Question

A population consists of the elements 3, 5, 7, 8, and 10. What is the population variance?

Round off your answer to the nearest hundredths.

Answer 1

Population mean:

$\mu=\cfrac{3+5+7+8+10}{5}=6.6.$

Population variance:

$\sigma^2=\sum(x_i-\mu)^2\cdot P(x_i),$

$X-\mu=\\ =\begin{Bmatrix} 3-6.6,5-6.6,7-6.6,8-6.6,10-6.6 \end{Bmatrix}=$

$=\begin{Bmatrix} -3.6, - 1.6,0.4,1.4,3.4 \end{Bmatrix},$

$\sigma^2=(-3.6)^2\cdot \cfrac{1}{5}+(-1.6)^2\cdot \cfrac{1}{5}+\\ +0.4^2\cdot \cfrac{1}{5}+1.4^2\cdot \cfrac{1}{5}+3.4^2\cdot \cfrac{1}{5}=5.84.$