Question

A test is given to a class of 3 students. Marks obtained are 2,4 and 6, assume the students constitute the population. Find the standard deviation of the population

List all possible samples of size 2 that can drawn with replacement from the population

Determine the mean of each sample

Determine the mean of the sample means

Determine the standard deviation of sample means

Answer 1

Lets list all such samples and calculate their means

(2, 4) $m={\frac {2+4} 2}=3$

(4, 2) $m={\frac {2+4} 2}=3$

(2, 6) $m={\frac {2+6} 2}=4$

(6, 2) $m={\frac {2+6} 2}=4$

(4, 6) $m={\frac {6+4} 2}=5$

(6, 4) $m={\frac {6+4} 2}=5$

Lets calculate M - mean of the sample means

$M={\frac{2*3+2*4+2*5} 6}=4$

Lets calculate $\sigma$ - the standard deviation of the sample means

$\sigma=\sqrt{{\frac {2(3-4)^2+2(4-4)^2+2*(5-4)^2} {6-1}}}=\sqrt{\frac 4 5}\approx 0.89$