Thirteen different numbers have a median of 18. Each of the six smallest of these numbers is decreased by 13. What is the median of the thirteen numbers after this change?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 18
For 13 numbers written in ascending order, the median is the 7th number. It depends only on which value stands in the middle, not on how far the other values are from it.
Write the numbers in ascending order; the 7th number is 18
The numbers that change are the six smallest numbers, all on one side of the middle
After the change they become still smaller, so they stay on the same side of the 7th number
The middle number itself does not change, so it is still the 7th number in order
The median stays 18; only the mean changes.
Why the other options are incorrect
Option (1) 31 — Incorrect. This adds 13 to the median; the six smallest numbers fall, and the middle one stays.
Option (2) 12 — Incorrect. This takes away the fall in the mean, \(6\times13\div13=6\), from the median; the median itself does not move.
Option (4) 5 — Incorrect. This subtracts 13 from the median, but the median (the 7th number) is not one of the six numbers that change.