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Chapter 10. Theorems of Statistics II

Higher category: 【Statistics】 Statistics Overview

1. Calculation of sample standard deviation with-replacement

2. Calculation of sample standard deviation without-replacement

1. Calculation of sample standard deviation with-replacement

⑴ theorem

for samples X₁, X₂, ···, X_n extracted from population, the definition of sample standard deviation is not 

but 

then why? when the sample group and the population are different, it is common not to know the value of the population standard deviation σ when estimating the population mean m, so the population standard deviation can be replaced by the sample standard deviation. when trying to replace σ, which is more appropriate between Sn and S?

⑵ proof

the following is the expected value of S_n², i.e. E(S_n²) 

because E(S_n²) = (n -1) / n × σ², S_n² tends to get smaller than the variance of population, σ². as the expected value of Sn2 multiplied by n/n/(n - 1) is the same with S², the following holds

in conclusion, S² is more appropriate than S_n² when replacing population variance

2. calculation of sample standard deviation without-replacement  

⑴ theorem

when we take a sample of size n without-replacement from a population with size N, population mean m, and population standard deviation σ, the variance of sample group is as follows: 

⑵ proof

variables in population can be expressed as a₁, a₂, ···, a_N. let’s say b_i = a_i - m, then

when extraction n samples x₁, x₂, ···, x_n without-replacement from the population, for all possible sample means, each a_i appears _N-1C_n-1 times  

the variance of sample mean is the mean of the squared deviation

at the above, b₁², b₂², ···, b_N² appear _N-1C_n-1 times respectively, and b₁b₂, ···, b_N-1b_N appear _N-2C_n-2 times respectively. there the conclusion is as follows:

Input : 2019.06.18 21:39