Question

A small accounting firm has
\[4\] accountants who each earn a different salary between
\[52{,}000\] dollars and
\[58{,}000\] dollars. For extra help during tax season, they hire a
\[5^{\text{th}}\] accountant who earns
\[10{,}000\] dollars.
How will hiring the
\[5^{\text{th}}\] accountant affect the mean and median?
Choose 1 answer:
A
Both the mean and median will decrease, but the median will decrease by more than the mean.
B
Both the mean and median will decrease, but the mean will decrease by more than the median.
C
The mean will decrease, and the median will increase.
D
The mean will increase, and the median will decrease.

Answer 1

B

Let the original salaries (in thousands) be a<b<c<d (each between 52 and 58). The original mean is S/4 where S=a+b+c+d, and the new mean is (S+10)/5; since 10 < S/4, the mean decreases. The original median is (b+c)/2, the new median (with 10 added at the low end) is b, so the median decreases by (c−b)/2 (at most 3). Meanwhile the mean decreases by (S−40)/20, and with 208 ≤ S ≤ 232 this is between 8.4 and 9.6. Thus both decrease, and the mean decreases by more.