Question

The mean deviation about the mean for the following data is:
 

 

• A. \(2\)
• B. \( \frac{15}{13} \)
• C. \( \frac{22}{13} \)
• D. \( \frac{20}{13} \)

Hint:

To compute mean deviation, find class midpoints, calculate the mean, take absolute deviations, and apply the formula.

Answer 1

The Correct Option is D

Step 1: Computing class midpoints.
The class midpoints are: \[ x_i = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \] \[ x_1 = 1, \quad x_2 = 3, \quad x_3 = 5, \quad x_4 = 7, \quad x_5 = 9 \] Step 2: Computing the mean.
The mean is given by: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] \[ = \frac{(1 \times 1) + (3 \times 3) + (5 \times 5) + (3 \times 7) + (1 \times 9)}{1+3+5+3+1} \] \[ = \frac{1 + 9 + 25 + 21 + 9}{13} = \frac{65}{13} = 5 \] Step 3: Computing absolute deviations.
\[ |x_i - \bar{x}| \] \[ |1 - 5| = 4, \quad |3 - 5| = 2, \quad |5 - 5| = 0, \quad |7 - 5| = 2, \quad |9 - 5| = 4 \] Step 4: Computing mean deviation.
\[ MD = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i} \] \[ = \frac{(1 \times 4) + (3 \times 2) + (5 \times 0) + (3 \times 2) + (1 \times 4)}{13} \] \[ = \frac{4 + 6 + 0 + 6 + 4}{13} = \frac{20}{13} \] Thus, the mean deviation about the mean is: \[ \mathbf{\frac{20}{13}} \]