Variance Calculator

Sample and population variance with standard deviation, standard error and the sum-of-squares steps, from any pasted data set.

The Variance Calculator computes sample variance s² = Σ(x − x̄)²/(n − 1) and population variance σ² = Σ(x − μ)²/n, and reports the standard deviation, standard error and coefficient of variation with the intermediate sum of squares.

Free, runs in your browser. Nothing you enter is uploaded.

How to use it

Paste numbers.
Choose sample or population.
Read variance and SD.

What you enter

A list of numbers

What you get

Sample and population variance
Standard deviation
Standard error
Sum of squares

Formula

s² = Σ(xᵢ − x̄)² / (n − 1)   |   σ² = Σ(xᵢ − μ)² / n

Variance is the mean squared deviation from the mean; dividing by n − 1 makes the sample estimate unbiased for the population variance.

Worked example

Input: 2 4 4 4 5 5 7 9

Result: Sample variance 4.571 · population variance 4.0 · SD 2.138 / 2.0

Mean 5; Σ(x − 5)² = 32; 32/7 = 4.571 (sample), 32/8 = 4.0 (population).

Questions

Why divide by n − 1?

Using the sample mean instead of the true mean makes deviations slightly too small on average; dividing by n − 1 (Bessel's correction) compensates and gives an unbiased estimate of the population variance.

Limitations

Variance is in squared units; report SD alongside it for interpretability.

References

Altman DG, Bland JM (2005). BMJ 331:903.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.