# What does variance measure?

Nov 10, 2015

As the name of the topic indicates variance is a "Measure of Variability"

#### Explanation:

The variance is a measure of variability. It means that for a set of data you can say: "The higher variance, the more different data".

Examples

• A set of data with small differences.

$A = \left\{1 , 3 , 3 , 3 , 3 , 4\right\}$

$\overline{x} = \frac{1 + 3 + 3 + 3 + 3 + 4}{6} = \frac{18}{6} = 3$

${\sigma}^{2} = \frac{1}{6} \cdot \left({\left(2 - 3\right)}^{2} + 4 \cdot {\left(3 - 3\right)}^{2} + {\left(4 - 3\right)}^{2}\right)$

${\sigma}^{2} = \frac{1}{6} \cdot \left(1 + 1\right)$

${\sigma}^{2} = \frac{1}{3}$

• A set of data with bigger differences.

$B = \left\{2 , 4 , 2 , 4 , 2 , 4\right\}$

$\overline{x} = \frac{2 + 4 + 2 + 4 + 2 + 4}{6} = \frac{18}{6} = 3$

${\sigma}^{2} = \frac{1}{6} \cdot \left(3 \cdot {\left(2 - 3\right)}^{2} + 3 \cdot {\left(4 - 3\right)}^{2}\right)$

${\sigma}^{2} = \frac{1}{6} \cdot \left(3 \cdot 1 + 3 \cdot 1\right)$

${\sigma}^{2} = \frac{1}{6} \cdot \left(6\right)$

${\sigma}^{2} = 1$

In set $A$ there are only 2 numbers other then the mean, and the difference is $1$. The variance is small.

In set $B$ there are no elements equal to mean, and this fact makes the variance bigger.