Unequal Averages

Age

11 to 14

Challenge level

Play around with sets of five numbers and see what you can discover about different types of average...

Being curious Being resourceful Being resilient Being collaborative

Problem

This problem follows on from M, M and M and About Average, so you may want to look at these first.

Here's an interesting set of five numbers: $$2, 5, 5, 6, 7$$ The mean, mode, median and range are all 5.

Can you find other sets of five positive whole numbers where:

Mean = Median = Mode = Range

Can you find sets of five positive whole numbers that satisfy the following properties?

A. Mode < Median < Mean

B. Mode < Mean < Median

C. Mean < Mode < Median

D. Mean < Median < Mode

E. Median < Mode < Mean

F. Median < Mean < Mode

Not all of these can be satisfied by sets of five numbers! Can you explain why?

Show that some of them can be satisfied with sets of just four numbers.

Show that all of them can be satisfied with sets of six numbers.

If you enjoyed this problem, you may be interested in Wipeout