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Standard Deviation Calculator

Enter your data to find the sample or population standard deviation, the variance, the mean and the sum of squares, with a step-by-step working.

Formulas last reviewed: September 2026

Disclaimer: Results are calculated with standard formulas and are meant for learning and quick checks. Please double-check important work, especially anything used for exams, engineering, finance or safety.

How spread out is your data?

The standard deviation measures how far the values in a data set typically sit from the mean. A small standard deviation means the numbers are clustered tightly; a large one means they are spread widely.

There are two versions. Use the population standard deviation (divide by n) when your data is every member of the group. Use the sample standard deviation (divide by n − 1) when your data is only a sample drawn from a larger group, which is the usual case.

How it is calculated

Mean = sum ÷ n
Variance = sum of (x − mean)² ÷ n (or n − 1 for a sample)
Standard deviation = √variance

Worked example

Data: 10, 12, 23, 23, 16, 23, 21, 16

Mean
18
Sum of squared deviations
192
Sample SD (n − 1)
5.2372
Population SD (n)
4.899

Reading the number

  • In a bell-shaped data set about 68% of values lie within one SD of the mean, 95% within two and 99.7% within three.
  • Standard deviation is in the same units as the data, while variance is in squared units.
  • The coefficient of variation (SD ÷ mean) lets you compare spread between data sets on different scales.

A quick way to read the number

For roughly bell-shaped data, about 68% of values sit within one standard deviation of the mean, and about 95% sit within two. So a class with a mean score of 70 and a standard deviation of 8 has most students scoring between 62 and 78, and nearly everyone between 54 and 86.

Frequently asked questions

When should I use sample instead of population?

Use sample whenever your numbers are a subset of a bigger group, such as a survey of 100 people out of a city. Use population only if you have every value.

Why divide by n − 1 for a sample?

A sample tends to underestimate the true spread, and dividing by n − 1 corrects that bias. It is called Bessel's correction.

Can the standard deviation be negative?

No. It is a square root, so it is always zero or positive. It is zero only when every value is identical.

What is the difference between variance and standard deviation?

Variance is the average squared deviation, and the standard deviation is its square root, which brings the answer back to the original units.