statistical tolerance intervals(1ISO 16269-6,2005,Statistical inlerpretation of data-Part 6:Determination of statistical tolerance intervals, IDT) Issued on October 15, 2009
General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China Standardization Administration of China
Implementation on December 1, 2009
Normative referenced documents
3 Terms, definitions and symbols
3.1 Terms and definitions
3.2 Symbols
4 Procedure
4. The case where the mean and variance of the upper normal population are both known 1.2 The case where the mean of the normal population is unknown but the variance is known 4.3 The case where the mean and variance of the stop population are both unknown 4.4 The case of an arbitrary continuous distribution with an unknown type 5 Examples
5.2 Example 1: One-sided statistical tolerance zone when variance is known 5.3 Example 2: Two-sided statistical tolerance interval when variance is known 5.4 Example 3: One-sided statistical tolerance zone when variance is unknown 5.5 Example 4: Two-sided statistical tolerance interval when variance is unknown 5.6 Example 5: Statistical tolerance interval of distribution deviation of continuous distribution Appendix A (Informative Appendix) Tolerance Interval Table Appendix (Normative Appendix)
Appendix C (Normative Appendix)
Appendix D (Normative Appendix)
Appendix E (Normative Appendix)
Appendix F (Normative Appendix)
Appendix G (Normative Appendix)
Appendix H (Informative Appendix)
Appendix I (Informative Appendix)
References
GB/T 3359—2009/IS0 16269-6:2005 One-sided statistical tolerance coefficient when variance a is known, (n;;l-u) Two-sided statistical tolerance coefficient when difference is known, (n;;1-) One-sided statistical tolerance coefficient when variance is unknown, (n;;!) Two-sided statistical tolerance coefficient when variance is unknown, k (n;p;l-a) One-sided tolerance interval with free distribution
Two-sided tolerance interval with free distribution www.bzxz.net
Construction and parameter methods of statistical tolerance interval with free distribution of any type of distribution Calculation of two-sided statistical tolerance interval coefficient 15
"Statistical processing and interpretation of data" includes the following national standards: GB/T3359
G3/T 3361
GB/T 4087
GB/T4088
G3/T 4089
GB/T 4882
GB/T 4883
GB/T 4885
GB/T 4889
G3/F4890
GB/T 8055
GB/T 6380
GB/I10092
GB/T 10094
CB/T 3359-2009/ISO 16269-6:2005Statistical processing and interpretation of data
Determination of statistical tolerance intervals
Statistical processing and interpretation of data
Statistical processing and interpretation of data
Statistical processing and interpretation of data
Statistical processing and interpretation of data
Statistical processing and interpretation of data
Comparison of two and values ​​in the case of paired observationsOne-sided lower confidence limit for the reliability of a distribution
Estimation and test of three distribution parameters
Estimation and test of the parameters of the Zhansong distribution
Tests of normality
Normal sample deviation Judgment and processing of outliers
Study of normal distributionReliability lower confidence limit of the whole sampleStatistical processing and interpretation of data
Statistical processing and interpretation of data
Estimation and test of mean and variance of normal distributionPower of test on mean and variance of normal distributionStatistical processing and interpretation of dataParameters of I distribution (Pearson type II distribution)Statistical processing and interpretation of calculated dataJudgment and processing of outliers in samples from exponential distributionJudgment and processing of outliers in samples from type I extreme value distributionStatistical processing and interpretation of data
Statistical processing and interpretation of dataComparison of test resultsConfidence limits of quantiles and coefficient of variation of normal distributionPart 6: Determination of statistical tolerance interval. This standard is equivalent to IS016269-6:2005 & Statistical processing and interpretation of data. For ease of use, this standard has made the following editorial changes: Except for the preface and introduction of the International Standard:
… The decimal point "\ is used instead of the comma \,\ as the decimal point. This standard replaces GB/T3359--1982 "Statistical processing and interpretation of data - Determination of statistical tolerance intervals". The main changes of this standard compared with (GBT3359-1982 are as follows: Change the strength variation coefficient C in GI3/T33591982 to Ce; Change the stress variation coefficient C in GI3/T 33591982 to Cs; Change GB/T The strength standard value Fk in 33591982 is changed to Fia. The stress standard value is modified accordingly; the two reliability coefficients 7 and 7k are changed to rr respectively, highlighting the ratio property of these two quantities, and the characteristics corresponding to the mean and quantile respectively:
Some annotations are added to eliminate some doubts. Appendix B, Appendix C, Appendix 1. Appendix E, Appendix F and Appendix G of this standard are normative appendices, and Appendix A, Appendix H and Appendix I are informative appendices.
This standard is proposed and coordinated by the National Technical Committee for Standardization of Statistical Methods (SAC/T21). The drafting units of this standard are: China Institute of Standardization and Chemical Technology, Beijing Institute of Technology, Peking University. The main contributors of this standard are: Xu Xingzhong, Zhao Shuran, Bing Yuliang, Xie Chuanfa, Yu Zhenfan, Fang Xiangzhong, Ding Wenxing. The certification version of the standard replaced by this standard is released as follows: GB/T 3359.-1982.
1 Scope
GB/T3359--2009/S016269-6:2005 Statistical processing and interpretation of data
Determination of statistical tolerance intervals
This standard gives the procedure for establishing tolerance intervals, which contain at least a specified proportion of the population with a certain confidence level. This standard considers the accuracy of one-sided and two-sided statistical tolerance intervals. Two determination methods are given. The parametric method is applicable to the case where the characteristics under study follow a normal distribution, and the distribution-free method is applicable to the case where only the distribution is known to be continuous. 2 Normative references
The provisions of the following documents become the provisions of this standard through reference in this standard. For any dated referenced document, all subsequent amendments (excluding errata) or revisions are not applicable to this standard. However, the parties to an agreement based on this standard are encouraged to study whether the latest versions of these documents can be used. For any undated referenced document, the latest version applies to this standard. G3/T 3358.1 Statistical vocabulary and symbols Part 1: General statistical terms and terms used in probability (GB/T 3358.12009, IS0 3534-1:2006, JDT)
GB/T3358.2 Statistical vocabulary and symbols Part 2: Statistical terms (GB/T3358.2--2009.ISO3534-2.2006, IDT)
3 Terms, definitions and symbols
3.1 Terms and definitions
The following terms and definitions established in GB/T3358.1 and GB/T3358.2 apply to this standard. 3.1.1
Statistical tolerance interval is an interval determined by a random sample that contains at least a specified proportion of the sample population with a specified probability. Note: The confidence level of such an interval is the frequency with which it contains at least a specified proportion of the sample population when repeated multiple times. 3.1.2
Statistical tolerance limit The statistic represents the endpoints of the statistical tolerance interval. Note: A statistical tolerance interval can be one-sided or two-sided. A one-sided interval has only one upper or lower tolerance limit. A two-sided interval has both upper and lower tolerance limits. 3.1.3
Coverage
The proportion of a population that falls within the statistical tolerance interval. Note: This concept should not be confused with the coefficient of variation in reference 5. 3.1. 4
Normal population
A population that follows a normal distribution.
3.2 Symbols
The following symbols apply to this standard.
Subscript of the observed value
,(:lα
ka(n;p;l -a)
When known, it is used to determine the endpoints of the one-sided tolerance interval rl. Or the coefficient of when known, it is used to determine the endpoints of the two-sided tolerance interval. And the coefficient of u1
GB/T 3359—2009/ISO 16269-6:2005,(n:p;1--a)
k,(n;p;l--α)
A procedure
is used to determine the endpoints of a one-sided tolerance interval when unknown or: the coefficient of the coefficient is used to determine the endpoints of a two-sided tolerance interval when known. The coefficient of the quantity
is the minimum proportion of units in the population that fall within the tolerance range of the normal distribution
the observed value (||1,2,n)
the most observed value: mmax,*.2)
the smallest observed value: mn… min(,,.1)
Lower limit of the statistical tolerance interval (lower tolerance limit) Upper limit of the statistical tolerance interval (upper tolerance limit) Sample mean, 2:
Sample standard deviation: 5=
Nn--i4
(r; --
(n—1)
Confidence level that claims that the proportion of the population falling within the tolerance interval is greater than or equal to the specified level Population mean
Population standard deviation
4.1 Case where the mean and variance of a normal population are known When the mean and variance of a normally distributed population are both known, the distribution of the characteristic under study is completely determined. The interval that contains exactly 100% of the population batch is:
a）-right side (one-sided interval)
b) = μ + to the left (one-sided interval)
) = + and = (two-sided interval)
Jiang: Since we know that the above statements are always true, they have a 100% confidence level. In the above formula, u is the quantile of the standard normal distribution. The value of u can be found from the last sum of Table B.1 to Table B.G and Table C.1 to Table C.6.
4.2 The case where the mean of a normal population is unknown and the variance is known Appendix Tables A.1 and A.2 in Appendix A apply to the case where the normal population variance is known but the mean is unknown. Table A.1 applies to the one-sided interval case, while Table A.2 applies to the two-sided interval case. 4.3 Case where both the mean and variance of a normal population are unknown Tables A.3 and A.4 in Appendix A apply to the case where both the variance and mean of a normal population are unknown. Table A.3 applies to the one-sided interval case, while Table A.1 applies to the two-sided interval case. 4.4 Case of any continuous distribution of unknown type If the characteristic under study is a continuous variable of unknown distribution type, and n independent random observations of this characteristic are combined into a sample, a statistical tolerance interval can be determined by sorting the observations. The procedures given in Tables A.5 and A.6 in Appendix A provide the coverage or sample size required to determine the tolerance interval for the sample extreme value or x at a given confidence level of 1-u. Note: A statistical tolerance interval that does not depend on the distribution type of the sampled population is called a distribution-independent tolerance interval. This standard does not provide procedures for other known types of distribution populations other than the static distribution population. However, for other continuous distributions, the distribution method can also be used. The references provided at the end of this standard are helpful in determining the tolerance intervals for other known types of distributions. 2
5 Examples
5.1 Data
GB/T 3359-2009/IS016269-6:2005 uses examples to illustrate how to use Tables A.1 to A.4 in the appendix. The data in this example are from ISO2854:1976. The data in Table 1 are 12 test values ​​of the breaking load r of cotton yarn. It should be noted that the number of test values ​​in this example is n12, which is much lower than that recommended by ISO2602. The units of data and calculated values ​​in each example are all %·Newton (see Table 1). Table 1 Data of Examples 1 to 4 228.6: 232.71 233.8
315.8275.1
Unit: hundredth of a ton
224.7251.2210.4
These test values ​​are from a batch of 2000 bobbins, 100 bobbins per box, 120 boxes in total. 12 boxes were randomly selected from this batch, and then one yarn was randomly selected from each box. A 50cm long section was cut from these yarns about 5m away from the active end of the yarn for testing. Each test was carried out in the middle of these test sections. The previous information shows that it is reasonable that the breaking load measured under these conditions actually obeys the normal distribution. The results are as follows:
Sample:
Sample mean:
Number = 3 024,1/12 = 252. 01
Sample standard deviation: 5
na*-(2)
/[G6 772. 77
\12×11
5.2 Example 1: One-sided statistical tolerance interval when variance is known 1263.4263=3G.645
Assume that the test values ​​obtained previously show that the degree of dispersion of different batches from the same supplier is constant and can be expressed by standard deviation = 33.150, while the mean of different batches is not well estimated. Here, the single tolerance interval given by the lower tolerance limit L is used. It is required to ensure with a confidence level of 1--a0.95 (95%) that the proportion of products in the batch with a breaking load above 1. is at least 0.95 (95%). From Table 1.4, we can find
k,(12;0.95;0.95) = 2.120
t. = —ki (n;p;l —) ×a -- 252.01 -2.120 × 33.150181.732 If a larger population proportion (such as α = 0.99) or a higher confidence level (such as 1--α = 0.9) is required, the lower tolerance limit obtained will be smaller.
5.3 Example 2. Two-sided statistical tolerance interval when variance is known The conditions are the same as those in Example 1. This example uses the two-sided tolerance limit and u, requiring that at least 0.90 (90%) of the breaking loads of this batch of yarns fall between 3L and u with a confidence level of 1 α0.95. From the table, we get
ka(12:0.90;0.95) - 1.889
zz--kz(n;p;1-α)Xa252.01-1.889X33.150189.390x:t: = r +k2(n;p;1 —a) X :- 252.01 +1.889 X 33.150314. Comparison with Example 630 clearly shows that ensuring that at least 90% of the population falls between 2. and : and ensuring that no more than 5% of the population falls outside each one-sided limit will give different results. 5.4 Example 3: One-Sided Statistical Tolerance Regions When the Variance is Unknown Assume that the standard deviation of the population is unknown and must be estimated from the sample. Use the same requirements as for the known standard deviation (Example 1), namely p0.95 and 1α0.95. The detailed calculation steps are given in the table below. 3
GB/T 3359—2009/IS0 16269-6.2005 Determination of the statistical tolerance zone of proportion: a) Lower tolerance interval
is determined as:
b) Selected population proportion of tolerance interval: = 0.95c) Selected confidence level: 10.95
d) Sample base: = 12
Table T). 4 Tolerance coefficient value of the book:
2(/p;1—0) = 2. 737
Calculation:
r: - Zr:/4 = 252. c1
(n≥r(2x)
—35.545
ks(n:p:1 —0) X y — 97, 286 7 Results: The lower penetration limit of the single-test interval with confidence level 1 and containing a population proportion of at least is i. ,(;;1 )×154.723
5.5 Example 4: Two-sided statistical tolerance interval when variance is unknown Under the same conditions as in Example 2, it is required to calculate the lower tolerance limit r. and the lower tolerance limit u so that the proportion of the breaking loads in this batch of yarns falling between and can be guaranteed to be at least -0.90 (90%) with a value of 1-α-.95. From Table E.1, we can obtain
ht(n;p;1-α) - 2.671
±t. 5 k.(np:1 -- o) × $ - 252.01 --- 2.671 × 35.545— 157.069ru - x [ k,(n:p; 1 _e) X s - 252.01 -f 2.671 X 35.545 = 346. 951 It should be noted that the value of s is smaller here than in Example 2 (where the variance is known) and the value of s is larger because the substitution requires a larger tolerance factor to take the additional unknown into account. The penalty of not knowing the variance of the population is necessary, and the widening of the tolerance zone takes into account the unknown variance. If the 33.150 used in Example 1 and Example 2 cannot be guaranteed to be correct, an estimated value of s and the corresponding Table 1).1 or Table E.1 should be used. 5.6 Example 5: Statistical tolerance intervals for continuous distributions A component of an aircraft engine was subjected to fatigue testing with rotating stresses. The test results (measured as endurance) for 15 samples are given in Table 2 in ascending order of value.
C.2000. 330
The data in Example 5
0. 9200. 950
Using a graphical test such as a probability plot to test normality shows that the normality assumption for the population of component combinations is unwritten (see GB/T 1882). Therefore, the method of Table A.5 in Appendix A should be used to determine the statistical tolerance interval. The extreme values ​​of the 15 sample measurements are:
0, 200,.. - 8. 800
Assuming a confidence level of 0. 95
a) What is the maximum proportion of the population of components that falls below - 0. 200? For n = 0. 9i, Table F.1 gives a value for the minimum proportion of the population that falls above rni = 0.200 that is greater than 0.75 (75%). Therefore, the maximum proportion of the population of parts that falls below 1-q is slightly lower than .25 (2; for)
b) How large a sample size must be taken to ensure, with a confidence level of 0.95, that at least a proportion of the population of parts falls below the maximum value of the sample? For 1-q, 0.95 and = 0.90, Table F.1 gives n=29. e) What is the minimum proportion of the population of parts that falls between mcenter = 0.2C0 and mx=8.800 at a confidence level of 0.95? For 1-a = 0.95 and n-1.5, Table G.1 gives a value slightly lower than 0.75 (75%). d) How large a sample size is needed to ensure, with a confidence level of 0.95, that at least a proportion of the population of parts falls between the minimum and maximum values ​​of the sample at p0.90 (90%)? For 1—α0.95 ​​and α0.90, Table G, 1 gives π=16. e) "In general, if the normality test shows that the population does not obey the normal distribution, it is recommended to transform the data into data that obeys the normal distribution based on the relevant information of the data. For example, fatigue data often approximately obey the lognormal distribution. In these cases, the data can be transformed into data that obeys the normal distribution, and the tolerance interval can be calculated according to the normal population, and finally transformed back to the tolerance interval of the original population.
Appendix H gives the construction method of the distribution-free statistical tolerance interval for any type of distribution. Appendix I gives the calculation of the coefficient of the two-sided statistical tolerance interval with parameter method. 5
GB/T 3359—2009/1SO 16269-6 :2005 Appendix A
(Informative Appendix)
Tolerance Interval Table
Table A, 1 One-sided statistical tolerance interval (variance is known) Confidence level is "Determination of one-sided statistical tolerance interval with coverage rate a) "One-sided interval to the left"
) "One-sided interval to the right"
Known quantity:
) Difference:
d) Standard coupon: $=
Exact value:
Selected population proportion for tolerance interval: -e
) Selected confidence level: 1—α
g) Sample size: n—
Tabulation coefficient:
(n:1)-
Corresponding to the series n, and 1, the coefficient values ​​are given in Appendix B. Calculation:
r -- r,/n =
t(nip:la)Xa
Result:
a) "The \unilateral region to the left
confidence level for 1-α coverage ratio is the one-sided statistical tolerance interval has an upper limit: --(n;p:>xa --
b) "The \unilateral region to the right
confidence level for 1-α coverage ratio is the one-sided statistical tolerance interval has a lower limit: lj. - ki(nsh;l )xa-
Two-sided statistical tolerance interval (variance known)
Set the book level to 1--αThe two-sided statistical tolerance interval with a coverage rate of is exactly what is the quantity:
h)Standard error: =
Confirmation quantity:
c)Selected population proportion for the tolerance interval: p{[)Selected confidence level,
e)Sample size:
Tabulation coefficient:
k(;lα)-
The appendix gives the values ​​corresponding to the range of, and, calculation:
k:(n;p;1-a)xa-
Result:
Confidence level is! . The upper and lower limits of the two-sided statistical tolerance interval with a micro-profit rate are F limits ri ——k(n;p;」—u)Xa
tLimit: +(n;p;-a)x
CB/T 3359—2009/1SO 16269-6:2005 Table A, 3 One-sided statistical tolerance interval (variance unknown) confidence level is 1.·Determination of the one-sided statistical tolerance interval with a coverage rate of α) "Left-sided\one-sided interval
) "Right-sided\one-sided interval
Determination quantity:
c) Selected population proportion for the interval: pd) Selected confidence level: 1α
e) Variance: -
Tabulation coefficient:
(;1 -)-
The appendix gives a certain range of values ​​corresponding to π, and -. Calculation:
x ​​-Eadn -
/n2z (2x22
n(n--1)
h,(n;p;la)Xs-
Result:
a) "Left" one-sided interval
Confidence level is 1 - One-sided statistical tolerance interval with coverage has upper limit Zu-iks(np;1-a)xs-
b) "Right" one-sided interval
Confidence level is 1 - One-sided statistical tolerance interval with coverage has lower limit: 21. -±-k(n:p:1-u)Xs -
Table A. 4 Two-sided statistical tolerance interval (variance unknown) Two-sided statistical tolerance interval with confidence level 1-α and coverage of Quantification:
a) Population proportion selected for tolerance interval: =b) Confidence level selected: 1-α…
c) Sample size:
Tabulation coefficient:
The Appendix gives the values ​​corresponding to the range of π, and 1-α. Calculation:
-Zx/n-
n22-(2x)
n(nl)
I(n:p:l-a)X--
Result:
A two-sided statistical tolerance interval with confidence level 1 and coverage of has a lower limit: F limit 21-(n:p;l-α)x$-
Upper limit ru--+ ,(n+p;1-a)x-
CB/T 3359.---2009/ISO 16269-6:2005 One-sided statistical tolerance intervals for arbitrary distributions
with compensation level 1-. The coverage rate of the distribution of is determined by the single-sided statistical penetration interval a) "to the left of the \single-sided interval,
) "to the right of the \single-sided interval;
determine the quantity:
c) the selected population proportion for the penetration interval: - d) the selected confidence level: 1
c) in the sample: let =
(and determine the difference between them)
tabulate the coefficients
for given n and 1-α
for given and 1-α.
F.1 gives the corresponding range of n, and 1. Calculation and results
Confidence level 1 One-sided statistical tolerance interval with coverage of has lower bound: —n
or upper bound: Zu — Ina -
Table A, 6 Two-sided statistical tolerance intervals for arbitrary distributions The confidence level is the distribution with coverage of The determination of the two-sided statistical tolerance interval from the distribution of ... Calculation and results
The two-sided statistical tolerance intervals with a confidence level of 1-α and a coverage rate of p are, I: lower limit: lower limit.
Upper limit
Appendix B
(Normative Appendix)
CB/T 3359—2009/IS0 16269-6:2005 One-sided statistical tolerance coefficient when variance is known, k, (n; p; 1-α) Table B.1 Confidence level 50, 0%
(1-a = 0.50)
0,000
0,000
1,645
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