Question

The coefficient of variation for the frequency distribution is:

• A. \( \frac{50}{\sqrt{3}} \)
• B. \( \frac{125}{2\sqrt{3}} \)
• C. \( \frac{100}{3\sqrt{2}} \)
• D. \( \frac{100}{\sqrt{3}} \)

Hint:

The coefficient of variation (CV) helps to compare the variability of different distributions by normalizing the standard deviation with respect to the mean.

Answer 1

The Correct Option is D

We are given the frequency distribution: \includegraphics[]{34.png} To calculate the coefficient of variation (CV), we use the formula: \[ CV = \frac{\sigma}{\mu} \times 100, \] where \( \sigma \) is the standard deviation and \( \mu \) is the mean. 1. First, calculate the mean \( \mu \): \[ \mu = \frac{\sum f_i x_i}{\sum f_i} = \frac{1(4) + 3(3) + 5(1)}{1 + 3 + 5} = \frac{4 + 9 + 5}{9} = \frac{18}{9} = 2. \] 2. Next, calculate the variance \( \sigma^2 \): \[ \sigma^2 = \frac{\sum f_i x_i^2}{\sum f_i} - \mu^2 = \frac{1(4^2) + 3(3^2) + 5(1^2)}{9} - 2^2 = \frac{16 + 27 + 5}{9} - 4 = \frac{48}{9} - 4 = \frac{48}{9} - \frac{36}{9} = \frac{12}{9} = \frac{4}{3}. \] Thus, \( \sigma = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \). 3. Now, calculate the coefficient of variation: \[ CV = \frac{\sigma}{\mu} \times 100 = \frac{\frac{2}{\sqrt{3}}}{2} \times 100 = \frac{1}{\sqrt{3}} \times 100 = \frac{100}{\sqrt{3}}. \] Thus, the correct answer is \( \frac{100}{\sqrt{3}} \).

Answer 2

Given the frequency distribution:
\[ \begin{array}{c|ccc} x_i & 4 & 3 & 1 \\ \hline f_i & 1 & 3 & 5 \\ \end{array} \] Calculate the coefficient of variation (CV).

Step 1: Calculate total frequency:
\[ N = 1 + 3 + 5 = 9 \]

Step 2: Calculate the mean \( \bar{x} \):
\[ \bar{x} = \frac{1 \times 4 + 3 \times 3 + 5 \times 1}{9} = \frac{4 + 9 + 5}{9} = \frac{18}{9} = 2 \]

Step 3: Calculate variance \( \sigma^2 \):
\[ \sigma^2 = \frac{1 \times (4 - 2)^2 + 3 \times (3 - 2)^2 + 5 \times (1 - 2)^2}{9} \] \[ = \frac{1 \times 4 + 3 \times 1 + 5 \times 1}{9} = \frac{4 + 3 + 5}{9} = \frac{12}{9} = \frac{4}{3} \]

Step 4: Calculate standard deviation \( \sigma \):
\[ \sigma = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \]

Step 5: Coefficient of variation (CV) is:
\[ \text{CV} = \frac{\sigma}{\bar{x}} \times 100 = \frac{\frac{2}{\sqrt{3}}}{2} \times 100 = \frac{1}{\sqrt{3}} \times 100 = \frac{100}{\sqrt{3}} \]

Therefore,
\[ \boxed{\frac{100}{\sqrt{3}}} \]