Question

For grouped data, if the class intervals are 10-20, 20-30, 30-40 with frequencies 5, 15, and 10, and the mean is 25, the variance is:

• A. 0%
• B. 0%
• C. 0%
• D. 0%

Answer 1

50

Answer 2

To calculate the variance for grouped data, 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 10-20: $x_1 = (10 + 20) / 2 = 15$
• For the interval 20-30: $x_2 = (20 + 30) / 2 = 25$
• For the interval 30-40: $x_3 = (30 + 40) / 2 = 35$

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

• $f_1 = 5$
• $f_2 = 15$
• $f_3 = 10$

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

• $\bar{x} = 25$

4. Calculate the total frequency ($\sum f_i$):

$$\sum f_i = 5 + 15 + 10 = 30$$

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 ($f_i$)	$x_i - \bar{x}$	$(x_i - \bar{x})^2$	$f_i (x_i - \bar{x})^2$
10-20	15	5	$15 - 25 = -10$	$(-10)^2 = 100$	$5 \times 100 = 500$
20-30	25	15	$25 - 25 = 0$	$(0)^2 = 0$	$15 \times 0 = 0$
30-40	35	10	$35 - 25 = 10$	$(10)^2 = 100$	$10 \times 100 = 1000$
Total		$\sum f_i = 30$			$\sum f_i (x_i - \bar{x})^2 = 500 + 0 + 1000 = 1500$

6. Calculate the variance ($\sigma^2$):

$$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} = \frac{1500}{30}$$ $$\sigma^2 = 50$$