Examples on Variance Formula
Example 1: Calculate the variance of the sample data: 7, 11, 15, 19, 24.
Solution:
We have the data, 7, 11, 15, 19, 24
Find mean of the data.
x̄ = (7 + 11 + 15 + 19 + 24)/5
= 76/5
= 15.2Using the formula for variance we get,
σ2 = ∑ (xi – 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 data is 12 and the sum of squared differences of data from the mean is 156.
Solution:
We have,
(xi – x̄)2 = 156
σ2 = 12
Using the formula for variance we get,
σ2 = ∑ (xi – x̄)2/n
12 = 156/n
n = 156/12
n = 13
Example 3: Calculate the variance for the given data
xi |
fi |
---|---|
10 | 1 |
4 | 3 |
6 | 5 |
8 | 1 |
Solution:
Mean (x̄) = ∑(fi xi)/∑(fi)
= (10×1 + 4×3 + 6×5 + 8×1)/(1+3+5+1)
= 60/10 = 6n = ∑(fi) = 1+3+5+1 = 10
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 8 Now,
σ2 = (∑in fi(xi – x̄)2/n)
= [(16 + 12 + 0 +8)/10]
= 3.6Variance(σ2) = 3.6
Example 4: Find the variance of the following data table
Class |
Frequency |
---|---|
0-10 | 3 |
10-20 | 6 |
20-30 | 4 |
30-40 | 2 |
40-50 | 1 |
Solution:
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 = 20σ2 = (∑in fi(xi – μ)2/n)
= [(2000)/(16)]
= (125)The variance of given data set is 125.
Variance
Variance is a measurement value used to find how the data is spread concerning the mean or the average value of the data set. It is used to find how the distribution data is spread out concerning the mean or the average value. The symbol used to define the variance is σ2. It is the square of the Standard Deviation.
The are two types of variance used in statistics,
- Sample Variance
- Population Variance
The population variance is used to determine how each data point in a particular population fluctuates or is spread out, while the sample variance is used to find the average of the squared deviations from the mean.
In this article, we will learn about Variance (Sample, Population), their formulas, properties, and others in detail.
Table of Content
- What is Variance?
- Variance Definition
- Types of Variance
- Variance Symbol
- Variance Example
- Variance Formula
- Sample Variance Formula
- Population Variance Formula
- Variance Formula for Grouped Data
- Variance Formula for Ungrouped Data
- Formula for Calculating Variance
- How to Calculate Variance?
- Variance and Standard Deviation
- Variance and Covariance
- Variance Properties
- Examples on Variance Formula
- Summary – Variance
- FAQs on Variance
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