Question

The mean deviation from the median for the following data is:
\[ \begin{array}{c|ccccc} x_i & 9 & 3 & 7 & 2 & 5 \\ f_i & 1 & 6 & 2 & 8 & 4 \\ \end{array} \]

• A. \(\frac{94}{21}\)
• B. \(\frac{12}{7}\)
• C. \(\frac{10}{7}\)
• D. \(\frac{100}{21}\)

Hint:

Arrange data in order, find median using cumulative frequency, then find mean deviation using absolute differences.

Answer 1

The Correct Option is C

Calculate total frequency \(n = 1 + 6 + 2 + 8 + 4 = 21\). Arrange data in ascending order of \(x_i\): 2, 3, 5, 7, 9 with frequencies 8, 6, 4, 2, 1 respectively. Median position \(= \frac{n+1}{2} = \frac{22}{2} = 11\). Cumulative frequencies: - Up to 2: 8 - Up to 3: 8 + 6 = 14 Median lies in class \(x = 3\). Mean deviation from median: \[ \frac{\sum f_i |x_i - \text{median}|}{n}. \] Calculate deviations: \[ |2 - 3| = 1,
|3 - 3| = 0,
|5 - 3| = 2,
|7 - 3| = 4,
|9 - 3| = 6. \] Sum: \[ (8 \times 1) + (6 \times 0) + (4 \times 2) + (2 \times 4) + (1 \times 6) = 8 + 0 + 8 + 8 + 6 = 30. \] Mean deviation: \[ \frac{30}{21} = \frac{10}{7}. \]