Standard Deviation Calculator

Standard deviation measures how far values typically sit from the mean. Choose “sample” when your numbers are a subset of a larger group.

Standard deviation of a list

Separate numbers with commas, spaces or new lines.

Sample standard deviation2.1380899353
Sample variance
4.5714285714
Mean
5
Count
8
Population standard deviation
2

Working

  1. Mean: 5
  2. Subtract the mean from each value and square the result.
  3. Add the squares and divide by n − 1 = 7: variance = 4.5714285714
  4. Take the square root: 2.1380899353

Formulas

population: σ = √( Σ(x − μ)² ÷ N )

sample: s = √( Σ(x − x̄)² ÷ (n − 1) )

Variance is the same calculation without the final square root. Its units are squared (cm², seconds²…), which is why standard deviation — back in the original units — is usually easier to interpret.

Worked example

Population SD of 2, 4, 4, 4, 5, 5, 7, 9

  1. Mean: 40 ÷ 8 = 5
  2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 — sum 32
  3. Population variance: 32 ÷ 8 = 4
  4. Standard deviation: √4 = 2

σ = 2. As a sample, s = √(32 ÷ 7) ≈ 2.138.

Sample or population?

Use population when you have every member of the group you care about (all 30 pupils in one class). Use sample when your data represents a larger group (30 pupils chosen from a whole school). Dividing by n − 1 (Bessel’s correction) compensates for a sample’s tendency to underestimate the spread.

Interpreting the result

For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two. A small standard deviation means values cluster tightly; a large one means they are spread out. The coefficient of variation (SD ÷ mean) lets you compare spread between data sets with different scales.

Common questions

Why does a single value give an error for a sample?

The sample formula divides by n − 1, which is zero when there is only one value. You need at least two values to estimate spread from a sample.