Activity: Averages Brain-Teaser

Here is a little puzzle about averages. Is it right?


Who is Better at Kicking Goals?

soccer teams

At practice last week:

Sam is better!


This week:

Sam is still better.


But let's add up the scores for BOTH weeks:

Hang on! YOU are better!


Sam was better last week and this week ... but you are better over both weeks?

Please explain.




Maybe make a table with all the data and do the calculations yourself

Last Week  
This Week  
Both Weeks  





... read on after you have thought about it ...







It is All True


Because you had SO MANY shots at goal this week, and did well at them, you lifted your two-week average above Sam's.


At practice last week:

This week:


For BOTH weeks:



To be fair, we should really compare the averages when your and Sam's attempts at goal is roughly the same.

If Sam had attempted 100 shots this week, he may have scored 60 out of 100, and his two-week average would have been about 57%, better than you.

So be careful when comparing two sets of data with widely different counts.


Note: only certain sets of data produce this "paradox", usually everything makes sense.