Statistical interpretation of data;Estimation of the mean;Confidence interval

This standard applies to the estimation of the mean of a population based on a series of tests on random samples drawn from a normal population, when the population variance is unknown; or to the calculation of an interval using the data obtained from the test so that the interval contains the population mean with a given probability. The assumption that the probability of the population partially follows the normal distribution can be widely met: under certain experimental conditions, the distribution of the test results is often normal or approximately normal. This standard assumes that the individuals used for the measurement constitute an independent random sample from the original population. GB 3360-1982 Statistical processing and interpretation of data Estimation and confidence interval of the mean GB3360-1982 standard download decompression password: www.bzxz.net
This standard applies to the estimation of the mean of a population based on a series of tests on random samples drawn from a normal population, when the population variance is unknown; or to the calculation of an interval using the data obtained from the test so that the interval contains the population mean with a given probability. The assumption that the probability of the whole population follows a normal distribution can be widely met: under certain test conditions, the distribution of the test results is often a normal distribution or an approximate normal distribution. This standard assumes that the individuals used for the measurement constitute an independent random sample from the original population.

National Standard of the People's Republic of China
Statistical interpretation of dataEstimation of the mean-Confidence intervalUDC519.25:620
GB 3360--82
This standard applies to the estimation of the mean of a population when the variance of the population is unknown, or to the calculation of an interval using the data obtained from the test, so that the interval contains the mean of the population with a given probability. The assumption that the probability distribution of the population follows the normal distribution can be widely met: under certain experimental conditions, the distribution of the test results is often normal or approximately normal. This standard assumes that the individuals used for the measurement constitute independent random samples from the original population.
Transforming the origin or unit of the test results can simplify the calculation. No observations may be excluded or doubtful observations may be corrected without experimental, technical or other obvious reasons. The test method may be subject to systematic errors (errors with identifiable causes). This standard assumes that there are no systematic errors. The presence of systematic errors may invalidate the methods described below. The calculated interval is called the confidence interval of the mean, and is associated with the confidence level. The confidence level is the probability that the confidence interval contains the population mean, usually expressed as a percentage. This standard only considers the 95% and 99% levels. This standard is formulated with reference to the international standard ISO2602 "Statistical interpretation of test results-estimates and confidence intervals of the mean" (second edition in 1980).
1 Estimation of the mean
1.1 Case of no grouping
After eliminating suspicious data, this batch of data contains n observations; (i=1,2,,n), some of which may take the same value. Use the arithmetic mean x of n data to estimate the mean μX-Ir of the normal distribution
1.2 Case of grouping
When the number of data is large (for example, more than 50), they can be grouped at equal intervals. In some cases, it is also possible to obtain grouped data directly.
n; represents the frequency of the i-th group, that is, the number of data in the i-th group. k represents the number of groups, then
represents the midpoint of the i-th group, and the weighted arithmetic mean of the midpoints of all groups is used as the estimate of the mean u. k
National Bureau of Standards 1982-12-30 Issued
1984-01-01 Implementation
2 Confidence interval of the meanWww.bzxZ.net
GB3360—82
The confidence interval of the population mean can be calculated using the estimators of the population mean and standard deviation. Another way to calculate the confidence interval is to use the range, which is given in Appendix A. 2.1 Estimation of standard deviation
2.1.1 Ungrouped case
The estimation formula for the standard deviation α is as follows:
Where: αi-
the ith observation value (i=1.2,,n),
the total number of observations,
the arithmetic mean of n observations.
For the convenience of calculation, it is recommended to use the following formula S=.
2.1.2 Grouping situation
The estimation formula of standard deviation? is as follows:
For the convenience of calculation, it is recommended to use the following formula S=
In the formula:—
Zn (yi -j)2
n-1-=1
Midpoint of the i-th group (i=1,2,,k)
Number of groups:
ni—Number of observations in the i-th group
-Total number of observations, n=≥ni
Also—Weighted arithmetic mean of the medians of each group. 2.2 Confidence interval of the mean
For the selected confidence level (95% or 99%), determine the two-sided or one-sided confidence interval according to the specific situation. 2.2.1 Two-sided confidence interval
The two-sided confidence interval of the population mean is determined by the following double inequality: a.
At the confidence level of 95%,
b. At the 99% confidence level,
one-sided confidence interval
GB 3360—82
t o.975 s
± 0. 975-s <μ<5 +
t 0.995-s <μ< +
The one-sided confidence interval of the population mean is determined by one of the following inequalities: At the 95% confidence level, the value of
1-|tt||Table 1t,
1-g and
Confidence level (two-sided case) Confidence level (one-sided case) 95%
Confidence level (two-sided case)
Confidence level (one-sided case)
GB 3360--82
Continued Table 1
Confidence level (two-sided case)Confidence level (one-sided case) 95%
At the confidence level of 99%,
For the case of data grouping, it can be replaced by to.99
Confidence level (two-sided case)Confidence level (one-sided case) 95%
t0.975, t0.995, t0.95 and to.99 are all values ​​of the t distribution with v=n1 degrees of freedom. These values ​​are given in Table 1, along with the values ​​of Vn
, 10.95 and '09. When n is greater than 60, Table 2 can be used to calculate the value of t by linear interpolation from 120 to .995.
For example:
n = 250
t o.995 = 2. 576
3+0.48（2.617
—2.576)
Expression of results
3.1 Give the expression of the mean value according to 1.1 or 1.2. GB336082
Express the confidence interval in the form of the double inequality in 2.2.1 or the inequality in 2.2.2, state the confidence level (95% or 99%), and indicate the number of data excluded due to suspicion and the reasons for exclusion. 22
GB 3360--82
Appendix A
Confidence interval of the mean determined by range
(Supplement)
If the observations are arranged from small to large, that is, 2n, then R= is defined as the sample range. The population is still assumed to be normally distributed. When the number of observations is small, for example less than or equal to 12, the sample range can be used to determine the confidence interval of the population mean. The advantage of this algorithm is that it is relatively fast, but the disadvantage is that it usually derives a wider confidence interval and is sensitive to deviations from the normal assumption.
Two-sided confidence interval,
The two-sided confidence interval of the population mean is determined by the following double inequality: At a confidence level of 95%
- q0. 975 R<μ< + q0.975 R
At a confidence level of 99%
One-sided confidence interval
-q0.995 Rq0.95R
μ<+q0.9g R
or, μ>q0.9gR
The coefficients 90.975, 90.995, 90.95, 90.99 are given in the attached table. Attachment
Confidence level (two-sided case)
Additional notes:
This standard is proposed by the Standardization Institute of the Ministry of Electronics Industry. Confidence level (one-sided case)
This standard is drafted by the Standardization Institute of the Ministry of Electronics Industry, the Institute of Systems Science of the Academy of Sciences, and Harbin Institute of Technology. 23
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