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: \[ \sigma^2 - M = 13.2 - 3 = 10.2 \]

Answer 2

Step 1: Finding the variance (\( \sigma^2 \))
The variance is calculated using the formula: \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] where \( \bar{x} \) is the mean of the data and \( x_i \) are the individual data points.

For the given data \( 2, 3, 5, 8, 12 \), first, calculate the mean \( \bar{x} \): \[ \bar{x} = \frac{2 + 3 + 5 + 8 + 12}{5} = \frac{30}{5} = 6 \] Now, calculate the variance: \[ \sigma^2 = \frac{1}{5} \left[ (2 - 6)^2 + (3 - 6)^2 + (5 - 6)^2 + (8 - 6)^2 + (12 - 6)^2 \right] \] \[ \sigma^2 = \frac{1}{5} \left[ 16 + 9 + 1 + 4 + 36 \right] = \frac{66}{5} = 13.2 \] Step 2: Finding the median and calculating the mean deviation from the median (M)
To find the median, arrange the data in ascending order: \( 2, 3, 5, 8, 12 \). Since there are 5 numbers, the median is the third number: \[ \text{Median} = 5 \] The mean deviation from the median is calculated as: \[ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \text{Median}| \] \[ M = \frac{1}{5} \left[ |2 - 5| + |3 - 5| + |5 - 5| + |8 - 5| + |12 - 5| \right] \] \[ M = \frac{1}{5} \left[ 3 + 2 + 0 + 3 + 7 \right] = \frac{15}{5} = 3 \] Step 3: Calculate \( \sigma^2 - M \)
Now, calculate \( \sigma^2 - M \): \[ \sigma^2 - M = 13.2 - 3 = 10.2 \] Step 4: Final Answer
The value of \( \sigma^2 - M \) is:

10.2