Standard Deviation Calculator

Standard deviation measures how spread out a set of numbers is. The equation for standard deviation, with SS the sum of squared deviations from the mean, Σ(x - mean)², and n the count:

s = sqrt(SS / (n - 1)) for a sample, and σ = sqrt(SS / n) for a population.

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the deviations are -3, -1, -1, -1, 0, 0, 2, 4, and their squares sum to 32. Divided by 8 that is a variance of 4 and a standard deviation of exactly 2.

The squaring is not arbitrary. The deviations always sum to zero (that is what makes the mean the mean), so averaging them directly gives 0 for every dataset. Squaring removes the signs, and the square root at the end puts the answer back into the units of the original data. Variance, the figure before the square root, is in units squared, which is why standard deviation is the one that gets quoted.

Now the divisor. Dividing by n gives the POPULATION standard deviation and is right when your numbers are every case you care about: the scores of all thirty students in a class, when the class is the whole question. Dividing by n-1 gives the SAMPLE standard deviation and is right when your numbers are a sample from something larger that you want to say something about.

The n-1 is Bessel's correction, and the reason for it is subtle. You are measuring spread around the sample mean, but the sample mean is itself computed from the same data, and it necessarily sits closer to those particular values than the true population mean does. So the squared deviations come out too small, systematically, and dividing by n would carry that understatement through. Dividing by n-1 corrects it exactly.

On the eight values above, the two answers are 2.00 and 2.14, about 7% apart. On ten values the gap is around 5%, on a hundred it is half a percent, and on a thousand it is negligible. So it matters most on small datasets, which is exactly where most people are working. There the difference is small enough to pass unnoticed while being large enough to change a conclusion. Most software defaults to the sample version; spreadsheet functions split, with STDEV.S and STDEV.P in Excel being the two.

What standard deviation is bad at is data with an extreme value in it. It squares the deviations, so a single outlier dominates: adding one value of 100 to the list above takes the standard deviation from 2.14 to 32.6. The interquartile range (the width of the middle half of the data) barely moves, which is why box plots and the 1.5 x IQR outlier rule exist and why median and IQR are the better summary for skewed data.

The empirical rule is the thing that makes a standard deviation interpretable: in roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. The caveat matters: it is a property of the normal distribution, not of data in general, and income, waiting times and most things with a hard lower bound are not remotely normal.