Question

For a dataset with midpoints 10, 20, and 30 and frequencies 4, 6, and 10, and the mean is 24, the variance is:

• A. 62
• B. 24
• C. 20
• D. 18

Answer 1

Answer: (A)

62

Answer 2

The problem asks us to calculate the variance of a dataset given its midpoints, frequencies, and the mean.

Given Data:

• Midpoints ($x_i$): 10, 20, 30
• Frequencies ($f_i$): 4, 6, 10
• Mean ($\bar{x}$): 24

Formula for Variance (for grouped data): The variance ($\sigma^2$) is given by the formula: $\sigma^2 = \frac{\sum_{i=1}^{n} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{n} f_i}$

Step-by-step Calculation:

1. Calculate the total frequency ($\sum f_i$): $\sum f_i = 4 + 6 + 10 = 20$

2. Calculate the deviation from the mean ($x_i - \bar{x}$) for each midpoint:

   • For $x_1 = 10$: $10 - 24 = -14$
   • For $x_2 = 20$: $20 - 24 = -4$
   • For $x_3 = 30$: $30 - 24 = 6$

3. Calculate the squared deviation $(x_i - \bar{x})^2$ for each midpoint:

   • For $x_1 = 10$: $(-14)^2 = 196$
   • For $x_2 = 20$: $(-4)^2 = 16$
   • For $x_3 = 30$: $(6)^2 = 36$

4. Calculate the product of frequency and squared deviation $f_i (x_i - \bar{x})^2$ for each midpoint:

   • For $x_1 = 10, f_1 = 4$: $4 \times 196 = 784$
   • For $x_2 = 20, f_2 = 6$: $6 \times 16 = 96$
   • For $x_3 = 30, f_3 = 10$: $10 \times 36 = 360$

5. Calculate the sum of $f_i (x_i - \bar{x})^2$: $\sum f_i (x_i - \bar{x})^2 = 784 + 96 + 360 = 1240$

6. Calculate the variance ($\sigma^2$): $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} = \frac{1240}{20} = 62$

The variance of the dataset is 62.