Variance is the average squared deviation from the mean, and it is the quantity that statistical theory is actually built on. Its units are the square of the data units, which makes it hard to interpret directly — a variance of 9 on a set of measurements in centimetres is 9 square centimetres — which is why the standard deviation, its square root, gets quoted instead.
Variance earns its place because it adds. The variance of the sum of two independent quantities is the sum of their variances, which is not true of standard deviations. That additivity underpins error propagation, portfolio risk, analysis of variance and the central limit theorem, so the awkward units are a price worth paying.
The both spread measures for sample datasets table follows the calculator: 5 rows spanning 3, 7, 7, 19, 24, 30, 8 through 6, 6, 6, 6, 6, 60, there so the result can be cross-checked against a value you already trust. Nothing has to be entered to see it work: the default for your numbers is 3, 7, 7, 19, 24, 30, 8, and the output recomputes the moment that changes. Alongside the number, 4 bands label it — very low variance through very high variance — because a bare figure rarely tells you whether it is unusual. Interquartile Range Calculator and Standard Deviation Calculator are the nearest neighbours to this one.
The formula
x̄- The mean
Σ(x − x̄)²- The sum of squared deviations
n − 1- Degrees of freedom for a sample
s²- Sample variance
How it works, step by step
- Paste your numbers, separated by commas or spaces.
- The gauge shows the sample variance.
- The population variance and the sum of squares appear below.
- The coefficient of variation puts the spread in proportion to the mean.
Worked examples
Seven values with a wide spread
The mean is 14.0000 and the squared deviations sum to 636.0000. Dividing by 6 gives a sample variance of 106.0000.
Why variance adds
Two independent processes with variances of 9 and 16 combine to a variance of 25, so the combined standard deviation is 5 — not the 3 plus 4 that adding the deviations would suggest. That is error propagation in one line.
Checking the 20, 21, 19, 20, 20 row
The reference table lists 20, 21, 19, 20, 20, giving Mean: 20.000, Sum of squares: 2.000, Sample variance: 0.5000, Sample s: 0.7071. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
Why does variance use squared units?
Because the deviations are squared before averaging. It makes the number hard to read directly, which is why the standard deviation is reported in practice.
When would I quote variance rather than standard deviation?
When you need to combine sources of variation, apportion them, or feed them into a statistical model. Variance adds for independent quantities; standard deviations do not.
What are degrees of freedom?
The number of independent pieces of information left after estimating the mean from the same data. Using the sample mean costs one, which is why the divisor is n minus one.
What is the coefficient of variation for?
Comparing spread between datasets with different scales. It expresses the standard deviation as a percentage of the mean, making the comparison dimensionless.
What do the bands under the sample variance mean?
They label the figure so it can be read without a reference point of your own: Very low variance from 0 to 5; Moderate variance from 5 to 50; High variance from 50 to 250; Very high variance from 250 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Variance Calculator use?
It computes s^2 = Σ(x − x̄)^2 ÷ (n − 1), where x̄ is the mean; Σ(x − x̄)² is the sum of squared deviations; n − 1 is degrees of freedom for a sample; s² is sample variance. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Variance and standard deviation together
| Dataset | Mean | Sum of squares | Sample variance | Sample s |
|---|---|---|---|---|
| 3, 7, 7, 19, 24, 30, 8 | 14.000 | 636.000 | 106.0000 | 10.2956 |
| 1, 2, 3, 4, 5 | 3.000 | 10.000 | 2.5000 | 1.5811 |
| 20, 21, 19, 20, 20 | 20.000 | 2.000 | 0.5000 | 0.7071 |
| 0, 10, 20, 30, 40 | 20.000 | 1000.000 | 250.0000 | 15.8114 |
| 6, 6, 6, 6, 6, 60 | 15.000 | 2430.000 | 486.0000 | 22.0454 |
The variance column is the square of the final column in every row.