Variance: The Square of Standard Deviation

What is variance?

Variance is the average of the squared deviations from the mean. It measures spread in squared units. Standard deviation is its square root.

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

Step-by-step: {10, 12, 14, 16, 18}

Step 1 — Mean: $\bar{x} = (10+12+14+16+18)/5 = 70/5 = 14$

Step 2 — Deviations: $-4, -2, 0, +2, +4$

Step 3 — Squared deviations: $16, 4, 0, 4, 16$

Step 4 — Average: $\sigma^2 = (16+4+0+4+16)/5 = 40/5 = 8$

$\boxed{\text{Variance} = 8, \quad \text{SD} = \sqrt{8} \approx 2.83}$

Why square?

Squaring ensures all deviations are positive (no cancellation) and penalizes large deviations more heavily. Variance = $\sigma^2$; SD = $\sigma$.

Shortcut formula

$\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 = \frac{10^2+12^2+14^2+16^2+18^2}{5} - 14^2 = \frac{1000}{5} - 196 = 200 - 196 = 4$

Wait — let me recheck: $100+144+196+256+324 = 1020$. $1020/5 = 204$. $204 - 196 = 8$ ✓.

Try it in the visualization

Squared deviations are drawn as literal squares with areas proportional to $(x_i - \bar{x})^2$. The variance is the average area.