The formula
s = √( Σ(x − x̄)² / (n − 1) )
What it means
Standard deviation says how far the values spread from their mean, and it is in the same units as the data: measure heights in centimetres and the deviation is in centimetres too. You subtract the mean from each value, square the differences so they do not cancel out, average them and take the square root. There are two versions, and the difference is the denominator: the population one divides by n and the sample one by n − 1.
How to work it out by hand
- Work out the mean of all the values
- Subtract the mean from each value and square each difference
- Add all those squares together
- Divide by n − 1 for a sample, or by n for a whole population, and take the square root
What is worth knowing
Bessel's correction — dividing by n − 1 instead of n — is not an arbitrary tweak. When you work the mean out from the data itself, those values sit slightly closer to their own mean than they would to the real population mean, so the variance comes out short. Taking one off the denominator corrects it. With many values the difference is negligible; with five it is twenty-five per cent, and there it matters which one you pick.