Standard Deviation Calculator

Standard deviation measures how spread out values are around the mean. A small standard deviation means values cluster tightly near the average; a large one means they are scattered widely. For example, test scores of 78, 80, 82 have a much smaller standard deviation than scores of 50, 80, 110, even though both sets share the same mean of 80.

Population standard deviation divides by n and is correct when your dataset contains every value in the group. Sample standard deviation divides by n - 1 (Bessel's correction) and is the right choice when your data is a subset drawn from a larger group, which fits most real surveys and experiments. This calculator defaults to sample mode and requires at least 2 values in that mode.

The calculator finds the mean, subtracts it from each value, squares each difference so negatives do not cancel positives, sums those squared differences, divides by n - 1 for a sample or n for a population, then takes the square root. Results are displayed to four decimal places.

For a roughly normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. If your dataset has a mean of 100 and a standard deviation of 15, you would expect about 95% of values to land between 70 and 130. This rule is a quick way to spot outliers, but it only holds when the data is approximately bell-shaped.

Variance is the average squared deviation and is easier to work with algebraically, since variances from independent sources add directly. Standard deviation is the square root of variance, which puts it back in the same units as your original data and makes it easier to interpret. This calculator reports both so you have each value available depending on what your analysis needs.