Standard Deviation Calculator

Paste a list of numbers to get the sample and population standard deviation, variance and mean, with steps.

Sample standard deviation (s)
2.13809
Distribution of the data (bar with the mean 5 highlighted)
01232 — Count: 123 — Count: 034 — Count: 345 — Count: 256 — Count: 067 — Count: 178 — Count: 089 — Count: 19
Population standard deviation (σ)
2
Sample variance (s²)
4.571429
Population variance (σ²)
4
Mean
5
Count (n)
8
Standard error of the mean
0.755929
Show the working
  1. n = 8, mean x̄ = 40 ÷ 8 = 5
  2. Squared deviations (x − x̄)²: (2 − 5)² + (4 − 5)² + (4 − 5)² + (4 − 5)² + (5 − 5)² + (5 − 5)² + (7 − 5)² + (9 − 5)²
  3. Sum of squares SS = 32
  4. Population variance σ² = SS ÷ n = 32 ÷ 8 = 4 → σ = 2
  5. Sample variance s² = SS ÷ (n − 1) = 32 ÷ 7 = 4.5714286 → s = 2.1380899

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Formula

s = √( Σ(x − x̄)² ÷ (n − 1) ) (sample)σ = √( Σ(x − μ)² ÷ N ) (population)SE = s ÷ √n

How it works

Standard deviation measures how spread out numbers are around their mean. A small standard deviation means the values cluster tightly; a large one means they are scattered. It is in the same units as the data, which makes it easier to interpret than the variance (its square).

To compute it, find the mean, subtract it from every value, square those deviations so negatives don’t cancel positives, add them up, divide, and take the square root. The only difference between the two versions is the divisor.

Use the population formula (divide by N) when your list contains every member of the group you care about. Use the sample formula (divide by n − 1) when your data is a sample drawn from a larger population — which is the usual case in science and surveys. Dividing by n − 1, known as Bessel’s correction, makes up for the fact that a sample tends to sit closer to its own mean than to the true population mean.

Frequently asked questions

Which one does my calculator or spreadsheet give?

In Excel and Google Sheets, STDEV.S (or STDEV) is the sample version and STDEV.P is the population version. On TI calculators, Sx is sample and σx is population.

What is the 68-95-99.7 rule?

For roughly normal data, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

What is the standard error?

The standard error of the mean, s ÷ √n, estimates how much the sample mean itself varies between samples. It shrinks as the sample grows, while the standard deviation of the data does not.

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