Variance Calculator

Published on: August 24, 2025

This Variance Calculator helps you find the population variance of a list of numbers and shows each step clearly. It works by finding the mean, measuring how far each value is from the mean, squaring those differences, and then averaging the squared differences. This makes it useful for checking answers, understanding spread, and practising statistics step by step.

Step-by-step method

  1. List the values and count how many there are. This gives n.
  2. Add the values and divide by n to find the mean, μ.
  3. Subtract the mean from each value to find each deviation, x − μ.
  4. Square each deviation.
  5. Add the squared deviations.
  6. Write the population variance formula.
  7. Substitute the values and divide to get the population variance.

Formula:

σ2=
Σ(x − μ)2
n

Example 1: Take the set of numbers below.

3, 1, 4, 2

Step 1 - List the values and count how many there are. This gives n.

In this problem: The values are 3, 1, 4, 2. There are 4 values, so n = 4.

n=4

Step 2 - Add the values and divide by n to find the mean, μ.

In this problem: Add the values: 3 + 1 + 4 + 2 = 10. Then divide by n = 4, so μ = 2.5.

μ=
3+1+4+2
4
=2.5

Step 3 - Subtract the mean from each value to find each deviation, x − μ.

In this problem: Subtract the mean μ = 2.5 from each value. The deviations are 0.5, -1.5, 1.5, -0.5.

xx − μ
30.5
1-1.5
41.5
2-0.5

Step 4 - Square each deviation.

In this problem: Square each deviation. The squared deviations are 0.25, 2.25, 2.25, 0.25.

x − μ(x − μ)2
0.50.25
-1.52.25
1.52.25
-0.50.25

Step 5 - Add the squared deviations.

In this problem: Add the squared deviations to get 5.

Σ(x − μ)2=0.25+2.25+2.25+0.25=5

Step 6 - Write the population variance formula.

In this problem: Now write the population variance formula.

σ2=
Σ(x − μ)2
n

Step 7 - Substitute the values and divide to get the population variance.

In this problem: Substitute Σ(x − μ) squared = 5 and n = 4, then divide.

σ2=
5
4
=1.25

Final answer: The population variance is:

σ² = 1.25

Example 2: Take the set of numbers below.

5, 6, 7, 8, 9

Step 1 - List the values and count how many there are. This gives n.

In this problem: The values are 5, 6, 7, 8, 9. There are 5 values, so n = 5.

n=5

Step 2 - Add the values and divide by n to find the mean, μ.

In this problem: Add the values: 5 + 6 + 7 + 8 + 9 = 35. Then divide by n = 5, so μ = 7.

μ=
5+6+7+8+9
5
=7

Step 3 - Subtract the mean from each value to find each deviation, x − μ.

In this problem: Subtract the mean μ = 7 from each value. The deviations are -2, -1, 0, 1, 2.

xx − μ
5-2
6-1
70
81
92

Step 4 - Square each deviation.

In this problem: Square each deviation. The squared deviations are 4, 1, 0, 1, 4.

x − μ(x − μ)2
-24
-11
00
11
24

Step 5 - Add the squared deviations.

In this problem: Add the squared deviations to get 10.

Σ(x − μ)2=4+1+0+1+4=10

Step 6 - Write the population variance formula.

In this problem: Now write the population variance formula.

σ2=
Σ(x − μ)2
n

Step 7 - Substitute the values and divide to get the population variance.

In this problem: Substitute Σ(x − μ) squared = 10 and n = 5, then divide.

σ2=
10
5
=2

Final answer: The population variance is:

σ² = 2