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 = S^{2} = (5746) / (6 – 1)

Variance = S^{2} = (5746) / 5

Variance = S^{2} = 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 = S^{2} = (5110) / (7)

Variance = S^{2} = 730

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