The least and greatest

Question

The least and greatest numbers in a list of 7 real
numbers are 2 and 20, respectively. The median of the
list is 6, and the number 3 occurs most often in the list.
Which of the following could be the average (arithmetic
mean) of the numbers in the list?

Answer 1

From the information given, one may conclude that four of the seven numbers are

2,3,3,20.     (the number 3 is the only number appearing twice.  It can appear only twice if the median is 6)

 

 

The only way that the median can be 6 is if the number 6 is in the list.  Thus the fifth number is 6.  This leaves two numbers to choose.   To get the minimum mean  one chooses 7 and 8 resulting in a mean of 49/7 = 7.   To get the maximum, one chooses 18 and 19 to get a mean of 71/7.     Thus the mean will fall between 7 and 71/7  inclusive.

 

 

Answer 2

the mean must have a value between 7 and 10.1