Standard Deviation Calculator
Paste your numbers — sample and population SD, with the steps shown.
How this is calculated
s = √( Σ(x − x̄)² ÷ (n − 1) ) σ = √( Σ(x − x̄)² ÷ n )
Worked example (the default set 2, 4, 4, 4, 5, 5, 7, 9): mean = 40 ÷ 8 = 5; squared deviations sum to (−3)² + (−1)²×3 + 0²×2 + 2² + 4² = 32. Population variance = 32 ÷ 8 = 4 → σ = 2. Sample variance = 32 ÷ 7 = 4.571 → s ≈ 2.138. Every number in that chain is on the result panel, ready to copy into working.
Sample vs population in one sentence each
Sample (s): your data represents a bigger group — divide by n−1 to correct the small-sample bias. Population (σ): your data IS the whole group — divide by n. When a test question gives you "the marks of a class" with no framing, sample is the safer answer and the one textbooks expect.
Frequently asked questions
Population or sample — which one do I want?
Sample (n−1) if your numbers are a subset standing in for something bigger — survey responses, a batch off a production line, class test scores generalized to teaching. Population (n) if the list IS everything — all 12 players on your team, every month of one year. Homework default: sample, unless the question says population.
Why divide by n−1 for a sample?
Bessel's correction: a sample's spread around its own mean slightly understates the true spread around the real (unknown) mean. Dividing by n−1 instead of n corrects that bias — and with big n the two answers converge, which is why the distinction matters most for small datasets.
What does the standard deviation actually tell me?
Typical distance from the mean, in the data's own units. For roughly bell-shaped data, about 68% of values sit within 1 SD of the mean and 95% within 2 SD. An SD of 2 around a mean of 5 says "most of this data lives between 3 and 7".
Why show the steps?
Because the steps are what the homework wants: mean → deviations → squared deviations → their sum → variance → square root. Every intermediate this page displays is one you can transcribe straight into working.