Variance

The average squared distance of each value from the mean — a measure of spread.

Same mean, different variance

Variance ignores where the mean sits and measures how far the values are from it, after squaring those distances.

Definition

Variance measures how spread out the values in a dataset are around the mean. A small variance means the values cluster tightly together; a large variance means they are widely scattered.

To compute variance:

Find the mean $\bar{x}$ .
Subtract the mean from each value to get the deviation.
Square each deviation (so negatives become positive).
Average the squared deviations.

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2

Computing variance

Dataset: 2, 4, 4, 4, 5, 5, 7, 9. Mean $= \bar{x} = 5$ .

Deviations: $-3, -1, -1, -1, 0, 0, 2, 4$

Squared deviations: $9, 1, 1, 1, 0, 0, 4, 16$

\sigma^2 = \frac{9+1+1+1+0+0+4+16}{8} = \frac{32}{8} = 4

Variance has squared units

If the data is in metres, variance is in metres². That makes variance hard to interpret directly. The standard deviation $\sigma = \sqrt{\sigma^2}$ brings things back to the original units.

Try it

Find the variance of: 1, 3, 5, 7, 9.

Solution

Mean $= 5$ . Deviations: $-4, -2, 0, 2, 4$ . Squared: $16, 4, 0, 4, 16$ .

$\sigma^2 = \frac{16+4+0+4+16}{5} = \frac{40}{5} = 8$

Related concepts

Statistics· Descriptive

MeanThe arithmetic average — the sum of all values divided by how many there are.

Statistics· Descriptive

Standard DeviationThe square root of variance, expressing spread in the same units as the data.