Standard Deviation Explained
Two data sets can have identical means and look nothing like each other. Standard deviation is the number that tells them apart. The Standard Deviation Calculator gives it along with the mean and variance.
What it measures
The average distance of each value from the mean - roughly. The method: find the mean, subtract it from each value, square the differences, average those, then take the square root.
The squaring is the part that needs explaining. Without it the positive and negative differences cancel to exactly zero every time, which is useless. Squaring makes them all positive, and the final square root undoes the squaring so the answer comes back in the same units as the data. That is the whole reason standard deviation is quoted rather than variance: variance is in squared units, which nobody can picture.
So a set of heights in centimetres has a standard deviation in centimetres, and a standard deviation of 7 cm means something you can imagine.
Why n-1 for a sample
Two versions exist, and the difference is not a technicality.
Population standard deviation divides by n - the number of values - and is correct when your data is the entire group you care about.
Sample standard deviation divides by n-1 and is correct when your data is a sample from a larger population you are trying to describe.
The reason is that a sample's own mean is always closer to its own values than the true population mean would be, so the squared differences come out slightly too small. Dividing by the smaller number corrects the bias. With small samples the difference is large - at n=5, dividing by 4 instead of 5 raises the answer by about 12 per cent. At n=1,000 it is negligible.
If you are not sure which you have, you almost certainly have a sample.
The 68-95-99.7 rule
For data that is roughly normally distributed, about 68 per cent of values lie within one standard deviation of the mean, 95 per cent within two, and 99.7 per cent within three.
That is what makes the number intuitive. Adult male height in the UK averages around 175 cm with a standard deviation of about 7, so roughly two-thirds of men are between 168 and 182, and 95 per cent between 161 and 189.
It also gives you a sense of what is unusual. Three standard deviations out is about one in 370 - which is why manufacturing and quality control are expressed in these terms.
The rule only holds for normal distributions. Income, for instance, is heavily skewed, and the mean and standard deviation describe it poorly - which is why incomes are reported as medians. The Average / Median Calculator covers that distinction.
Common questions
Can standard deviation be negative?
No - it is a square root of an average of squares. Zero means every value is identical.
What is the coefficient of variation?
Standard deviation divided by the mean, as a percentage. It lets you compare spread between data sets with very different scales.
Is a high standard deviation bad?
Only in context. In manufacturing, yes. In a portfolio, it is risk, which some people want.
Got a data set to measure?