Question

If the variance of the data \( 2,3,5,8,12 \) is \( \sigma^2 \) and the mean deviation from the median for this data is \( M \), then \( \sigma^2 - M \) is:

• A. \( 10.2 \)
• B. \( 5.8 \)
• C. \( 10.6 \)
• D. \( 8.2 \)

Hint:

Variance measures spread, while mean deviation measures absolute dispersion.

Answer 1

The Correct Option is A

Given observations: \( 2, 3, 5, 8, 12 \). 
1. Calculate Mean: \[ {Mean} = \frac{2 + 3 + 5 + 8 + 12}{5} = 6 \] 
2. Calculate Variance: \[ \sigma^2 = 13.2 \] 
3. Find Median: Since the number of observations is odd, the median is the middle value: \[ {Median} = 5 \] 
4. Calculate Mean Deviation about Median: \[ M = \frac{|2 - 5| + |3 - 5| + |5 - 5| + |8 - 5| + |12 - 5|}{5} = 3 \] 
5. Final Calculation: \[ \sigma^2 - M = 13.2 - 3 = 10.2 \]