If \(a\), \(b\) and \(c\) are the median, mode and range respectively of the data 6, 14, 9, 11, 6, 17, 12, 6, 19, 10, 8, 15, then what is the value of \(3a-2b+c\)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 32.5
Put the data in order first: the median is the middle value, the mode the most frequent value and the range the largest minus the smallest.
In ascending order: 6, 6, 6, 8, 9, 10, 11, 12, 14, 15, 17, 19
12 values, so the median is the average of the 6th and 7th values: \(a=\displaystyle\frac{10+11}{2}=10.5\)
6 occurs 3 times, more than any other value, so \(b=6\)
Range \(c=19-6=13\)
\(3a-2b+c=3\times10.5-2\times6+13=32.5\)
The value of \(3a-2b+c\) is 32.5.
Why the other options are incorrect
Option (2) 38.5 — Incorrect. This takes the range as the largest value 19; the range is largest minus smallest, 13.
Option (3) 56.5 — Incorrect. This adds \(2b\) instead of subtracting it.
Option (4) 44.5 — Incorrect. This takes the median from the list as written (14.5) instead of from the ordered list.