Question

Let \( x \) be the median of the data: \[ \{20, 50, 60, 53, 77, 88, 90, 40, 30, 70, 25, 45, 64, 72, 8, 15, 85, 60, 55, 28\} \] If \( y \) is the median of the same dataset when 25 and 28 are replaced by 52 and 82 respectively, then what is the value of \( |x - y| \)?

• A. 2.5
• B. 3
• C. 3.5
• D. 4

Hint:

When data changes, always re-sort and recompute the median. For even-sized datasets, median is the average of the two central elements.

Answer 1

The Correct Option is C

Original dataset (sorted): \[ \{8, 15, 20, 25, 28, 30, 40, 45, 50, 53, 55, 60, 60, 64, 70, 72, 77, 85, 88, 90\} \] Size = 20 ⇒ Even number ⇒ \[ x = \text{Median} = \frac{10^\text{th} + 11^\text{th}}{2} = \frac{53 + 55}{2} = 54 \] Replace 25 and 28 with 52 and 82: New dataset (sorted): \[ \{8, 15, 20, 30, 40, 45, 50, 52, 53, 55, 60, 60, 64, 70, 72, 77, 82, 85, 88, 90\} \] \[ y = \text{Median} = \frac{10^\text{th} + 11^\text{th}}{2} = \frac{55 + 60}{2} = 57.5 \] \[ |x - y| = |54 - 57.5| = 3.5 \] Correct value: 3.5