Some standard content:
GB/T 175601998
This standard is equivalent to ISO8595:1989 "Statistical processing and interpretation of data - Estimation of the median". The main differences from ISO8595 are as follows:
is changed to:
-2-la/2
2. An example without truncated data is added. 2*a/2
>2\a/2
3. Add the median estimation method of Wilcoxon signed rank test>. 4. Add the application conditions for censored data. 5. Delete the content of Chapter 4 of IS08595:19891. 6. There are changes in the arrangement. Appendix A of this standard is a prompt appendix. This standard is proposed and managed by the National Technical Committee for the Application of Statistical Methods. The drafting units of this standard are: China Institute of Standardization and Information Classification and Coding, Metal Products Research Institute of the Ministry of Metallurgy, Beijing University of Humanities, Institute of Systems Science of the Chinese Academy of Sciences, and Graduate School of the University of Science and Technology of China. The main drafters of this standard are: Yu Zhenfan, Liu Qiong, Chu Anjing, Sun Shanze, Ma Yilin, Zhang Jianfang, and Li Renliang. GE/T 17560—1998
ISO Foreword
ISO (International Organization for Standardization) is a worldwide federation of national standardization bodies (ISO members). The work of developing international standards is carried out through ISO's various technical committees. Each ISO member has the right to participate in the work of the relevant technical committee established for the subject in which it is interested. Other international organizations, both official and unofficial, that have ties with ISO may also participate in this work.
Draft international standards developed by technical committees are sent to group members before being accepted as international standards by the ISO Council. According to ISO procedures, these draft standards can only be approved as formal international standards if at least 75% of group members vote in favor. International standard ISO8595 was drafted and formulated by IS0/TC69 (Technical Committee for Standardization of Statistical Methods). 1 Scope
National Standard of the People's Republic of China
Statistical Processing and Interpretation of Data
Estimation of Median
Interpretation of statistical dala-Estlmation of a nedian
GB/T17560— 1998
egv IS0 8595: 1989
This standard gives the procedures for point estimation and interval estimation of the median of a population probability distribution by randomly selecting sample units from the population. These procedures give a non-parametric estimation method. The method described in this standard is applicable to any continuous distribution population. Note: If the population distribution can be considered to obey a stationary distribution, then the median is equal to the mean, and its confidence interval should be calculated according to CB336; if the probability distribution function of the population is known, other methods can be used to estimate the median; this method is only used to estimate the median when the distribution parameters are not well understood.
2 Referenced standards
The clauses contained in the following standards constitute the clauses of this standard through reference in this standard. When this standard is revised, the versions shown are valid. All standards will be revised. Parties using this standard should explore the possibility of using the latest versions of the following standards. GB/T3360-1982 Statistical processing and interpretation of data Estimation and confidence interval of the mean (ENV1S02854:1976) GB/T3358.1-1993 Statistical terminology Part 1: General statistical terms GB/T3358.2-1993 Statistical terminology Part 2: Statistical quality control terms 3 Definitions and special numbers
3. 1 Definitions
This standard adopts the definitions in GB/T3358.1-1993 and GB/T3358.2-1993. 3.1.1 Estimation
Inferring unknown population distribution parameters based on samples. (3.39 in GB/T3358.1-1993) 3.1.2 Estimator
is a statistic used to estimate the known parameters of the population distribution. (3.10 in GB/T3358.11993) 3.13 Estimate
is the result of the most accurate calculation of the estimated value based on the sample observations. (3.41 in GB/T3358.1--1993) 3.1.4 Two-sided confidence interval twu-sided confidence interval is the unknown quantity of the population distribution to be estimated, T, and T: are two statistics (T, T), so that the interval [TT, contains t with a certain probability. Then this interval is called a two-sided confidence interval. T, and T1 are called the upper and lower limits of the confidence interval respectively. (3.47 in GB/T3358.1-1993)
3.1.5 One-sided confidence interval one-sided confidence interval In the confidence interval [T:T, when the upper limit T: is , or the upper limit of the unknown quantity; or the lower limit 7, is x or the lower limit of the unknown quantity, the confidence interval is called a one-sided confidence interval. At this time, for the former, T: is called the lower confidence limit, and for the latter, T: is called the upper confidence limit. (3.48 in GB/T 3358. 1-- 1993)3.1.6 Confidence level, confidence level [T1,] is a one-sided confidence interval, 1-α is a constant between 0 and 1, if for all 8, there is PT>T)1-a, then 1- is called the confidence level of the confidence interval. When P(T, ≤6T)-1a, 1--α is also the confidence coefficient or confidence level. (3.49 in GB/T3358.1-1993) 3.1.7 Order statistic orderstatist1c8 Arrange the components of the sample in ascending order (2), called (1)2), \.() is the order statistic.) is called the th order statistic. (3.24 in GB/T3358.1-1993) 3.1.8 Median of a contnuous probability distribution When the population distribution is a continuous distribution F(), the median is the value M that makes F(M)=1/2. In this standard, M is called the population median.
3.2 SymbolswwW.bzxz.Net
This standard adopts the definitions and symbols in CB/T 3358.1-1993 and GB/T 3358.2-1993: F()
4 Point estimate
Value of the distribution function at
Population median
Sample size (3.7 in GB/T 3358. 1--1993) distribution parameter
Lower limit of the confidence interval
Upper limit of the confidence interval
Value of the th order statistic
1-α quantile of the standard normal distribution
1:6/2 quantile of the standard normal distribution
Confidence level (3.49 in GB/T 3358.1-1993) The point estimate of the population median M is given by the sample median. When it is an odd number, the point estimate of
M is: 2+a
5interval estimate
5. 1Confidence interval of M
When it is an even number,
The two-sided confidence interval of the population median M is a closed interval of the form _T,,,, where TT, and T are called the upper and lower limits of the confidence interval respectively.
One-sided confidence interval is T) or (-, T1. For the former, T is called the lower confidence limit; for the latter, T, is called the upper confidence limit. The actual meaning of the confidence interval of M is to make this interval contain the median M with a certain probability. 5.2 General method
The upper and lower limits of the two-sided confidence interval with a confidence level of 1-α are given by a pair of order statistics [±), (-+], where the integer table is determined by the following two formulas:
2ra/2
In the case of one side, α/2 should be replaced by.
GB/T 17560 - 1998
（2）
Note: The confidence coefficient of the confidence interval determined in this way is generally slightly greater than 1, unless the equality sign of the first formula above holds. Table 1 gives the corresponding values of the one-sided or two-sided confidence interval when 5≤30 and the confidence level 1-2 is 0.95 and 0.99. The confidence level is given by the following formula:
The confidence level is given by the following formula:
T,T<-+1)
here, a is the order statistic in the sample. When the sample size is too small, it is not possible to determine an appropriate confidence limit. 5.3 Approximate method
For the values of n assumed in Table 1, the approximate value of is given by the following formula y=(. 5(n+1-/n-0.5)
is the integer part of y, and is the quantile of the standard normal distribution determined by the following formula: Case of one-sided limit: -u. Case of two-sided limit: -41-4
For common values, this approximation is quite accurate. (3)
If a computer program is used, if n≤30, the value calculated by formula (4) is consistent with that in Table 1: -, when n>30, the value calculated by formula (4) is consistent with that calculated by formula (3). y - 0.5(m + 1 - 4 + 0.5 - 0.25u) is the integer part of y.
6 Application examples
6.1 When 5m30
Example 1 Test: Tensile strength of a batch of steel wires: Randomly select 15 steel wires and measure the resistance values such as F: (unit: N> (4)
1185.1188.1190,1194.1196,1200,1210,1210,1215,1220,1225,1228,1230,1235,1240 The median tensile strength value of the batch of steel wires is calculated by the point The value is: 3=1210, from the double test situation column in Table 1 -15. Find the corresponding zero value. This =4, the double-sided confidence interval of M with a confidence level of 0.96 is as follows:
The lower limit of the confidence interval: 7=T(4)=1194: Since n—+1-15-4+1=12, so
The upper limit of the confidence interval: T-242)-1228 Example 2; Perform an accelerated life test on a small device to simulate functional inspection. 24 life data were obtained in the experiment, and the data values are as follows: (unit: h (the 7 data marked with "\" are truncated data) 57.5,
The point estimate of the median life span is:
(::18, —2(11;,)/2=(105. 4+122. 5)/2=11498.4,
162.7\,
The lower limit of the one-sided confidence interval of the median with a confidence level of almost 0.95 can be found from the one-sided limit situation column in Table 1 when =24. The corresponding life span,
GB/T 17560— 1998
When the data is truncated at the upper end, the data r(L,+*,2+1)/z (when n is an odd number) must not be truncated, or the data (1),*aas+1)(when n is an even number) must not be truncated. When part of the data is truncated at the lower end, the data (C+1>>,\,r(m (when n is an odd number) must not be intercepted, or the data a, (when is an even number) must not be truncated. For interval estimation [(-*-, when part of the data is truncated at the upper end, the data s+*\cn-++ must not be truncated, and when part of the data is truncated at the lower end, the data)，,(m) must not be truncated. When this condition is not met, this activity fails. This note also applies to the example in 6.2.
6.2 When n≥30
In an accelerated life test, 34 transistors were selected from a batch of transistors, and their service life data values are as follows (unit, week) (the three data marked with "*\" are truncated data)
The point estimate of the median life is
(x(1n+xa))/2=(13+13)/2=13
The sample size n>30, in order to obtain the lower limit of the one-sided confidence interval of the median of M with a confidence level of 0.95, an approximate method is used. Since 1 —α- 0.95 = u1 - = gs = 1,645, so y = 0.5n + 1-un-0.5)
= 0.5 (34 + 1-1.645 / 34--0.5) 12.74
is the integer part of y, so table = 12. So we get the lower confidence limit T, a r (1a> -10. For the upper and lower limits of the bilateral confidence interval of M with a confidence level of 0.95, since 1α = 0.95, 1a/20.975 = 1-2 = %o 975-1.960. So
y -0.5 [ #+1-t Vn-0.5
=0.5(34+1-1.960~34-0.5)
Take =11,z—+1=34—11+1=24, so the double-sided confidence interval of M is [T1,T]=[z(3 +2r+]=[9,19 Standard: The data in this example are provided by the experiment done by wk et al. GB/T 17560—1998
The value of the number in the table is
The case of double-sided limits
The case of single-sided limits
The confidence level
Confidence level
Proof: "0\ means and under this confidence level, the sample size cannot be determined. The median estimation method of the Wllcoxon signed rank test (Wllcoxonsignedranktest) is shown in Appendix A (Suggested Appendix).
.A1 Scope of application
GB/T 17560 -- 1998
Appendix A
(Suggested Appendix)
The method of estimating the median and the sum using the Wilcoxon signed rank test is only applicable to the case where the population distribution is symmetric about a certain point. A2 Symbol uy
=,1n. That is, calculate
A2.1 According to the sample observations + construct all possible average values u,-++
a+ aa+a
tn-i+tn Ko
's value.
42. 2 Arrange this sequence in increasing order and there are N=n numbers in total, recorded as u,ue;uns2
A3 Point estimate
The median of the sequence is the point estimate of the population median. A4 Interval estimation
A4.1 Determine the value
According to the determined confidence level 1-α in Table A1, the town with the same industry can be obtained. A4.2 Determine the one-sided zone gate
A4.2.1 The lower limit of the one-sided confidence interval
The lower limit of the one-sided confidence interval is a monotonically increasing sequence, the th value. A4.2.2 Upper limit of one-sided confidence interval
The upper limit of one-sided interval is the \+1)2+1 value N of monotonically increasing sequence u-
A4.3 Determine two-sided confidence interval
The lower limit of confidence interval is the \+1)2+1 value N of monotonically increasing sequence u, and the upper limit of interval is the \+1)2+1 value N of monotonically increasing sequence, and the confidence interval is [)+.
A5 Application example
The data of 10 sample units drawn from a population are as follows: GB/T17560—1998
n is as follows
a) Calculate,
#2+=25.20
x+±=33.10
x+26=28.75
32+ 310=27.00
of=27.75
2±228.75
.40.45
3+21=33.40
2,+=27. 95
2+410= 34. 35
3-40=28.80
2g211 -27. 80
b) Arrange , into a monotonically increasing series as follows:
c) One-sided confidence interval
Serial number,
GB/T 17560-1998
Determine a=0.05,n10,N-10X11/2-=55From Table A1, it can be found that=12.
Serial number "
Serial number un
So the lower limit of the one-sided confidence interval is the 12th value of the monotonically increasing sequence u, 4(18 28.50, the upper limit of the one-sided confidence interval is the 10×11th value of the monotonically increasing sequence u.
1..-12-+-144th value)=34.85.
d) Two-sided confidence interval
Determine a=0. 05,210,N=10×11/2=55. From Table A1 4, we can find k=9. So the lower limit of the confidence interval is the 9th value of the monotonically increasing sequence, (9)-27. 75, and the upper limit of the confidence interval is the 10×11th value of the monotonically increasing sequence, 2(4)=35. 05.2
The confidence interval is [27.75,35.05].
Table A1 Values of Wilerxon estimated median Single-sided limit case
Confidence level
Double-sided limit case
Setting level
GB/T 17560—1998
A1(Complete)
Double-sided limit case
One-sided limit case
Setting level
Confidence level
Note: "0\ means that at this confidence level, the sample size cannot determine the confidence interval and confidence limit.
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