Standard deviation measures how spread out numbers are around their mean. A small value means the data is tightly clustered; a large value means it is widely spread.
Choose “sample” when your data is a subset of a larger group (the usual case in surveys and experiments), and “population” when you have every value.
Formula
s = √( Σ(xᵢ − x̄)² ÷ (n − 1) )
For a whole population divide by n instead of n − 1: σ = √( Σ(xᵢ − μ)² ÷ n ). The variance is the value before taking the square root.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations add up to 32. As a population, the variance is 32 ÷ 8 = 4 and σ = 2. As a sample, the variance is 32 ÷ 7 ≈ 4.571 and s ≈ 2.138.
Frequently asked questions
Why divide by n − 1 for a sample?
A sample’s values sit closer to their own mean than to the true population mean, so dividing by n underestimates the spread. Dividing by n − 1 (Bessel’s correction) fixes that bias.
What does a standard deviation tell me?
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two.
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