Question

For a grouped frequency distribution with intervals 5-15, 15-25, and 25-35, with frequencies 10, 20, and 15 and mean 20, what is the variance?

• A. 50
• B. 56
• C. 120
• D. 110

Answer 1

Answer: (B)

56

Answer 2

To find the variance for a grouped frequency distribution, we use the formula:

$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$

Where:

• $f_i$ is the frequency of each class.
• $x_i$ is the midpoint of each class.
• $\bar{x}$ is the mean of the distribution.
• $\sum f_i$ is the total frequency (N).

Given data:

Class Interval	Frequencies ($f_i$)
5-15	10
15-25	20
25-35	15

The mean ($\bar{x}$) is given as 20.

Step 1: Calculate the midpoints ($x_i$) for each class interval.

• For 5-15: $x_1 = \frac{5 + 15}{2} = 10$
• For 15-25: $x_2 = \frac{15 + 25}{2} = 20$
• For 25-35: $x_3 = \frac{25 + 35}{2} = 30$

Step 2: Create a table for calculations.

Class Interval	Frequency ($f_i$)	Midpoint ($x_i$)	$(x_i - \bar{x})$	$(x_i - \bar{x})^2$	$f_i (x_i - \bar{x})^2$
5-15	10	10	$10 - 20 = -10$	$(-10)^2 = 100$	$10 \times 100 = 1000$
15-25	20	20	$20 - 20 = 0$	$0^2 = 0$	$20 \times 0 = 0$
25-35	15	30	$30 - 20 = 10$	$10^2 = 100$	$15 \times 100 = 1500$
Total	$\sum f_i = 45$				$\sum f_i (x_i - \bar{x})^2 = 1000 + 0 + 1500 = 2500$

Step 3: Calculate the variance.

$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} = \frac{2500}{45}$

Simplify the fraction:

$\sigma^2 = \frac{2500 \div 5}{45 \div 5} = \frac{500}{9}$

Convert to decimal:

$\sigma^2 \approx 55.555...$

Step 4: Compare with the given options.

The calculated variance is approximately 55.56. Among the given options, 56 is the closest value.

Note: If we were to calculate the mean from the given data, it would be $\bar{x}_{calc} = \frac{\sum f_i x_i}{\sum f_i} = \frac{(10 \times 10) + (20 \times 20) + (15 \times 30)}{45} = \frac{100 + 400 + 450}{45} = \frac{950}{45} = \frac{190}{9} \approx 21.11$. Since the problem explicitly states "mean 20", we use this given value for the variance calculation.