Standard Deviation Calculator
Paste numbers and choose whether they are a sample or the whole population. The two methods use different denominators; the page does not guess.
How it works
Standard deviation is the square root of variance. Variance is the average squared distance from the mean. A sample uses n − 1 (Bessel’s correction) so the estimate is unbiased for a larger population.
A population uses n because every member is already in the list. For 1, 2, 3 the sample SD is 1 and the population SD is √(2/3). One number has no sample SD.
Formula
s = √[ Σ(x − x̄)² / (n − 1) ] (sample) σ = √[ Σ(x − μ)² / n ] (population)
Worked examples
Sample SD of 1, 2, 3
1, 2, 3 with sample method → 1
Population SD of 1, 2, 3
1, 2, 3 with population method → √(2/3) ≈ 0.8165
Useful notes
- This is not the standard error of the mean (that divides s by √n).
- The average calculator reports only the mean of a list.
FAQ
- Should I use sample or population?
Use sample when the list is a subset of a larger group you care about. Use population when the list is every value in the group.
- Why divide by n − 1?
A sample mean is fitted to the same data, so dividing by n would understate spread. n − 1 corrects that.
- What if I enter only one number?
Population SD is 0. Sample SD is undefined — there is no n − 1 spread.
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