Statistics

Standard Deviation Calculator

Find the mean, variance and standard deviation of a data set.

Calculated Output

Std deviation (population) 1.72

Variance 2.96

Mean 5.2

Formula: σ = √( Σ(xᵢ − mean)² ÷ n )

Step-by-Step Breakdown

1. ▶ Data: 4, 8, 6, 5, 3 (n = 5)

3. ▶ Step 1 — Mean (μ): μ = ( 4 + 8 + 6 + 5 + 3 ) ÷ 5 = 5.2

5. ▶ Step 2 — Deviations (xᵢ − μ):

6. x₁ − μ = 4 − 5.2 = -1.2

7. x₂ − μ = 8 − 5.2 = 2.8

8. x₃ − μ = 6 − 5.2 = 0.8

9. x₄ − μ = 5 − 5.2 = -0.2

10. x₅ − μ = 3 − 5.2 = -2.2

11.

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

13. (x₁ − μ)² = (-1.2)² = 1.44

14. (x₂ − μ)² = (2.8)² = 7.84

15. (x₃ − μ)² = (0.8)² = 0.64

16. (x₄ − μ)² = (-0.2)² = 0.04

17. (x₅ − μ)² = (-2.2)² = 4.84

18.

19. ▶ Step 4 — Sum of squares Σ(xᵢ − μ)²: 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8

20.

21. ▶ Step 5 — Variance σ² = Σ(xᵢ − μ)² ÷ n = 14.8 ÷ 5 = 2.96

22.

23. ▶ Step 6 — Standard deviation σ = √(σ²) = √(2.96) = 1.72

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The math behind it

Find the mean, then the average of the squared differences from the mean (variance). The standard deviation is the square root of the variance.

4, 8, 6, 5, 3 → mean 5.2, variance 2.96, σ ≈ 1.72.

Population vs sample SD?

Population divides by n (used here); sample divides by n − 1 for an unbiased estimate from a sample.