Variance Calculator

Variance is the average squared distance from the mean: the step right before standard deviation. Enter your numbers to get sample variance s² and population variance σ², with every squared deviation shown.

6 values read.

Sample variance s²
3.5
Population variance σ²
2.9167
Std deviation
1.8708
Mean
5.5
Sum of squares
17.5
n
6
±1 SD345678
▲ mean 5.5 · ◆ median 5.5 · shaded band = mean ± 1 SD (3.6292 to 7.3708)

Step-by-step working

  1. Count the valuesn = 6
  2. Add the values and divide by the count to get the meanx̄ = (4 + 8 + 6 + 5 + 3 + 7) ÷ 6 = 33 ÷ 6 = 5.5
  3. Subtract the mean from each value, then square the result (see the table)(x − x̄)² for each x, e.g. (4 − 5.5)² = 2.25
  4. Add up the squared deviations (the sum of squares)SS = Σ(x − x̄)² = 17.5
  5. Divide by n − 1 for the sample variance (Bessel’s correction)s² = 17.5 ÷ 5 = 3.5
Deviation table
xx − x̄(x − x̄)²
4−1.52.25
82.56.25
60.50.25
5−0.50.25
3−2.56.25
71.52.25
Σx = 33≈ 0SS = 17.5

Formula s² = Σ(x − x̄)² ÷ (n − 1) σ² = Σ(x − μ)² ÷ N

Ask the tutor about your result

An AI tutor reads your inputs and results above and explains them in plain English. The numbers come from the calculator; the explanation is AI-written, so check it against your notes.

See why standard deviation works

Drag any dot (or focus it and use ← →). Each square’s side is that value’s distance from the mean, so its area is the squared deviation. Standard deviation is the side of the “average” square.

012345678910mean 5.17

Σ(x − x̄)² = 10.03 + 1.36 + 1.36 + 0.03 + 3.36 + 14.69 = 30.83
s = √(30.83 ÷ 5) = 2.48

Why n − 1? A sample’s values sit closer to their own mean than to the true population mean, so squared deviations from x̄ run slightly small. Dividing by one less (Bessel’s correction) makes the variance unbiased. Notice the difference shrinks as n grows.

How to find variance

  1. Compute the mean.
  2. Subtract it from each value and square the result.
  3. Add those squares to get the sum of squares (SS).
  4. Divide by n − 1 (sample) or N (population).

Shortcut formula, handy with a calculator: SS = Σx² − (Σx)² ÷ n. It gives the same answer but can lose precision with large numbers, which is why this calculator uses the deviation method.

Why square the deviations?

Raw deviations always add to zero, because values above the mean cancel values below it. Squaring makes every term positive and gives extra weight to values far from the mean. Variances also add neatly: for independent variables, Var(X + Y) = Var(X) + Var(Y), which is why variance, not standard deviation, appears inside ANOVA, regression and the standard error formula.

Frequently asked questions

What units is variance in?

Squared units of the original data. If heights are in cm, variance is in cm². Take the square root to get back to cm (the standard deviation).

Is a higher variance bad?

Not inherently. It simply means more spread. In manufacturing, lower variance means more consistent parts; in a test designed to separate students, some variance is the point.

What is the variance of a single number?

The population variance of one value is 0. The sample variance is undefined because you would divide by n − 1 = 0.