GB/T 10094-2009 Confidence limits of quantiles and coefficient of variation of normal distribution GB/T10094-2009 Standard download decompression password: www.bzxz.net
This standard specifies the method for determining the confidence interval of the quantile of normal distribution and the upper confidence limit of the coefficient of variation at a given confidence level. This standard applies to the population of normal distribution.
Statistical processing and interpretation of data include the following national standards:
---GB/T3359 Statistical processing and interpretation of data Determination of statistical tolerance interval
---GB/T3361 Statistical processing and interpretation of data Comparison of two means in the case of paired observations
---GB/T4087 Statistical processing and interpretation of data One-sided lower confidence limit of reliability of binomial distribution
---GB/T4088 Statistical processing and interpretation of data Estimation and test of parameters of binomial distribution
---GB/T4089 Statistical processing and interpretation of data Estimation and test of parameters of Poisson distribution
---GB/T4882 Statistical processing and interpretation of data Normality test
---GB/T4883 Statistical processing and interpretation of data Judgment and treatment of outliers in normal samples
---GB/T4885 Lower confidence limit of reliability of complete sample of normal distribution
---GB/T4889 Statistical processing and interpretation of data Estimation and test of mean and variance of normal distribution
---GB/T4890 Statistical processing and interpretation of data Power of test of mean and variance of normal distribution
---GB/T6380 Statistical processing and interpretation of data Judgment and treatment of outliers in samples of type I extreme value distribution
---GB/T8055 Statistical processing and interpretation of data Parameter estimation of Γ distribution (Pearson type III distribution)
---GB/T8056 Statistical processing and interpretation of data Judgment and treatment of outliers in samples of exponential distribution
---GB/T10092 Statistical processing and interpretation of data Multiple comparison of test results
---GB/T10094 Confidence limits for quantiles and coefficient of variation of normal distribution
This standard replaces GB/T10094-1988 "Confidence intervals for quantiles of normal distribution", GB/T11791-1989 "Upper confidence limits for coefficient of variation of normal distribution" and GB/T14438-1993 "Lower confidence limits for normal probability within limits".
Compared with GB/T10094-1988, GB/T11791-1989 and GB/T14438-1993, the main changes of this standard are as follows:
--- The standard format has been revised according to the requirements of GB/T1.1-2000 "Guidelines for standardization work Part 1: Structure and writing rules of standards";
--- The examples in Appendix A of GB10094-1988 are put into the main text.
This standard is proposed and managed by the National Technical Committee for Standardization of Statistical Methods (SAC/TC21).
The main drafting units of this standard are: University of Science and Technology of China, Peking University, China National Institute of Standardization. ||tt
||The main drafters of this standard are: Wu Yaohua, Sun Shanze, Yu Zhenfan, Ding Wenxing, Zhou Zhengfa, etc.
The previous versions of the standards replaced by this standard are:
---GB/T10094-1988;
---GB/T11791-1989;
---GB/T14438-1993.

The clauses in the following documents become the clauses of this standard through reference in this standard. For all dated referenced documents, 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 the latest versions of these documents can be used. For any un-dated referenced documents, the latest version shall apply to this standard.
GB/T3358.1 Statistical vocabulary and symbols Part 1: General statistical terms and terms used in probability (GB/T3358.1-2009, ISO3534-1:2006, IDT)
GB/T3358.2 Statistical vocabulary and symbols Part 2: Applied statistics (GB/T3358.2-2009, ISO3534-2:2006, IDT)
GB/T4086.1 Numerical tables of statistical distributions Normal distribution
GB/T4086.2 Numerical tables of statistical distributions χ2 distribution
GB/T15932-1995 Quantile tables of non-central t-distribution
GB/T4885-2009 Lower confidence limits for complete sample reliability of normal distribution


Foreword Ⅰ
1 Scope1
2 Normative references1
3 Terms, definitions and symbols1
3.1 Terms and definitions1
3.2 Symbols3
4 Confidence interval for quantiles of normal distribution3
4.1 One-sided lower confidence limit3
4.2 One-sided upper confidence limit3
4.3 Two-sided upper and lower confidence limits3
4.4 Example4
5 Upper confidence limit for the coefficient of variation of normal distribution4
5.1 Exact upper confidence limit4
5.2 Approximate method for the upper confidence limit5
5.3 Example5
Appendix A (Normative Appendix) K coefficient table6

ICS 03. 120.30
National Standard of the People's Republic of China
GB/T10094-2009
Replaces GB/T 100941988, GB/T 11791—1989, GB/T 11138993Confidence limits of quantile and coefficient of variationfor normal distribution
Issued on October 15, 2009
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
Implementation on February 1, 2010
GB/T10094—2009
1 Scope
2 Normative references
3 Terms, definitions and symbols
3. 1 Terms and definitions
3.2 Symbols
4 Confidence interval of quantile of normal distribution
4.1 One-sided lower confidence limit
4.2 One-sided upper borrowed limit
.3 Effect-side upper and lower confidence limits
4.4 Example
5 Upper confidence limit of coefficient of variation of normal distribution
5.1 Exact upper confidence limit
Approximate method for upper confidence limit
Appendix A (Normative Appendix)
K coefficient table
Entire product
“Statistical processing and interpretation of data\Includes the following national standards: Statistical processing and interpretation of data
GB/T 3359
-GB/T3361
--GB/T A087
GB/T 4088 | |tt | |GB/T 8056
Determination of statistical tolerance zones
GB/T10094—2009
Statistical processing and interpretation of data
: Comparison of two means with different observed valuesStatistical processing and interpretation of data
One-sided lower confidence limit for the reliability of binomial distribution
Statistical processing and interpretation of data
Estimation and test of binomial distribution parameters
Statistical processing and interpretation of data
Statistical processing and interpretation of data
Statistical processing and interpretation of data
Parameters of Poisson distribution Estimation and test
Normality test
Judgment and treatment of outliers in normal samples
Statistical treatment and interpretation of the lower confidence limit of the reliability of normal distribution complete samples
Estimation and test of the mean and variance of normal distributionStatistical treatment and interpretation of dataStatistical treatment and interpretation of power data of the mean and variance test of normal distribution
Judgment and treatment of outliers in samples of type I extreme value distributionStatistical treatment and interpretation of data: parameter estimation of type I distribution (Pearson type III distribution)Statistical treatment and interpretation of outliers in samples of index distribution-CB/T 10052
Statistical processing and interpretation of data Multiple comparisons of test results - GB/T10194 Confidence limits for quantiles and coefficient of variation of normal distribution This standard replaces (GB/T10094--1988 "Normal distribution quantile:: Confidence interval", (GB/T11791:1989 "Upper confidence limit for coefficient of variation of normal distribution" and (GB/T144381993 "Lower confidence limit for normal probability within fixed limits". Compared with GB/T10094-1988, GB/T11791-1989 and GB/T14438-1993, the main changes of this standard are as follows: - The standard format has been revised according to the requirements of GB/T1.1-2000 Standardization Work Guidelines Part 1: Structure and Editing Rules of Standardization;
The examples in Appendix A of CB10094-1988 are incorporated into the text. This standard was proposed and exported by the National Technical Committee for Standardization of Statistical Methods (SAC/IC.21). The main drafting units of this standard are: University of Science and Technology of China, Peking University, China National Institute of Standardization. The main drafters of this standard are: Hao Yaohua, Sun Shanze, Yu Zhenfan, Ding Wenxing, Zhou Zhengfa, etc. This standard replaces the following versions: GB/T 10094—1988;
G/T 11791—1989,
(GB/T 144381993,
|tt||1Range
Confidence limits of quantile and coefficient of variation of normal distribution CB/T 10094--2009
This standard specifies the method for determining the confidence interval of the quantile of the normal distribution and the upper confidence limit of the coefficient of variation under the specified confidence level. This standard applies to the general population of normal distribution. 2 Normative referenced documents
The clauses in the following documents become the clauses of this standard through reference in this standard. For any dated referenced documents, all subsequent amendments (excluding errata) or revisions are not applicable to this standard. However, 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 documents, the latest versions apply to this standard, GJ3/T 3358. 1 Statistical Vocabulary and Symbols Part 1: General Statistical Terms and Terms Used in Probability (GB/T 3358. 1-2009, IS0 3534-1:2006,ID1)
CB/T3358.2 Statistics and symbols Part 2: Applied statistics (GB/T3358.22009,ISO3531-2.2006,IDT)
G3/14085.1 Statistical distribution numerical table Normal distribution GB/T4086.2 Statistical distribution numerical table x distribution GB/T159321995 Non-central t distribution Quantile table GB/T4885—2009i: Normal distribution Constitution Full sample reliability confidence limit 3 Terms, definitions and symbols
3. 1 Terms and definitions
The terms and definitions established in GB/T3358.1 and GB/T3358.2 and the following terms and definitions apply to this standard. For ease of reference, some terms are directly quoted from the above standards. 3. 1. 1
p-quantile p-quantile; p-fractileX,,x
for 0<<1, the infimum of all distribution function F() greater than or equal to. Example 1: Consider, binomial distribution, Table| gives the probability primes of binomial distribution with parameter = 6, P-C.3. Some quantiles of the distribution are: Xa.i --0
Xns—2
Xu.-5 = 3
Xo. 9c = 3
Xu. 95 - 4
Xe.9g5 — 5
X5. 9g - 5
From the above, binomial distribution is discrete and its quantiles are all integers. 1
Partner
GB/T 0.11 7 649
0.420 175
0.929 530
0. 989 065
0.999 271
1.000 000
0.882 351
0.070 470
0.01c 935
0. 000 729
0.090:000
Example 2: For standard normal distribution, Table 2 gives some numerical distribution functions and corresponding P quantiles (see GB/T 4086.1 Normal distribution quantile table),
Table 2 Standard normal distribution example
0.811 344 75
Satisfying P(Xs)-
Since the distribution of × is continuous, the title of the second column can also be written as: Satisfying P(X)-Note 1: For continuous distribution, if the force is 0.5, the 0.5 quantile is the median (2.14). For 0.25, the corresponding 0.The 25th quantile is called the lower quartile. For a continuous distribution, 25% of the distribution is below the 0.25th quantile and 75% of the distribution is above the 0.25th quantile. When the median is equal to 0.75, the corresponding 0.75th quantile is called the upper quartile. NOTE 2: In general, 100% of the values ​​in a distribution are below the quantile and 100(1)% of the values ​​in a distribution are above the quantile. However, it is difficult to determine the median of a discrete distribution because there will be many values ​​that satisfy the definition.
Note 3: If F is continuous and strictly inversely increasing, the force quantile is the solution of F () - force, in which case the "infimum" in the definition can be replaced by "minimum". Note 4: If the distribution quantile is equal to a constant P on a certain interval, then all values ​​on this interval are P quantiles of this distribution. Note 5: The definition of force quantile only applies to one-dimensional distributions. 3. 1.2
Coefficient of variation CV
The standard deviation of a positive random variable divided by a non-zero mean. Note 1: The coefficient of variation is usually expressed as a percentage. Note 2: The previous term "sum to standard deviation" cannot be used. 2
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Confidence intervalconfidcnceinlerval
GB/T 10094—2009
Interval estimation of parameter 0 () where the statistic T as the interval limit satisfies P[TT,α. Note 1: The confidence level reflects the proportion of the true value of the parameter contained in the confidence interval under the same conditions of repeated random sampling of a long sequence. The confidence interval cannot reflect the probability that the observed interval contains the true value of the parameter (the observed interval can only be known to be included or not included). Note 2: The characteristic associated with the confidence interval is 100(1-α)%, where. is a small number. This characteristic is called the confidence coefficient or confidence level, usually taken as% or 99%. For any determined but unknown population 9 value, P[T,T,=1. 3.2 Symbols
The following symbols apply to this standard.
n: sample size
: population mean
s: population standard deviation
main: observed value of normal random variable X
none: sample mean
s: sample standard deviation
1α: set multiple
T: p-quantile of probability distribution
Cu, upper confidence limit of
CL: lower confidence limit of,
CV-a/u: population coefficient of variation
CV-5/z: sample coefficient of variation
t: quantile of non-central distribution with non-centrality parameter CVu: upper confidence limit of population coefficient of variation
4Confidence interval of quantile of normal distribution
4. 1 One-sided confidence lower limit
Normal distribution quantile, the one-sided confidence lower limit CL with a confidence level of 1 is determined by formula (1): a
[— sK,- when ≤ 0. 50
yuan + sK. When 0.50
(1)
Where K and K1- can be found in Appendix A. In the K coefficient table, corresponding to the sample size of n, R=1-p, K can be found. Corresponding to the sample size of t, R-,-1--α, K1-4.2 One-sided confidence upper limit
Normal distribution quantile. The one-sided confidence upper limit C with a confidence level of 1-α is determined by formula (2): Cu
When 0.50
x sK---, when > 0.50, K and K in H can be found in Appendix A. In K coefficient Table 4, corresponding to sample size n, K=1, =α, we can find K, and the corresponding sample size is R-,-1-n, we can find,-4. 3 Upper and lower confidence limits of normal distribution quantiles. The confidence level of the two-sided confidence lower limit CL of 1-u is determined by formula (3): [sK1-u/2 when 0.50
l 1 sK, when 0. 50
The two-sided confidence limit Cn is determined by formula (4):
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. (3)
GB/T 10094—2009
[-K/2when ≤0.50
[-1 sK1-a/2,when 0. 50
(3).(4)In the two formulas, K and KI/ can be found in Appendix A. In the K coefficient table, the corresponding sample size is n, R-1p, and Ki-a can be found.
It can be found that K, corresponding to the detailed version, R=,1-2
4.4 Example
Example 1: The meteorological station of a certain city measured the annual rainfall data of 72 mu in the city as follows (unit: mru): 1 063, 8, 1 004. 9, 1 086.2, 1 022.5, 1 330.9, 1 439. 4, 1 236.5, 1 088.3, ​​1 288. 7, 1 115, 8, 1 217, 5, 1 320.7, 1 078.1, 1 203.4, 1 180.0. 1 269. 9, 1 049.2, 3 318. 1, 3 192. 0, 1 016. 0, 1 508. 2, 1 159. 5,1 021.3,986. 1,794. 7,1 318, 3,1 171.2,1 161. 7,791.2,1 143. 8,1 602.0,951.4,1 003.2.840. 4,1 061. 4,558, 0,1 025. 2,1 265. 0,1 195. 5,1 120.7.1 659. 3,912.7,1 123.3,910, 2,1 398. 5,1 208. 6,1 305.5,1 242. 3,1 572. 3,1 416. 9,1 256. 1,1 285, 9+984.8,1 390. 3,1 062. 2,1 287. 3,1 477. 0,1 017.9,1 217.7,1 197.1.1143.0,10188,1243.7,909.3,3030.3.1124.4,811.4,820.9,1184,1,31c7.5991.4,901.7.
After inspection, the annual rainfall X follows a normal distribution. Find the two-sided confidence limits of the 10% quantile 30.10 and the 90% quantile 10 when the confidence level is 0.90. The sample gap value and sample standard deviation calculated from the data are: x = 1 154, 78,: = 159. 562
According to formula (3) and formula (4), when the confidence level is 0.90, the upper and lower limits of the bilateral confidence level of the quantile. 1 are: C- - sKe.s, C = -- sKa.o
When the confidence level is -0. 9, the lower and upper limits of the bilateral confidence level of the quantile, ss are; C. - i + sKe.a, C. - α + sKa. s corresponding to n-72.R-5. 90, check Appendix A to get: Ke,ns - 1. 043, Ko.ss = 1. 57. The upper and lower limits of the bilateral confidence level of the quantile. m are: C).- 847.012, Cu 951.228
The lower and upper limits of the bilateral confidence level of the quantile are Ct 1 358. 336.Cu 1 462. 552 Example 2: The life of a certain high temperature alloy steel is a log-normal distribution. Twelve specimens were taken for life tests at 650°C and 4kgf/mm2 (39.2266MPa). The data are as follows (unit: h): 935.1 025.1 (081, 1 180, 1 197, 1 234, 1 328, 1 521, 1 621, 1 621.1 694, 1 933. Calculate the lower confidence limit of the 1% quantile 0.ar of the life distribution when the confidence level is 0.90. Take the logarithm of the test data t, (i-1,,12); = lt, t,, + t: 12 is a sample of normal distribution. The sample mean and sample standard deviation are calculated as follows: -- 7. 194 9, = 0. 225 8
According to formula (1), the confidence lower limit of the 1% quantile a01 of the state distribution with a confidence level of 0.90 is: C. - ± — sKe.90 = 7.194 00.225 8 × 3.371 = 6.433 7 where Kn.gc = 3, 371 is the corresponding t—12, R-1—=0.99, Y= 1 --α—0.9 as shown in Appendix A. Therefore, the lower limit of the error of the 1% quantile of the population distribution with a confidence level of 0.0 is: e.37 -.622.47(h)
5 Upper confidence limit of coefficient of variation of normal distribution
5. 1 Exact upper confidence limit
The upper confidence limit of coefficient of variation CV of normal population (V:) is determined by formula (5): [(n-IWh(CV)=AK
GB/T 10094—2009
where K=2/s-1/cu, (CV:). The value can be obtained by linear interpolation using (I3/T159321995). 5.2 Approximate upper confidence limit method
When CV<0.30, n≥6, the approximate formula of the upper confidence limit of coefficient of variation CV: of normal population is: CV.
[x.(n-1)(1+cu\. (n-12)]--(n1)cua
where (-1) represents the quantile of the distribution with degree n-1, and its value is obtained from the quantile table (see GB/T4086.2),
5.3 Example
A batch of carbon epoxy shells, 9 pieces of which were randomly selected for strength test, and the measured damage values ​​were (unit: h): 7, 92, 1, 25, 7.858, 7, G.67, 6.75, 6.87, 6.92. Calculate the upper confidence limit of the coefficient of variation of the strength of this batch of shells when the confidence level is 0.90. The sample mean and sample standard deviation calculated from the data are: x=7.2178, 5=0.629 6
) Exact method
From /nK=/×7.2178/0.629 6=34.3923, check the non-central t distribution table (see GB/T15932-1995). Use linear trapping to get:
n(CV-1).=22.6409
CVu -- 1/(CV-), -- 0. 132 5
b) Approximate method
Check the × quantile table (see GB/T4086.2): ​​x. (8)
H Formula (6) calculates:
CVu . 0. 131 6
Product Partner Network
GB/T10094-2009
The parameter range and table distance of this coefficient table are: Appendix A
(Normative Appendix)
K Coefficient Table
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