σ vs s: population or sample standard deviation?

The difference between population standard deviation σ and sample standard deviation s, when to use each, and a worked example with the data 2, 4, 4, 4, 5, 5, 7, 9. Also how they map to Excel STDEV.P and STDEV.S.

The two formulas

Standard deviation measures how spread out data is around its mean. Both versions square each deviation from the mean and add them up; they differ only in what you divide by.

• Population SD σ = √( Σ(x − x̄)² ÷ n )
• Sample SD s = √( Σ(x − x̄)² ÷ (n − 1) )

Calculators usually label them σx and sx. In Excel, STDEV.P is σ and STDEV.S (formerly STDEV) is s.

Worked example

The data 2, 4, 4, 4, 5, 5, 7, 9 has n = 8 and mean 40 ÷ 8 = 5.

• σ² = 32 ÷ 8 = 4, so σ = 2
• s² = 32 ÷ 7 ≈ 4.571428571, so s = √(32/7) ≈ 2.138089935

Open this data on the statistics screen ▸

Why divide by n − 1?

A sample’s mean x̄ is, by construction, the value closest to that sample, so the sum of squared deviations around it tends to come out a little smaller than it would around the true population mean. Dividing by the slightly smaller n − 1 corrects for that and gives an unbiased estimate of the population variance. This is called Bessel’s correction.

Which one should you use?

• σ: you have every member of the group you care about, such as all 30 test scores in a class or 12 months of this year’s sales.
• s: you measured part of a larger group to learn about the whole, such as 10 items pulled from a production line or a survey of 1,000 people.
• Textbook exercises often want σ; lab reports and statistical tests almost always use s. If the question says “sample”, use s.

The gap shrinks as n grows. With n = 100, s is √(100/99) ≈ 1.005 times σ, only about 0.5% larger.