Standard Deviation Calculator

What standard deviation tells you

Standard deviation measures how spread out a set of numbers is around its mean. A small value means the data clusters tightly; a large value means it is dispersed. Because it is expressed in the same units as the original data, it is directly interpretable in a way variance is not.

Two data sets can share an identical mean and behave completely differently. The sets {48, 49, 50, 51, 52} and {10, 30, 50, 70, 90} both average 50, but the first has a standard deviation near 1.6 and the second near 31.6. Reporting a mean without a measure of spread hides most of what is interesting about the data.

A worked example

Take the eight values 2, 4, 4, 4, 5, 5, 7, 9. Their mean is 5. Subtract the mean from each value, square the differences, and add them up; the population variance is that sum divided by 8, and the standard deviation is its square root. The sample version divides by 7 instead.

Sample versus population, and why n−1

The population formula divides the sum of squared deviations by n. The sample formula divides by n−1. Use the population formula only when your data covers every member of the group you care about — the test scores of all 30 students in a class, if the class is the whole subject of interest. Use the sample formula when your data is a subset used to estimate a larger group, which is the more common situation.

The n−1 adjustment is Bessel's correction. A sample's own mean is, by construction, the value that minimizes the squared deviations within that sample, so deviations measured from it are systematically smaller than deviations from the true population mean. Dividing by n−1 rather than n compensates for this bias. The difference matters most with small samples: at n=5 it inflates the estimate by about 12%, while at n=1000 it is negligible.

The empirical rule

For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is what makes standard deviation useful for judging whether an individual observation is unusual.

With a mean of 100 and a standard deviation of 15, a value of 130 sits two standard deviations above the mean, placing it above roughly 97.5% of the distribution. The rule assumes approximate normality, so it should not be applied to strongly skewed data or distributions with heavy tails.

Related measures

Variance is the square of the standard deviation. It has useful mathematical properties — variances of independent variables add — but its units are squared, so a variance of 25 square-dollars is hard to interpret directly.

The coefficient of variation, standard deviation divided by the mean and expressed as a percentage, allows comparison of spread between data sets with different scales or units. It is only meaningful for data with a true zero and a positive mean. For skewed data, the interquartile range often describes spread better than standard deviation, because it is not pulled around by outliers.