Spread and standard deviation

Two classes both average 75% on a test. In one, everyone scored between 72 and 78. In the other, half scored 95 and half scored 55. Same center, completely different stories. Spread is what tells them apart.

Same center, different spread

The center tells you where the data sits; the spread tells you how tightly it clusters there. Three measures, from crude to essential:

• Range — the largest value minus the smallest. Simple, but one outlier ruins it.
• Variance — the average of the squared distances from the mean.
• Standard deviation — the square root of the variance, back in the original units.

The standard deviation

The standard deviation σ is the one you'll use most. It measures the typical distance of a value from the mean:

$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}$

Reading it from the inside out: take each value's distance from the mean (xᵢ − μ), square it (so distances don't cancel out and big gaps count extra), average those, then take the square root to undo the squaring.

A small σ means values huddle near the mean; a large σ means they scatter widely.

Why square, then root?

Squaring does two jobs: it makes every distance positive (so values above and below the mean don't cancel), and it punishes big deviations more than small ones. The final square root returns the answer to the original units — so a standard deviation of test scores is measured in points, not points².

Why this matters

Standard deviation is the ruler of statistics. It's what "68% within one σ" (next module) is measured in, what a z-score counts, and how we say whether a value is ordinary or surprising. An average without a standard deviation is only half the story.