What is a standard deviation calculator?
A standard deviation calculator measures how spread out a set of values is around its arithmetic mean. It can describe a complete population or estimate variability from a sample. To calculate the mean or weighted average separately, use the Average Calculator.
How does the standard deviation calculator work?
The calculator uses Welford's incremental algorithm to update the mean and the sum of squared deviations as each value is processed. This avoids a less stable formula that subtracts two potentially large, nearly equal quantities.
What formulas are used?
The Square Root Calculator can be used for a focused square-root calculation.
Population vs. sample standard deviation
Use Population when the values are the complete group you want to describe. Use Sample when the values are a sample from a larger population and the goal is to estimate that population's variability. The sample formula uses n − 1 in the denominator.
How do you use this standard deviation calculator?
- Select Population or Sample.
- Enter the data values.
- Select Calculate Standard Deviation.
- Review the standard deviation, mean, variance, count, and range.
Standard deviation example
For the population 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5, the population variance is 4, and the population standard deviation is 2. For sorted center comparisons, see the Median Calculator.
How should you interpret standard deviation?
A value of zero means all observations are identical. Larger standard deviations indicate more dispersion around the mean, but whether a value is “large” depends on the units and context of the data. Percentage comparisons can be calculated with the Percentage Calculator.
What are the calculator's limits?
Up to 10,000 finite values are accepted. Sample mode requires at least two observations. Browser double-precision arithmetic is used, so extremely large values or very fine differences near the limits of floating-point precision can reduce accuracy.
Frequently asked questions about standard deviation
Methodology and calculation scope
The calculation uses Welford's online algorithm for the running mean and M2 value. Population variance is M2/N; sample variance is M2/(n−1). Small negative variance values caused only by floating-point noise are clamped to zero before taking the square root.