How to Calculate Variance for a Given Distribution
Introduction:
In this post, we will explore the step-by-step process of calculating the variance and standard deviation for a given frequency distribution. Understanding these concepts is crucial for analyzing data variability and distribution spread.
Example:
Given the following frequency distribution, we will calculate both the variance and the standard deviation.
| \( X \) | 4.5 | 14.5 | 24.5 | 34.5 | 44.5 | 54.5 | 64.5 |
|———-|——|——|——|——|——|——|——|
| \( f \) | 1 | 5 | 12 | 22 | 17 | 9 | 4 |
Step-by-Step Solution:
Define Parameters:
– \( x_i \) is the value of the distribution.
– \( f_i \) is the frequency corresponding to each \( x_i \).
– \( h = 10 \) (class interval).
– \( \bar{x} = 34.5 \) (assumed mean).
Calculation of Deviation and Frequency Deviations:
First, we calculate the deviation \( d_i = x_i – 34.5 \) and then find \( u_i = \frac{d_i}{h} \). Next, we calculate \( f_i u_i \) and \( f_i u_i^2 \).
| \( x_i \) | \( f_i \) | \( d_i = x_i – 34.5 \) | \( u_i = \frac{d_i}{10} \) | \( f_i u_i \) | \( u_i^2 \) | \( f_i u_i^2 \) |
|————|———–|————————|—————————-|—————|————-|——————|
| 4.5 | 1 | -30 | -3 | -3 | 9 | 9 |
| 14.5 | 5 | -20 | -2 | -10 | 4 | 20 |
| 24.5 | 12 | -10 | -1 | -12 | 1 | 12 |
| 34.5 | 22 | 0 | 0 | 0 | 0 | 0 |
| 44.5 | 17 | 10 | 1 | 17 | 1 | 17 |
| 54.5 | 9 | 20 | 2 | 18 | 4 | 36 |
| 64.5 | 4 | 30 | 3 | 12 | 9 | 36 |
\( N = \sum f_i = 70, \quad \sum f_i u_i = 22, \quad \sum f_i u_i^2 = 130 \)
Formula for Variance:
The variance \( \text{Var}(X) \) is given by:
\[
\text{Var}(X) = h^2 \left[ \frac{1}{N} \sum f_i u_i^2 – \left( \frac{1}{N} \sum f_i u_i \right)^2 \right]
\]
Plugging in the values:
\[
\text{Var}(X) = 100 \left[ \frac{130}{70} – \left( \frac{22}{70} \right)^2 \right] = 100 \left[ \frac{13}{7} – \frac{11}{35} \right]
\]
\[
\text{Var}(X) = 100 \left[ 1.857 – 0.098 \right] = 100 \times 1.759 = 175.822
\]
Standard Deviation (S.D.):
The standard deviation is the square root of the variance:
\[
\text{S.D.} = \sqrt{175.822} = 13.259
\]
Video
Conclusion:
From the given data, the variance is 175.822, and the standard deviation is 13.259. This detailed approach can be applied to any similar frequency distribution to determine how spread out the data is from its mean.