Measures of Spread

The standard deviation is simply a measure of how far values typically sit from their average: small means tightly bunched, large means widely scattered. The formula looks like a lot, but read it from the inside out: $x - \bar x$ (how far one value is from the mean), squared (so values below the mean don't cancel those above), added up with $\sum$, divided by $n - 1$ (roughly an average), and finally a square root (to get back to the original units). It's an average distance from the mean with a couple of small adjustments, so bear with it.

Why square, and why the square root? You might wonder why we don't just average the distances $x - \bar x$. Try it: the deviations always sum to $0$, because the mean sits exactly where the values above and below balance. Squaring makes every deviation count as positive. But squaring also changes the units (dollars become dollars squared), so at the end we take the square root to get back to dollars.

Computing $s$:

1. Find each deviation $x - \bar x$. (They always sum to 0.)
2. Square them and add.
3. Divide by $n - 1$ to get the variance $s^2$.
4. Take the square root.

For a whole population, divide by $N$ instead and call it $\sigma$. Why $n - 1$ for a sample? The sample mean $\bar x$ is computed from the same data, so it sits closer to the data than the true mean does, and the squared deviations come out a little too small. Dividing by $n - 1$ instead of $n$ makes up for that.

The empirical rule. For bell-shaped data, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

z-scores put different scales on equal footing: a score with $z = 2$ is unusual whatever the test. Values with $|z| > 2$ are commonly called unusual. Notice how this ties back to the empirical rule: only about 5% of bell-shaped data are more than 2 standard deviations out.

A common mistake. It's tempting to compare raw scores across different tests, so an 88 beats an 82. But an 88 on an easy test with a wide spread can be less impressive than an 82 on a hard one. z-scores take the mean and the spread into account, so they're the fair comparison.

Manufacturers track the standard deviation of part sizes to control quality; investors call the standard deviation of returns "volatility"; growth charts for children are z-scores.