Standard Deviation Calculator

Sample or population, with the full working.

What it measures

Standard deviation is the typical distance of a value from the mean, in the same units as the data. A small value means the data clusters tightly; a large one means it spreads. Variance is its square, which is easier to work with algebraically but has unhelpful units — squared metres, squared marks.

The method

1. Find the mean.
2. Subtract the mean from each value to get its deviation.
3. Square each deviation. This removes the signs, which would otherwise sum to zero.
4. Add the squares.
5. Divide by n for a population, or n − 1 for a sample.
6. Take the square root, returning to the original units.

Sample or population — the choice that changes the answer

Use the population formula only when your data is the entire group you care about: every student in the class, when the class is the whole question. Use the sample formula when the data is a subset standing in for something larger.

Dividing by n − 1 is Bessel's correction. A sample's own mean is closer to that sample than the true population mean would be, so the squared deviations come out too small and the estimate is biased downward. Dividing by a smaller number corrects it. On the eight values above, the population figure is exactly 2 while the sample figure is about 2.138 — same data, different question.

The empirical rule

For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It is a useful sanity check and nothing more — it does not hold for skewed distributions.

Standard deviation and outliers

Because deviations are squared, a single distant value has an outsized effect. If one point is driving the whole result, report the interquartile range alongside it.