Variance Formulas
Sample Variance (s²) = Σ(xᵢ − x̄)² / (N − 1)
Standard Deviation = √Variance
The key difference: population variance divides by N (all data points); sample variance divides by N−1 (Bessel's correction) to produce an unbiased estimate of the true population variance from a sample.
When to Use Each Type
| Type | Use when | Divisor |
|---|---|---|
| Population Variance (σ²) | You have ALL data for the group | N |
| Sample Variance (s²) | Your data is a sample from a larger group | N − 1 |
Example
Dataset: 4, 8, 6, 5, 3, 2, 8, 9, 2, 5
Mean = 52 / 10 = 5.2
Sum of squared deviations = 50.4
Population variance = 50.4 / 10 = 5.04
Sample variance = 50.4 / 9 = 5.6
Population std dev = √5.04 = 2.245
Frequently Asked Questions
What does variance tell you?
Variance measures how spread out the data is from the mean. A variance of 0 means all values are identical. A high variance means data points are widely scattered. Because variance is in squared units, standard deviation (√variance) is often preferred for interpretation since it is in the same units as the data.
Why does sample variance use N−1?
Using N in a sample systematically underestimates the true population variance. Dividing by N−1 (Bessel's correction) corrects for this bias, making the sample variance an unbiased estimator of the population variance. This is why most statistics textbooks and software default to N−1 for sample data.
What is the relationship between variance and standard deviation?
Standard deviation is simply the square root of variance. Variance is useful mathematically (additive for independent variables) but standard deviation is more interpretable because it is in the same units as the original data.
Can variance be negative?
No. Variance is always ≥ 0 because it is the average of squared deviations. Squared values are always non-negative. Variance is 0 only when all data points are identical.