Standard Deviation Calculator
Paste a list of numbers and get the sample and population standard deviation, variance and mean.
How it works
σ = √(Σ(x − mean)² ÷ n) · s = √(Σ(x − mean)² ÷ (n − 1))
Standard deviation measures how spread out the values are around the mean: square each value’s distance from the mean, average those squares (that’s the variance), and take the square root. The population version divides by n; the sample version divides by n − 1 (Bessel’s correction), which compensates for a sample underestimating the true spread.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40 ÷ 8 = 5. The squared deviations sum to 32, so population variance = 32 ÷ 8 = 4 and population SD = √4 = 2. The sample SD divides by 7 instead: √(32 ÷ 7) ≈ 2.1381.
Frequently asked questions
Sample or population — which one do I need?
If your data is every member of the group you care about (all 30 students in one class), use population. If it is a sample standing in for something larger (30 students representing all students), use sample — that’s the more common case.
What does one standard deviation actually mean?
For roughly bell-shaped data, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD — the empirical rule.
Why square the deviations instead of just averaging distances?
Squaring keeps everything positive so spread doesn’t cancel out, and it weights big deviations more heavily. The plain average of absolute distances exists too (mean absolute deviation) but has less convenient math.
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This calculator is for educational purposes. Double-check important results before acting on them.