Standard Deviation Calculator
Calculate population or sample standard deviation from a list of numbers. Results also include variance, arithmetic mean, count and sum so you can review the main statistics together.
How to use the Standard Deviation Calculator
- Enter numbers separated by commas, spaces, semicolons or line breaks.
- Choose Population when the entered values represent the full population or Sample when they are a sample from a larger population.
- The calculator finds the arithmetic mean of the data.
- Each value's difference from the mean is squared and the squared differences are summed.
- Population variance divides by n, while sample variance divides by n − 1. Standard deviation is the square root of variance.
Examples
Population standard deviation
For the population data 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5 and the population standard deviation is 2.
Sample calculation
When the values are treated as a sample, the denominator is n − 1 rather than n, producing a slightly larger variance and standard deviation.
Identical values
If every value is identical, every deviation from the mean is zero, so the variance and standard deviation are both zero.
Frequently asked questions
What does standard deviation measure?
Standard deviation measures how spread out values are around their arithmetic mean. A smaller value indicates tighter clustering around the mean.
What is the difference between population and sample standard deviation?
Population standard deviation divides the squared-deviation total by n. Sample standard deviation uses n − 1 to estimate variability in a larger population.
What is variance?
Variance is the average squared distance from the mean under the chosen population or sample formula. Standard deviation is the square root of variance.
Can standard deviation be negative?
No. Squared deviations are nonnegative, so variance and standard deviation cannot be negative.
Does order matter when calculating standard deviation?
No. Reordering the same values does not change the mean, variance or standard deviation.
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