Variance calculator
Variance is the average squared distance from the mean. Each squared deviation is drawn as an actual square, so you can see which values dominate.
Each square has sides equal to one value's distance from the mean. Variance is their average area.
Step by step
- Mean: x̄ = 1495 / 20 = 74.75
- Squared deviations (x − x̄)², added up: Σ = 1963.75
- Divide by n − 1 = 19: variance = 1963.75 / 19 = 103.3553
- Square root: standard deviation = √103.3553 = 10.1664
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 62 | −12.75 | 162.5625 |
| 71 | −3.75 | 14.0625 |
| 74 | −0.75 | 0.5625 |
| 58 | −16.75 | 280.5625 |
| 88 | 13.25 | 175.5625 |
| 91 | 16.25 | 264.0625 |
| 67 | −7.75 | 60.0625 |
| 73 | −1.75 | 3.0625 |
| 79 | 4.25 | 18.0625 |
| 84 | 9.25 | 85.5625 |
| 70 | −4.75 | 22.5625 |
| 65 | −9.75 | 95.0625 |
| 77 | 2.25 | 5.0625 |
| 81 | 6.25 | 39.0625 |
| 69 | −5.75 | 33.0625 |
| 95 | 20.25 | 410.0625 |
| 72 | −2.75 | 7.5625 |
| 76 | 1.25 | 1.5625 |
| 60 | −14.75 | 217.5625 |
| 83 | 8.25 | 68.0625 |
| Σ | 0 | 1963.75 |
Variance, drawn as squares
Variance is the mean of the squared deviations. In the chart above, every value's distance from the mean is the side of a square, so the variance is literally the average area of those squares. Notice how one far-away value produces a square much bigger than all the others: variance is very sensitive to outliers.
s² = Σ(x − x̄)² / (n − 1) σ² = Σ(x − μ)² / N
A shortcut formula
For hand calculation, Σ(x − x̄)² = Σx² − (Σx)²/n. It gives the same answer with fewer subtractions, though computers avoid it because it can lose precision when the numbers are large.
Why statisticians like variance
For independent random variables, variances add: Var(X + Y) = Var(X) + Var(Y). Standard deviations do not. That property is behind the √n in the standard error σ/√n that appears in confidence intervals and the central limit theorem.
To get back to the original units, take the square root: that is the standard deviation.
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Questions students ask
How do you find variance?
Subtract the mean from each value, square the results, add them up, and divide by n − 1 for a sample or N for a population.
Why do we square the deviations?
Deviations from the mean always add up to zero, so they have to be made positive. Squaring does that and also gives extra weight to values far from the mean, which gives variance useful mathematical properties (variances of independent variables add).
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units (cm²), SD is back in the original units (cm).
Can variance be negative?
No. It is a sum of squares divided by a positive count, so the smallest possible value is 0, which happens only when every value is identical.