Standard Deviation Calculator

What is standard deviation?

Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster close to the average; a large one means they are spread far apart. It is the square root of the variance, which is the average of the squared distances from the mean. There are two versions: population standard deviation, used when your data is the whole group, and sample standard deviation, used when your data is a sample from a larger group. This calculator returns the mean, variance and both standard deviations.

How it's calculated

First find the mean. Then, for each value, subtract the mean and square the result. The population variance is the average of those squared differences (divide by n); the sample variance divides by n - 1 instead, which corrects for using a sample. The standard deviation is the square root of the variance. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the population variance is 4 and the population standard deviation is 2, while the sample variance is about 4.571429 and the sample standard deviation about 2.138090.

Example

For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 divided by 8 = 5. The squared differences from 5 are 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32. Dividing by 8 gives a population variance of 4, so the population standard deviation is the square root of 4 = 2. Dividing by 7 (n - 1) gives a sample variance of about 4.571429 and a sample standard deviation of about 2.138090.