Five students take a test on which any integer score from 0 to 100 inclusive is possible. What is the largest possible difference between the median and the mean of the scores?

Step 1: Let the scores of the five students in non-decreasing order be $x_1\le x_2\le x_3\le x_4\le x_5$, where $0\le x_i\le 100$.

Step 2: The median of the scores is the middlemost value, which is $x_3$. The mean of the scores is:

Step 3: We want to maximize the absolute difference $|x_3-\mu |$. There are two cases to consider:

• Case 1: Mean is larger than Median (Maximize $\mu -x_3$)

To make $\mu$ as large as possible and $x_3$ as small as possible:

• Set $x_3=0⟹x_1=x_2=x_3=0$.

• To maximize $\mu$, set the remaining scores to their upper bound: $x_4=x_5=100$.

• The scores are $\{0,0,0,100,100\}$.

• The median is $0$, the mean is $\mu =\frac{0+0+0+100+100}{5}=40$.

• The difference is $\mu -x_3=40-0=40$.

• Case 2: Median is larger than Mean (Maximize $x_3-\mu$)

To make $x_3$ as large as possible and $\mu$ as small as possible:

• Set $x_3=100⟹x_3=x_4=x_5=100$.

• To minimize $\mu$, set the remaining scores to their lower bound: $x_1=x_2=0$.

• The scores are $\{0,0,100,100,100\}$.

• The median is $100$, the mean is $\mu =\frac{0+0+100+100+100}{5}=60$.

• The difference is $x_3-\mu =100-60=40$.

Step 4: In both cases, the maximum difference is exactly $40$.