Variance vs Standard Deviation

Both measure spread; the difference is units. Variance is the average of squared deviations from the mean — so its units are squared (grams², points²). Standard deviation is the square root of variance, which puts spread back into the data’s own units. That’s the entire distinction.

Why keep both?

Variance is the mathematician’s tool: variances of independent variables add cleanly, which makes it the engine inside ANOVA, regression and portfolio theory. Standard deviation is the communicator’s tool: "±5 points" means something; "25 points²" doesn’t. You compute with variance, you report with standard deviation — and you standardize with it too, since σ is the denominator of every z-score.

Worked example

Data: 4, 6, 8, 10, 12 (mean 8). Squared deviations: 16, 4, 0, 4, 16 → sum 40 → variance 40/5 = 8 → standard deviation √8 ≈ 2.83. Both describe the same spread; only one speaks the data’s language.

One caution before you compute anything: check whether you have the whole population or just a sample — the divisor changes, and that N vs n−1 difference is its own classic exam trap.