Statistics

Variance Calculator

Find population and sample variance.

Calculated Output

Population variance 200

Sample variance 250

Mean 30

Formula: σ² = Σ(x−mean)² ÷ n ; s² uses n−1

Step-by-Step Breakdown

1. ▶ Data: 10, 20, 30, 40, 50 (n = 5)

3. ▶ Step 1 — Mean (μ): μ = ( 10 + 20 + 30 + 40 + 50 ) ÷ 5 = 30

5. ▶ Step 2 — Deviations (xᵢ − μ):

6. x₁ − μ = 10 − 30 = -20

7. x₂ − μ = 20 − 30 = -10

8. x₃ − μ = 30 − 30 = 0

9. x₄ − μ = 40 − 30 = 10

10. x₅ − μ = 50 − 30 = 20

11.

12. ▶ Step 3 — Squared deviations (xᵢ − μ)²:

13. (x₁ − μ)² = (-20)² = 400

14. (x₂ − μ)² = (-10)² = 100

15. (x₃ − μ)² = (0)² = 0

16. (x₄ − μ)² = (10)² = 100

17. (x₅ − μ)² = (20)² = 400

18.

19. ▶ Step 4 — Sum of squares Σ(xᵢ − μ)²: 400 + 100 + 0 + 100 + 400 = 1000

20.

21. ▶ Step 5 — Population variance σ² = Σ(xᵢ − μ)² ÷ n = 1000 ÷ 5 = 200

22. ▶ Sample variance s² = Σ(xᵢ − μ)² ÷ (n − 1) = 1000 ÷ 4 = 250

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The math behind it

Average of squared deviations from the mean. Population divides by n; sample divides by n−1.

10,20,30,40,50 → population variance 200.

Population or sample?

Use sample (n−1) when your data is a sample of a larger group.