Math and measurement tools
Standard Deviation Calculator
Calculate sample and population standard deviation, variance and mean. Paste a list, compare both conventions and see the calculation with private printable results.
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How to use this tool
1. Paste up to 1,000 equally weighted numbers. Separate them with commas, spaces, semicolons or new lines; commas are not thousands separators.
2. Choose Sample for observations from a larger group, or Population for the complete group you are describing. Sample calculations need at least two values.
3. Calculate to see standard deviation, variance, mean, count and both conventions side by side. Copy and print omit the source list.
Standard deviation and variance
Variance measures the squared distance of values from their mean. Standard deviation is its square root and returns to the original units. Population variance divides the sum of squared deviations by n; sample variance divides by n − 1.
Worked example
For 2, 4, 4, 4, 5, 5, 7 and 9, the count is 8, mean is 5 and sum of squared deviations is 32. Population variance is 32 ÷ 8 = 4, so population standard deviation is 2. Sample variance is 32 ÷ 7, giving sample standard deviation approximately 2.13809.
Questions and assumptions
Which convention should I choose?
Population describes the entire set of interest. Sample uses the n − 1 denominator when estimating variance from a sample of a larger population. The choice depends on what the data represents, not just its size. This tool cannot determine whether your sample is representative.
Precision and unusual lists
Each value may have up to six decimal places and magnitude up to 1 billion. Fixed-point integer sums and squared sums avoid cancellation when large values differ only slightly. Square roots and displayed results are approximate to 12 significant digits. Identical values have zero spread. A one-value population has zero standard deviation; sample deviation is unavailable.
What is not inferred?
The tool reports descriptive spread. It does not assume a normal distribution, assign confidence intervals, remove outliers or infer that a percentage of values lies within a given number of standard deviations. Variance has squared units; standard deviation has the same units as the input.
Sources and further reading
Built and maintained by Custom Logic. Page updated .