Question

If \( M_1 \) is the mean deviation from the mean of the discrete data \( 44, 5, 27, 20, 8, 54, 9, 14, 35 \) and \( M_2 \) is the mean deviation from the median of the same data, then \( M_1 - M_2 = \):

• A. \( \frac{7}{9} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{5}{9} \)
• D. \( \frac{4}{9} \) \bigskip

Hint:

The mean deviation from the mean is calculated using \( M_1 = \frac{1}{n} \sum |x_i - \bar{x}| \), while the mean deviation from the median follows \( M_2 = \frac{1}{n} \sum |x_i - \text{median}| \). These measures provide insights into data dispersion.

Answer 1

The Correct Option is D

We are given the following set of numbers: \[ 44, 5, 27, 20, 8, 54, 9, 14, 35 \] 

Step 1: Compute the Mean Deviation from the Mean ( \( M_1 \) ) The mean deviation from the mean is given by: \[ M_1 = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| \] where \( \bar{x} \) represents the mean of the dataset, and \( x_i \) are the individual values. First, we calculate \( \bar{x} \): \[ \bar{x} = \frac{44 + 5 + 27 + 20 + 8 + 54 + 9 + 14 + 35}{9} = \frac{216}{9} = 24 \] Now, compute \( M_1 \): \( M_1 = \frac{1}{9} \left( |44 - 24| + |5 - 24| + |27 - 24| + |20 - 24| + |8 - 24| + |54 - 24| + |9 - 24| + |14 - 24| + |35 - 24| \right) \) \[ M_1 = \frac{1}{9} \left( 20 + 19 + 3 + 4 + 16 + 30 + 15 + 10 + 11 \right) \] \[ M_1 = \frac{128}{9} \approx 14.22 \] 

Step 2: Compute the Mean Deviation from the Median ( \( M_2 \) ) To determine \( M_2 \), we first find the median by arranging the data in ascending order: \[ 5, 8, 9, 14, 20, 27, 35, 44, 54 \] Since there are 9 values, the median is the middle value: \[ \text{Median} = 20 \] Now, compute \( M_2 \): \[ M_2 = \frac{1}{9} \left( |5 - 20| + |8 - 20| + |9 - 20| + |14 - 20| + |20 - 20| + |27 - 20| + |35 - 20| + |44 - 20| + |54 - 20| \right) \] \[ M_2 = \frac{1}{9} \left( 15 + 12 + 11 + 6 + 0 + 7 + 15 + 24 + 34 \right) \] \[ M_2 = \frac{124}{9} \approx 13.78 \] 

Step 3: Compute \( M_1 - M_2 \) The difference between the mean deviation from the mean and the mean deviation from the median is: \[ M_1 - M_2 = 14.22 - 13.78 = 0.44 \] Approximating this value: \[ M_1 - M_2 = \frac{4}{9}. \] Thus, the final result is: \[ \boxed{\frac{4}{9}} \]