Solved Examples of Sample & Population Variance

In this post, we look at some solved examples for Sample & Population Variance.

Recommended Reading: What is Variance? Explained with its types & calculations

Example 1: For Sample Variance

Calculate the variance of the sample data 12, 45, 23, 86, 43, 97

Solution:
Let x = 12, 45, 23, 86, 43, 97
Total terms = n = 6

Formula:

Step 1: Find the mean of the sample data

Mean = x̅ = (x) / n
Mean = x̅ = (12 + 45 + 23 + 86 + 43 + 97) / 6
Mean = x̅ = 306 / 6
Mean = x̅ = 51

Step 2: Find the difference between the values and the mean.

12 – 51 = – 39
45 – 51 = – 6
23 – 51 = – 28
86 – 51 = 35
43 – 51 = – 8
97 – 51 = 46

Step 3: Find the square of the difference.

(– 39)² = 1521
(– 6)² = 36
(– 28)² = 784
(35)² = 1225
(– 8)² = 64
(46)² = 2116

Step 4: Sum up the squared values.

Sum of squares = 1521 + 36 + 784 + 1225 + 64 + 2116
Sum of squares = 5746

Step 5: Divide the calculated value by “n – 1” – because we are finding the variance of the sample data

Variance = S² = (5746) / (6 – 1)
Variance = S² = (5746) / 5
Variance = S² = 1149.5

Example 2: For Population Variance

Calculate the variance of the population data 92, 54, 71, 20, 67, 34, 12

Solution:
Let X = 92, 54, 71, 20, 67, 34, 12
Total terms = N = 7

Formula:

Step 1: Find the mean of the population data µ

Mean =µ = (X) / N
Mean = µ= (92 + 54 + 71 + 20 + 67 + 34 + 12) / 7
Mean = µ = 350 / 7
Mean = µ = 50

Step 2: Find the difference between the values and the mean (deviation)

92 – 50 = 42
54 – 50 = 4
71 – 50 = 21
20 – 50 = – 30
67 – 50 = 17
34 – 50 = – 16
12 – 50 = – 38

Step 3: Find the square of the difference.

(42)² = 1764
(4)² = 16
(21)² = 441
(– 30)² = 900
(17)² = 289
(– 16)² = 256
(– 38)² = 1444

Step 4: Sum up the squared values. (Sum of squares)

Sum of squares = 1764 + 16 + 441 + 900 + 289 + 256 + 1444
Sum of squares = 5110

Step 5: Divide the calculated value by “N” – because we are finding the variance of the population data

Variance = S² = (5110) / (7)
Variance = S² = 730