Standard Deviation Calculator

Find the standard deviation for population or sample data.

How standard deviation works

The standard deviation measures how spread out the values are around the mean. A small standard deviation means the values are close to the mean; a large one means they are widely scattered.

Population and sample use different formulas. The population standard deviation divides by N; the sample standard deviation divides by n−1 (Bessel's correction) because it estimates the population value from a sample.

The variance is the standard deviation squared. Both are reported so you can see the relationship between the two measures of spread.

The formulas

Population: σ = √(Σ(x−μ)² ÷ N) · Sample: s = √(Σ(x−x̄)² ÷ (n−1))

Subtract the mean from each value, square the differences, sum them, divide by N (population) or n−1 (sample), and take the square root. Use the population formula when your data covers the entire group, and the sample formula when it is a sample.

Worked example

Values 2, 4, 4, 4, 5, 5, 7, 9

The mean is 5. The sum of squared deviations is 32. As a population: √(32 ÷ 8) = 2. As a sample: √(32 ÷ 7) ≈ 2.14.

Frequently asked questions

What is the difference between population and sample standard deviation?

Population divides by N and describes the entire group. Sample divides by n−1 (Bessel's correction) and estimates the population value from a sample, which avoids underestimating spread.

Why do I need at least two values for a sample?

With n−1 as the denominator, a single value would divide by zero. You need at least two observations to estimate a sample standard deviation.

What does a standard deviation of zero mean?

Every value is identical to the mean — there is no spread at all.

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