Question

Consider the 10 observations \(2, 3, 5, 10, 11, 13, 15, 21, a\) and \(b\) such that the mean of observations is \(9\) and variance is \(34.2\). Then the mean deviation about median is:

• A. \(3\)
• B. \(5\)
• C. \(6\)
• D. \(7\)

Hint:

For statistics problems involving unknown observations:

Use mean to form a linear equation.
Use variance to form a quadratic equation.
Always arrange data before finding median-based measures.

Answer 1

The Correct Option is B

Concept:

Mean \(= \dfrac{\text{Sum of observations}}{n}\)
Variance \(= \dfrac{1}{n}\sum (x_i - \bar{x})^2\)
Mean deviation about median: \[ \text{MD}_{\text{median}} = \frac{1}{n}\sum |x_i - \text{Median}| \]
Step 1: Using the given mean. \[ \text{Mean} = 9 \Rightarrow \text{Sum of 10 observations} = 10 \times 9 = 90 \] Sum of known observations: \[ 2+3+5+10+11+13+15+21 = 80 \] \[ \Rightarrow a + b = 10 \quad \cdots (1) \]
Step 2: Using the given variance. \[ \text{Variance} = 34.2 \Rightarrow \sum (x_i - 9)^2 = 10 \times 34.2 = 342 \] For known observations: \[ \sum (x_i - 9)^2 = 302 \] \[ \Rightarrow (a-9)^2 + (b-9)^2 = 342 - 302 = 40 \quad \cdots (2) \] Solving equations (1) and (2), we get: \[ a = 3,\quad b = 7 \]
Step 3: Arrange the complete data in ascending order: \[ 2, 3, 3, 5, 7, 10, 11, 13, 15, 21 \] Since \(n = 10\), the median is: \[ \text{Median} = \frac{7 + 10}{2} = 8.5 \]
Step 4: Compute mean deviation about median. \[ \sum |x_i - 8.5| = 50 \] \[ \text{Mean Deviation} = \frac{50}{10} = 5 \] \[ \Rightarrow \text{Mean deviation about median} = 5 \]