Variance Calculator: Sample (s²) and Population (σ²) Variance with Steps
Enter your numbers to get the variance with the sum of squared deviations worked out, plus the mean and standard deviation.
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Sample variance s²
12
- Sample SD s
- 3.464102
- Mean
- 15
- Count n
- 6
- Population variance (if whole population)
- 10
Sample formula divides by n − 1 (Bessel’s correction): use it when the data is a sample from a bigger group.
Show working
- n = 6, Σx = 90
- x̄ = Σx / n = 90 / 6 = 15
- Σ(x − x̄)² = (12 − 15)² + (15 − 15)² + (11 − 15)² + (18 − 15)² + (14 − 15)² + (20 − 15)² = 60
- s² = Σ(x − x̄)² / (n − 1) = 60 / 5 = 12
- s = √12 = 3.464102
Deviations table
| x | x − mean | (x − mean)² |
|---|---|---|
| 12 | −3 | 9 |
| 15 | 0 | 0 |
| 11 | −4 | 16 |
| 18 | 3 | 9 |
| 14 | −1 | 1 |
| 20 | 5 | 25 |
How to use
- Type or paste your numbers.
- Choose sample (divide by n − 1) or population (divide by n).
- Read the variance and check the working.
Good to know
Variance = average squared distance from the mean. Population: σ² = Σ(x − μ)²/n. Sample: s² = Σ(x − x̄)²/(n − 1). The standard deviation is its square root.
Because deviations are squared, variance is in squared units (marks², ₹²). That is why the standard deviation is usually reported, but variance is what adds up neatly: for independent variables, Var(X + Y) = Var(X) + Var(Y).
A short-cut that gives the same population variance: σ² = (Σx²)/n − (mean)². The calculator uses the two-pass method shown in the steps, which avoids rounding errors with large numbers.
Questions
What is the variance of 12, 15, 11, 18, 14, 20?
Mean 15, sum of squared deviations 60. Sample variance = 60/5 = 12; population variance = 60/6 = 10.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance, so it is in the same units as the data.
Can variance be negative?
No. It is a sum of squares, so it is 0 or more. It is 0 only when all values are equal.
Sample or population: which should I choose?
Population if you have every value in the group; sample if the data is a part used to estimate a larger group.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.