Sample and Population Standard Deviation Calculator

Population vs sample standard deviation

Population variance counts every member of the universe: σ² = Σ(xᵢ − μ)² / N, with σ = √σ². Sample variance instead estimates σ² from a subset, applying Bessel's correction: s² = Σ(xᵢ − x̄)² / (n − 1). The division by n − 1 (degrees of freedom) offsets the downward bias you get from using x̄ in place of the unknown μ. Take 2,4,4,4,5,5,7,9 as an example: mean = 5, σ ≈ 2.00, s ≈ 2.14. Reach for σ when you actually hold the full population (a census, a deterministic system), and for s when you're sampling from something larger.

Applications

It shows up in Six Sigma and statistical process control (Cp, Cpk, control charts at ±3σ), in financial volatility and risk work (the annualized σ of returns), in measurement uncertainty in metrology, in A/B test design, in normal-distribution confidence intervals, and in pretty much any quality assurance workflow that has to put a number on dispersion.

FAQ

Why divide by n − 1 instead of n? That's Bessel's correction at work. Because x̄ stands in for the real μ, the spread comes out too small on average; dividing by n − 1 cancels that bias and gives you an unbiased estimator of σ².

Variance or standard deviation, which to report? Variance lives in squared units and adds up nicely across independent variables. Standard deviation stays in the original units, so it's easier to read. Most reports go with σ or s.

What does Excel's STDEV vs STDEVP return? STDEV (or STDEV.S) gives you the sample s, dividing by n − 1; STDEVP (or STDEV.P) gives the population σ, dividing by N. Pick one based on whether your data is the whole universe or just a sample of it.