How this calculator works
Standard deviation is the square root of variance and therefore uses the original measurement unit. Sample variance divides squared distances from the sample mean by n - 1; population variance divides by n. A sample needs two observations, while a one-member population has SD zero. The additional SD/sqrt(n) is the standard error of a mean only when the independent-sampling assumptions and SD choice support that interpretation.
Formula
Sample s = sqrt(Σ(x - mean)²/(n - 1)); population sigma = sqrt(Σ(x - mean)²/n). Given variance V, SD = sqrt(V).
Worked example
For 1, 2, 3, 4, 5, the mean is 3 and squared deviations total 10. Sample SD is sqrt(10/4) ≈ 1.581139; population SD is sqrt(10/5) ≈ 1.414214.
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Frequently asked questions
Which divisor should I choose?
Use n for the complete population being described, or n - 1 for the usual sample estimate of population variance.
Can standard deviation be negative?
No. It is a non-negative square root. Negative variance inputs are invalid.
Is standard deviation the uncertainty of the mean?
It measures observation spread. Under an independent sample-mean model, divide the sample SD by sqrt(n) to estimate the mean standard error.