standard deviation (SD)

The standard deviation (SD) measures how far values typically sit from the mean. A small SD means the values are packed tightly around the mean; a large one means they are scattered. Unlike the variance, it is in the same units as the data (hours, dollars, orders), which is why it is the spread measure people actually report: "delivery takes 24 ± 8 hours".

Step by step, for a sample of $n$ values:

1. Compute the mean $\bar{x}$.
2. Subtract it from every value to get the deviations $x_i - \bar{x}$. They always sum to zero, so you can't simply average them.
3. Square each deviation and add the squares up.
4. Divide by $n - 1$. That is the sample variance $s^2$.
5. Take the square root.

$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}}$

If your data is the whole population rather than a sample, divide by $n$ instead. In product analytics you almost always have a sample, so $n - 1$ is the default: STDEV.S in Sheets or Excel, as opposed to STDEV.P.

For roughly bell-shaped data (see normal distribution) about 68% of values fall within one SD of the mean and about 95% within two. On skewed data that rule breaks down, and because deviations are squared, a few outliers can inflate the SD a lot. There the interquartile range is the safer partner of the median. To compare spread across metrics on different scales, divide the SD by the mean to get the coefficient of variation. And don't confuse the SD, the spread of individual values, with the standard error, the uncertainty of a mean.