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Variance Calculator

Find population and sample variance.

Population variance 200
Sample variance 250
Mean 30

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

Step-by-step with your numbers:
1. ▶ Data: 10, 20, 30, 40, 50 (n = 5)
2.
3. ▶ Step 1 — Mean (μ): μ = ( 10 + 20 + 30 + 40 + 50 ) ÷ 5 = 30
4.
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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Variance measures how spread out your data is (the square of standard deviation).

How the Math Works

Variance measures how spread out data points are from the mean. To calculate it, first find the mean (average) of the dataset. Then subtract the mean from each value to get deviations, square those deviations to eliminate negatives, and sum them all. For population variance (σ²), divide this sum by the total number of data points (n). For sample variance (s²), divide by n−1 to account for bias in estimating population variance from a sample. This process quantifies variability: higher variance means data is more dispersed, while lower variance indicates data points cluster closely around the mean.

Practical Applications

Variance is essential in statistics for understanding data reliability and risk. In finance, it measures investment volatility; in manufacturing, it assesses product quality consistency; and in research, it evaluates experimental result stability. By comparing variances across datasets, analysts identify trends, test hypotheses, and make data-driven decisions. For example, a low variance in crop yields suggests predictable harvests, while high variance in stock returns signals higher risk. This calculation underpins tools like standard deviation and statistical models used in academia and industry.

Day-to-Day Use

Variance helps you gauge consistency in everyday activities. It explains why some days you spend more time commuting than others, or why your grocery bills fluctuate. In health, tracking variance in daily steps shows exercise patterns. For students, it reveals performance spread across tests, guiding study strategies. Even in cooking, variance in recipe measurements affects dish outcomes. By understanding variance, you can spot unusual patterns, set realistic expectations, and improve routines by identifying what’s consistent versus unpredictable in your life.

Worked example

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

FAQ

Population or sample?

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

Related calculators

Standard Deviation Calculator Find the mean, variance and standard deviation of a data set. Mean, Median, Mode Calculator Find the mean, median, mode and range of a data set. Coefficient of Variation Calculator Express variability relative to the mean.