Population Standard Deviation

The population standard deviation is a measure of how widely the members of a whole population are spread around their average. It is written with the Greek letter sigma and is the square root of the population variance:

Here $x_i$ are the values, $\mu$ is the population mean and $N$ is the number of members. For the values 2, 4 and 6 the mean is 4, the squared deviations are 4, 0 and 4, their average is 2.67 and the standard deviation is 1.63. The formula divides by $N$ because every member is included. When only a sample is available, the sample standard deviation divides by one less than the sample size to correct a downward bias, and gives 2 for the same three numbers. In finance the standard deviation of returns is the usual measure of volatility. Since return histories are samples, the sample formula is normally the right one, though with hundreds of observations the two differ very little.