Variance Calculator
Calculate sample and population variance from a list of numbers.
How to use this tool
- Enter numbers (comma-separated) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your sample variance (s²) and the full breakdown beneath it.
Calculate sample variance (s²) and population variance (σ²) from a comma-separated list of numbers.
Formula
Sample variance: s2 = Σ(xi − x̅)2 / (n − 1)
Population variance: σ2 = Σ(xi − x̅)2 / n
How it works
Variance measures the average squared deviation of each value from the mean, expressing the overall spread of a dataset in squared units of the original data. The calculator parses a comma-separated list, computes the mean, accumulates the sum of squared deviations, and then divides by n − 1 for the sample estimate or n for the population value. Standard deviation is simply the square root of variance.
Worked example
Worked example
- Data: 2, 4, 4, 4, 5, 5, 7, 9. Count n = 8; mean = 40/8 = 5.
- Squared deviations: (2−5)²=9, (4−5)²=1 (three times), (5−5)²=0 (twice), (7−5)²=4, (9−5)²=16. Sum = 32.
- Sample variance: s² = 32 / (8−1) = 32/7 ≈ 4.5714.
- Population variance: σ² = 32 / 8 = 4.0.
Sample variance = 4.5714; population variance = 4.0; count = 8.
Key terms
- Variance
- The average of the squared differences from the mean; it measures how far values are spread from the mean in squared units.
- Sample variance (s²)
- An unbiased estimate of the population variance computed from a sample, using n−1 in the denominator (Bessel's correction).
- Population variance (σ²)
- The true variance of an entire population, computed by dividing the sum of squared deviations by n.
- Squared deviation
- The square of the difference between an individual data point and the mean: (xᵢ − x̅)². Squaring ensures all contributions are positive.
- Standard deviation vs. variance
- Variance is in squared units, making it less intuitive; standard deviation is its square root and shares the same unit as the original data.
Frequently asked questions
- What is variance?
- Variance measures the average squared deviation from the mean. Sample variance uses n−1 in the denominator; population variance uses n. Standard deviation is the square root of variance.