Mean Equals Median Problem

THE PROBLEM

5, 11, 19, 22, x
If the mean and median of these 5 numbers are equal, determine all possible values of x.

ANSWER
-2, 14.25, 38

REFLECTIONS

• If the question asked for one possible value for x, this could be an SAT Grid-In question of above-average difficulty. I chose to ask for ALL possible values not only to make it more challenging but also to encourage students to probe more deeply.

• Thinking of many strategies how YOU would solve this? But that's not the focus of this blog. After WE figure out how to solve the problem ourselves and, yes, we should try to find more than one method, we should be asking ourselves as educators:

What questions need to be asked to enable our students to solve it for themselves?

What learning environment best facilitates this?

• Do you believe some of your students would reason that the median could only be 11, 19, or x itself? Listen to their discussions. Give them time but if no one is thinking that way we might ask:

Which numbers could NOT be the median? Why can't 5 be the median? 22?

• Beyond solving this problem is an even bigger question:
What must be true about a set of data values if their mean (avg) equals their median? We know there are sufficient conditions like:
The data is normally distributed about the mean, in which case, the mean, median and mode coincide.
But this is NOT NECESSARY!

In the above example, you might want to ask students to examine the case where x=14.25:

"What is the mean of the numbers if x is NOT in the set? Why does this make sense?

[If you have an 80 average and one of your scores was exactly 80, what would happen to the mean if you removed that score?]

So what does it 'mean' if the mean and median of a set of data are equal?

Now that's an open-ended assessment question!