Variance Formula
Calculate variance with σ² = Σ(x - μ)² / N.
Understand how much data values deviate from the mean on average.
The Formula
Sample: s² = Σ(x - x̄)² / (n - 1)
Variance measures the average of squared differences from the mean. It tells you how spread out your data is. Variance is simply the standard deviation squared.
Variables
| Symbol | Meaning |
|---|---|
| σ² | Population variance |
| s² | Sample variance |
| x | Each individual data value |
| μ | Population mean |
| x̄ | Sample mean |
| N | Population size |
| n - 1 | Degrees of freedom (for sample data) |
Example 1 — Population Variance
Find the variance of: 2, 4, 6, 8, 10 (entire population)
Step 1: Find the mean — μ = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6
Step 2: Squared differences:
(2 - 6)² = 16, (4 - 6)² = 4, (6 - 6)² = 0, (8 - 6)² = 4, (10 - 6)² = 16
Step 3: Sum = 16 + 4 + 0 + 4 + 16 = 40
Step 4: Divide by N — 40 / 5 = 8
σ² = 8 — The average squared deviation from the mean is 8.
Example 2 — Sample Variance
Heights (in cm) from a sample: 160, 165, 170, 175, 180
Step 1: Find the mean — x̄ = (160 + 165 + 170 + 175 + 180) / 5 = 850 / 5 = 170
Step 2: Squared differences:
(160 - 170)² = 100, (165 - 170)² = 25, (170 - 170)² = 0, (175 - 170)² = 25, (180 - 170)² = 100
Step 3: Sum = 100 + 25 + 0 + 25 + 100 = 250
Step 4: Divide by (n - 1) — 250 / 4 = 62.5
s² = 62.5 — The sample variance is 62.5 cm².
When to Use It
Use the variance formula when:
- You need a precise numerical measure of data spread
- Performing advanced statistical tests that require variance (like ANOVA)
- Comparing the variability of two or more data sets
- Calculating standard deviation (just take the square root of variance)