Variance Practice Problems

Variance is a measure of dispersion that indicates how far the values in a dataset are spread out from the mean.

Example 1: Calculate the variance of the sample data: 7, 11, 15, 19, 24.

We have the data, 7, 11, 15, 19, 24

Find mean of the data.

x̄ = (7 + 11 + 15 + 19 + 24)/5
   = 76/5
   = 15.2

Using the formula for variance we get,

s² = ∑ (x_i - x̄)²/(n - 1)
     = (67.24 + 17.64 + 0.04 + 14.44 + 77.44)/(5 - 1)
     = 176.8/4
     = 44.2

Example 2: Calculate the number of observations if the variance of the dataset is 12 and the sum of squared differences of data from the mean is 156.

We have,

(x_i - x̄)² = 156
σ² = 12

Using the  formula for variance we get,

σ² = ∑ (x_i - x̄)²/n
12 = 156/n
12n = 156
n = 156/ 12
n = 13

Example 3: Calculate the variance for the given data

xi	fi	fixi	(xi - x̄)	(xi - x̄)2	fi(xi - x̄)2
10	1	10	4	16	16
4	3	12	-2	4	12
6	5	30	0	0	0
8	1	8	2	4	4

Mean (x̄) = ∑(f_i x_i)/∑(f_i)

              = (10×1 + 4×3 + 6×5 + 8×1)/(1+3+5+1)
              = 60/10 = 6

n = ∑(f_i) = 1+3+5+1 = 10

Now,

σ² = (∑_iⁿ f_i(x_i - x̄)²/n)
= (16 + 12 + 0 +4)/10
= 3.2

Variance(σ²) = 3.2

Example 4: Find the variance of the following data table

Class	Xi	fi	f×Xi	Xi - μ	(Xi - μ)2	f×(Xi - μ)2
0-10	5	3	15	-15	225	675
10-20	15	6	90	-5	25	150
20-30	25	4	100	5	25	100
30-40	35	2	70	15	225	450
40-50	45	1	45	25	625	625
Total		16	320			2000

Mean (μ) = ∑(fi xi)/∑(fi)
               = 320/16 = 20

σ² = (∑_iⁿ f_i(x_i - μ)²/n)

    = [(2000)/(16)]
    = (125)

The variance of given data set is 125.