If the median of the data 30, 8, 7, 3, 17, 15, 21, 24, 29, 23 is x and the…
2025
If the median of the data 30, 8, 7, 3, 17, 15, 21, 24, 29, 23 is x and the median of the data is obtained by replacing 3 by 33 and 8 by 18 in the above data is y, then what is the difference between y and x?
- A.
2
- B.
3
- C.
4
- D.
1
Attempted by 16 students.
Show answer & explanation
Correct answer: B
For a data set with an EVEN number of values, once it is arranged in ascending order the median is not a single term but the average of the two middle terms — the (n/2)th and (n/2 + 1)th terms. Whenever any value in the set is replaced, the set must be re-sorted from scratch before this same rule is reapplied, since a replacement can shift which terms land in the middle.
Arrange the original 10 values — 30, 8, 7, 3, 17, 15, 21, 24, 29, 23 — in ascending order: 3, 7, 8, 15, 17, 21, 23, 24, 29, 30.
With n = 10 (even), the median x is the average of the 5th and 6th terms: (17 + 21) / 2 = 19.
Replace 3 with 33 and 8 with 18 in the original data, giving 30, 18, 7, 33, 17, 15, 21, 24, 29, 23.
Re-sort this updated data in ascending order: 7, 15, 17, 18, 21, 23, 24, 29, 30, 33.
The median y is again the average of the 5th and 6th terms of this newly sorted list: (21 + 23) / 2 = 22.
The required difference is y − x = 22 − 19 = 3.
Both replaced values (3 and 8) sat in the lower half of the original ordering, and both replacements (33 and 18) land in the upper half of the new ordering, so exactly two terms move from below the middle to above it — this consistently pulls the middle pair upward, confirming that y should exceed x rather than staying the same or falling below it.