A data set consists of positive integers. Its mean is , and its range is . What is the greatest possible value of the median of the data set?
For maximum or minimum questions involving a mean, first turn the mean into a fixed total. Then assume the target statistic is one step beyond the value you expect and use ordering, the median definition, and the range to force a minimum possible total. If that total is too large, the target is impossible. Finally, construct a data set that reaches the largest remaining value to prove it is attainable.
Hints
Use the mean
Convert the mean into the total sum of all values.
Test the next possible median
To determine whether the median can be at least , count how many values must be at least .
Use the range as a lower bound
If the maximum is at least and the range is , determine the smallest possible value of every entry in the lower half of the data set.
Desmos Guide
Prefer a bounds argument
Algebraic reasoning is faster here because Desmos cannot by itself prove that no larger median is possible. First calculate the required total by entering 9(12).
Check the total forced by a median of
Enter 5(20)+4(20-16). This is the smallest possible total if five values are at least and the range is . Compare this result with the required total.
Verify a candidate data set
Enter L=[3,3,3,4,19,19,19,19,19]. Then enter mean(L), median(L), and max(L)-min(L) to verify the mean, median, and range.
Step-by-step Explanation
Find the total of the data set
Because the mean is for integers, the total is
Rule out a median of or greater
If the median were at least , then at least of the values would be at least .
Also, the maximum value would be at least . Since the range is , the minimum value would be at least . Therefore, each of the other values would be at least .
The total would have to be at least
But the actual total is only , so the median cannot be or greater.
Show that the greatest remaining value is possible
Consider the data set
Its sum is , so its mean is . Its range is , and its fifth value, the median, is . Therefore, the greatest possible median is .