When should I use population vs sample standard deviation?

Use the population version when your data is every member of the group — all exam scores in a class, every transaction this month. Use the sample version when your data is a subset — a survey of 200 out of 10,000 customers, a handful of measurements from a production line.

What does a standard deviation of zero mean?

Every value in the list is the same. There is no spread at all. The mean, the median, the min and the max are all equal.

How does standard deviation relate to variance?

Variance is the standard deviation squared. It is useful in formulas but hard to interpret because its unit is the square of the original unit — dollars squared, seconds squared. The standard deviation is the square root of the variance, which brings the unit back to the original one.

Math

Standard Deviation Calculator

Paste or type your numbers to see how spread out they are.

The formula

population σ = √(Σ(xᵢ − μ)² / N)

sample s = √(Σ(xᵢ − x̄)² / (N − 1))

mean = sum ÷ count

Worked example

The list 2, 4, 4, 4, 5, 5, 7, 9

mean = 40 ÷ 8 = 5
deviations²: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
population σ = √(32/8) = √4 = 2
sample s = √(32/7) ≈ 2.138

Where this goes wrong

Dividing by N when you should divide by N−1

If your data is the entire population, divide by N to get the population standard deviation (σ). If your data is a sample drawn from a larger group, divide by N−1 to get the sample standard deviation (s). The N−1 correction (Bessel's) compensates for the fact that a sample underestimates spread. This page shows both so you can pick the right one.

Questions

When should I use population vs sample standard deviation?: Use the population version when your data is every member of the group — all exam scores in a class, every transaction this month. Use the sample version when your data is a subset — a survey of 200 out of 10,000 customers, a handful of measurements from a production line.
What does a standard deviation of zero mean?: Every value in the list is the same. There is no spread at all. The mean, the median, the min and the max are all equal.
How does standard deviation relate to variance?: Variance is the standard deviation squared. It is useful in formulas but hard to interpret because its unit is the square of the original unit — dollars squared, seconds squared. The standard deviation is the square root of the variance, which brings the unit back to the original one.