Question

Mean deviation about the mean for the following data is: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-6 & 1 \\ 6-12 & 2 \\ 12-18 & 3 \\ 18-24 & 2 \\ 24-30 & 1 \\ \hline \end{array} \]

• A. \( 5 \)
• B. \( \frac{16}{3} \)
• C. \( 6 \)
• D. \( \frac{19}{3} \)

Hint:

To calculate the mean deviation, first find the midpoints of the class intervals, then calculate the mean, and finally use the formula for mean deviation to find the answer.

Answer 1

The Correct Option is B

To calculate the mean deviation about the mean for the given data, we first need to find the mean. 
Step 1: Find the midpoints of each class interval. The midpoints \( x_i \) of the class intervals are calculated as the average of the lower and upper limits of the intervals. \[ \text{Midpoint of } 0-6 = \frac{0 + 6}{2} = 3 \] \[ \text{Midpoint of } 6-12 = \frac{6 + 12}{2} = 9 \] \[ \text{Midpoint of } 12-18 = \frac{12 + 18}{2} = 15 \] \[ \text{Midpoint of } 18-24 = \frac{18 + 24}{2} = 21 \] \[ \text{Midpoint of } 24-30 = \frac{24 + 30}{2} = 27 \] 
Step 2: Find the weighted mean. To find the mean \( \bar{x} \), use the formula: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] where \( f_i \) is the frequency and \( x_i \) is the midpoint of each class interval. \[ \sum f_i x_i = 1(3) + 2(9) + 3(15) + 2(21) + 1(27) = 3 + 18 + 45 + 42 + 27 = 135 \] \[ \sum f_i = 1 + 2 + 3 + 2 + 1 = 9 \] Thus, the mean is: \[ \bar{x} = \frac{135}{9} = 15 \] 
Step 3: Find the mean deviation. The mean deviation is given by: \[ MD = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i} \] Now, calculate \( |x_i - \bar{x}| \) for each midpoint: \[ |x_1 - \bar{x}| = |3 - 15| = 12 \] \[ |x_2 - \bar{x}| = |9 - 15| = 6 \] \[ |x_3 - \bar{x}| = |15 - 15| = 0 \] \[ |x_4 - \bar{x}| = |21 - 15| = 6 \] \[ |x_5 - \bar{x}| = |27 - 15| = 12 \] Next, calculate the weighted sum of deviations: \[ \sum f_i |x_i - \bar{x}| = 1(12) + 2(6) + 3(0) + 2(6) + 1(12) = 12 + 12 + 0 + 12 + 12 = 48 \] Thus, the mean deviation is: \[ MD = \frac{48}{9} = \frac{16}{3} \]