Variance Calculator
Calculate the variance, standard deviation, and sum of squared deviations for a data set. See also Standard Deviation Calculator and Mean Calculator.
How to Calculate Variance
Variance measures how far each value in a data set is from the mean. To calculate: (1) find the mean of the data, (2) subtract the mean from each value to get the deviation, (3) square each deviation, (4) find the average of the squared deviations. For a sample, divide by n−1 (Bessel's correction); for a population, divide by n.
Variance Formulas
Population Variance: σ² = Σ(xᵢ − μ)² / N
Sample Variance: s² = Σ(xᵢ − x̄)² / (n − 1)
Standard Deviation = √Variance
Example Calculation
Data: 6, 7, 3, 9, 5, 8
Mean = 38 / 6 ≈ 6.333333
Deviations²: 0.111, 0.444, 11.111, 7.111, 1.778, 2.778
Σ(xᵢ − x̄)² ≈ 23.333
Sample Variance = 23.333 / 5 ≈ 4.666667
Sample Std Dev ≈ 2.160247
Variance Reference Table
| Concept | Formula | Measures |
|---|---|---|
| Variance | Σ(xᵢ−x̄)²/(n−1) | Squared spread |
| Std Deviation | √Variance | Spread in original units |
| Range | Max − Min | Total spread |
| IQR | Q3 − Q1 | Middle 50% spread |
Frequently Asked Questions
Why do we square the deviations?
Squaring ensures all deviations are positive (negative deviations don't cancel out positive ones) and gives more weight to larger deviations, making variance sensitive to outliers.
Why divide by n−1 for sample variance?
Dividing by n−1 (Bessel's correction) corrects the bias that occurs when estimating population variance from a sample. It produces an unbiased estimate of the population variance.
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units, while standard deviation is in the same units as the original data, making it easier to interpret.
Can variance be zero?
Yes. Variance is zero when all values in the data set are identical, meaning there is no spread at all.