Variance Calculator
Calculate the population variance, sample variance, standard deviation, mean, sum of squares and coefficient of variation for any data set, with a deviation table showing every step. Example: for 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the population variance is 4 and the standard deviation is 2.
Show steps
How to use the Variance Calculator
- Type or paste your numbers.
- Press Calculate to get population and sample variance, standard deviation and the deviation table.
Formula and method
Population variance = sum of (x – mean)^2 / n. Sample variance = sum of (x – mean)^2 / (n – 1). Standard deviation is the square root of the variance. Coefficient of variation = standard deviation / mean x 100%.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
| Value | Deviation | Squared |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
| Total | 0 | 32 |
Population versus sample variance
| Population | Sample | |
|---|---|---|
| Divide by | n | n – 1 |
| This example | 32 / 8 = 4 | 32 / 7 = 4.571 |
| Use when | You have every member | You have a sample |
Why divide by n – 1
A sample’s mean is fitted to the sample itself, so deviations around it come out slightly too small. Dividing by n – 1 (Bessel’s correction) removes that bias in the variance estimate.
Variance versus standard deviation
Variance is in squared units, which makes it hard to read. Standard deviation is its square root and is in the same units as the data.
Frequently asked questions
How do you calculate variance?
Find the mean, square each deviation from it, add them up and divide by n (population) or n – 1 (sample).
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance.
Why divide by n – 1?
It corrects the bias that comes from estimating the mean from the same sample.
Can variance be negative?
No. It is a sum of squares, so it is zero or positive.
What does a variance of 0 mean?
Every value is identical.
What is a high variance?
It depends on the units; compare using the coefficient of variation.
Sources
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