Standard Deviation Calculator

Standard deviation measures how spread out a set of numbers is around its average. This calculator gives you both the population and sample standard deviations, the matching variances, the mean and the count — everything a statistics class or a spreadsheet check requires.

Separate numbers with commas or spaces, e.g. 4, 8, 6, 5, 3

Results

Formula

Standard deviation is the square root of the average squared distance from the mean:
σ = √( Σ(x − x̄)² ÷ N )  ·  s = √( Σ(x − x̄)² ÷ (n − 1) )
The sample version divides by n − 1 (Bessel’s correction) because a sample tends to understate the true spread.

How to use this calculator

  1. Paste or type your numbers, separated by commas, spaces or new lines.
  2. Press Calculate.
  3. Read the population SD if your data is the entire group, or the sample SD if it is a sample of something larger.
Standard Deviation Calculator — how the calculation works, illustrated
Standard Deviation Calculator — how the calculation works, illustrated

Worked example

Data set: 4, 8, 6, 5, 3.

  1. Mean: (4 + 8 + 6 + 5 + 3) ÷ 5 = 5.2.
  2. Squared deviations: 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8.
  3. Population variance: 14.8 ÷ 5 = 2.96 → σ ≈ 1.72.
  4. Sample variance: 14.8 ÷ 4 = 3.7 → s ≈ 1.92.

Result: Population SD ≈ 1.7205; sample SD ≈ 1.9235.

Frequently asked questions

Should I use the sample or population standard deviation?

Use population SD when your numbers include every member of the group you care about. Use sample SD when they are a subset meant to estimate a larger group — the default in most research.

What does a standard deviation of zero mean?

Every value in the set is identical, so there is no spread at all.

Why is variance shown as well?

Variance is the squared version of the same spread measure. Some formulas (ANOVA, portfolio risk) need variance; standard deviation is easier to interpret because it uses the original units.

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