Answer to Question #152807 in Statistics and Probability for Khan

Question

Let A = 5 Consider the population A , A+2 ,A+4 , A+6 , A+8 Draw all posible sample of size 3 without replacement.If A is odd number verily the biasedness property of sample variance S^2

Answer 1

Samples of size 3 without replacement:

{5,7,9}; {5,7,11}; {5,7,13}; {5,9,11};{5,9,13}; {5,11,13}; {7,9,11}; {7,9,13}; {7,11,13}; {9,11,13}

If A is odd number verily the biasedness property of sample variance S².

$S^2_n=\frac{1}{n(n-1)}\sum_{i\neq j} \frac{(X_i-X_j)^2}{2} = \frac{1}{n(n-1)}((n-1)\sum_{i=1}^n X_i^2-\sum_{i\neq j} X_iX_j)$

The expectation of sample variance:

$E[S^2_n]=\frac{\sum_{i=1}^n E[X_i^2]}{n}-\frac{\sum_{i\neq j} E[X_i*X_j]}{n(n-1)}$

Since we have independent variables:

$E[X_iX_j]=E[X_i]*E[X_j]$

Let assume that:

$E[X_i^2]=\mu_2$ and $E[X_i]=\mu_1$

Then:

$\sum_{i=1}^n E[X_i^2]=n*\mu_2$ and $\sum_{i\neq j} E[X_i]E[X_j]=n(n-1)\mu_1^2$

Finally:

$E[S_n^2]=\mu_2-\mu_1^2=Var[X]$

So, S² is unbiased.