# Reverse Mean–The Answers Revealed!

The hat-trick was on this week, and the teacher members of my Diagnostic Questions website certainly did not bottle it. Once again, you correctly predicted the most common misconception that students had with the following reverse mean / backwards averages question:

The most common misconception was indeed a) 18, which attracted half of the incorrect answers. What reasons did students give for this? Well, let’s have a read of some of their explanations:

*58-40=18*

*The total is 58, so if one of the numbers is 40, the others must add up to 18.*

However, I think you will be slightly surprised about the second most common misconception that students have. Whilst only 6% of teachers went for c) 162, it in fact elicited 26% of incorrect student answers. Can you see why someone would get an answer of 162? I must admit, I couldn’t. But then I read their explanations:

*You need to multiply by how many numbers there are to do these sort of questions. The difference is 18 and there are 9 numbers, so you get 162*

*The difference is 18, and there are 9 numbers, so the mean must be 18 x 9*

*18×9*

I hope all of this serves to illustrate what I believe is the most important point of Diagnostic Questions. It is not simply the case that students are either right or wrong. They can be wrong for very different reasons, and each reason requires a specific form of intervention. Would you agree that an answer of c) is slightly better than an answer of a)? And further still, an answer of d) is the “best” wrong answer of all? Or am I talking nonsense? We will return to this more, I am sure, in the coming weeks.

For now, check the blog on Sunday for the next round of “Guess the Misconception”!