Standard Deviation Calculator

Measure how tightly values cluster

Paste numbers separated by commas, whitespace, or line breaks. Choose population when the list is the full group of interest; choose sample when it represents observations from a larger group.

What standard deviation tells you

Standard deviation describes spread in the same unit as the original data. A small value means observations sit relatively close to their mean. A larger value means they are more dispersed. It does not say whether the data are good, bad, accurate, or normally distributed.

Calculation steps

The calculator finds the arithmetic mean, subtracts that mean from every value, squares those differences, and adds them. Population variance divides the sum by N, the number of values. Sample variance divides by N − 1 to correct the typical downward bias when estimating spread in a larger population. Standard deviation is the square root of variance.

Worked example

For the population 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5. The squared differences add to 32. Divide by eight to get a population variance of 4, then take the square root: the population standard deviation is 2. Treating the same numbers as a sample divides by seven instead, producing variance about 4.5714 and standard deviation about 2.1381.

Population or sample?

Use population mode when the values are every item you intend to describe—for example, all scores in one completed small class if that class alone is the subject. Use sample mode when those observations stand in for a broader process or group. The choice depends on the question, not merely the number of rows.

Outliers and interpretation

Squaring gives large deviations extra influence, so one extreme value can move the result sharply. Look at the source data and mean alongside the standard deviation. For skewed data such as incomes or response times, the median and percentile range may explain typical behavior better. Comparing deviations across unlike units or very different means can also mislead.

Accuracy and limitations

Input must contain finite ordinary numbers. Sample mode needs at least two values. Results are rounded for display, although calculations use JavaScript floating-point numbers. The page does not calculate confidence intervals, weighted deviation, grouped-frequency data, statistical significance, or data quality. Browser floating-point rounding can appear in extremely large, tiny, or high-precision values.

Standard Deviation FAQ

Can standard deviation be negative?

No. Squared differences and their square root cannot produce a negative result.

What does zero mean?

Every entered value is identical.

Why are sample and population results different?

Sample variance divides by N − 1, producing a larger estimate unless all deviations are zero.