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 calculate the variance for a grouped frequency distribution, we use the formula: $\sigma^2 = \frac{\sum_{i=1}^{n} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{n} f_i}$ where:

• $x_i$ are the midpoints of the class intervals.
• $f_i$ are the frequencies of the class intervals.
• $\bar{x}$ is the mean of the distribution.

1. Determine the midpoints ($x_i$) for each interval:

• For the interval 5-15: $x_1 = (5 + 15) / 2 = 10$
• For the interval 15-25: $x_2 = (15 + 25) / 2 = 20$
• For the interval 25-35: $x_3 = (25 + 35) / 2 = 30$

2. List the given frequencies ($f_i$):

• $f_1 = 10$
• $f_2 = 20$
• $f_3 = 15$

3. Note the given mean ($\bar{x}$):

• $\bar{x} = 20$

4. Calculate the total frequency ($\sum f_i$): $\sum f_i = 10 + 20 + 15 = 45$

5. Calculate the deviation from the mean ($x_i - \bar{x}$), the squared deviation $(x_i - \bar{x})^2$, and $f_i (x_i - \bar{x})^2$ for each class:

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

6. Calculate the variance ($\sigma^2$): $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} = \frac{2500}{45}$ $\sigma^2 = \frac{500 \times 5}{9 \times 5} = \frac{500}{9}$ $\sigma^2 \approx 55.555...$

Rounding to the nearest whole number or checking the options, 55.555... is closest to 56.