Variance
The range
looks only at the two most extreme values. Variance does something far more
honest: it asks how far every data point sits from the
mean, and averages
those distances. It is the workhorse measure of spread.
For data x_1, x_2, \dots, x_n with mean \bar{x}:
\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}\left(x_i - \bar{x}\right)^2.
Each term (x_i - \bar{x}) is a deviation — how far
that one point strays from the centre. Variance is the average squared
deviation.
Why square? Because raw deviations cancel
It is tempting to just average the deviations themselves. But that always gives zero — the
mean sits exactly at the balance point, so the points below it cancel the points above:
\sum_{i=1}^{n}\left(x_i - \bar{x}\right) = 0 \quad\text{always.}
Squaring fixes the sign: a deviation of -3 and one
of +3 both contribute 9 instead of
cancelling. Squaring also punishes big misses — a point twice as far away
contributes four times as much — so variance is especially sensitive to far-flung values.
See it: variance is the average square
The five points share a mean of 7 (the dashed line). Above each
point sits a square whose side is that point's distance from the mean, so its
area is the squared deviation (x_i - \bar{x})^2. Drag the
slider to spread the points out: the squares grow, and the reported variance — their average
area — climbs with them.
Notice the point sitting on the mean has a square of side zero: it adds nothing. Only
distance from the centre counts.
The units are squared too
Because we squared the deviations, the variance carries squared units. If the
data are heights in centimetres, the variance is in \text{cm}^2 — a
slightly awkward quantity to interpret. That single inconvenience is exactly what the
standard
deviation is built to repair.
- Variance is the average squared deviation from the mean: \sigma^2 = \dfrac{1}{n}\sum (x_i - \bar{x})^2.
- Raw deviations always sum to zero; squaring removes the sign so they no longer cancel, and makes large deviations count disproportionately.
- Its units are the square of the data's units — the reason the standard deviation takes a square root.
- A larger variance means the data are more spread out around the mean; a variance of 0 means every value equals the mean.