# What is the variance of the following numbers?: {2,9,3,2,7,7,12}

Apr 10, 2016

$\text{Variance"_"pop.} \approx 12.57$

#### Explanation:

Given the terms: $\left\{2 , 9 , 3 , 2 , 7 , 7 , 12\right\}$

Sum of terms: $2 + 9 + 3 + 2 + 7 + 7 + 12 = 42$
Number of terms: $7$
Mean: $\frac{42}{7} = 6$

Deviations from Mean: $\left\{\left\mid 2 - 6 \right\mid , \left\mid 9 - 6 \right\mid , \left\mid 3 - 6 \right\mid , \left\mid 2 - 6 \right\mid , \left\mid 7 - 6 \right\mid , \left\mid 7 - 6 \right\mid , \left\mid 12 - 6 \right\mid\right\}$

Squares of Deviations from Mean: $\left\{{\left(2 - 6\right)}^{2} , {\left(9 - 6\right)}^{2} , {\left(3 - 6\right)}^{2} , \left(2 - {6}^{2}\right) , {\left(7 - 6\right)}^{2} , {\left(7 - 6\right)}^{2} , {\left(12 - 6\right)}^{2}\right\}$

Sum of Squares of Deviations form Mean: ${\left(2 - 6\right)}^{2} , + {\left(9 - 6\right)}^{2} + {\left(3 - 6\right)}^{2} + \left(2 - {6}^{2}\right) + {\left(7 - 6\right)}^{2} + {\left(7 - 6\right)}^{2} + {\left(12 - 6\right)}^{2} = 88$

Population Variance $= \left(\text{Sum of Squares of Deviations from Mean")/("Count of Terms}\right) = \frac{88}{7} \approx 12.57$