Activity: Averages Brain-Teaser
Here is a little puzzle about averages. Is it right?
Who is Better at Kicking Goals?
At practice last week:
- You scored 2 of 10 shots at goal
- Sam scored 3 of 10 shots
Sam is better!
This week:
- You scored 53 of 100 shots
- Sam scored 6 of 10 shots
Sam is still better.
But let's add up the scores for BOTH weeks:
- You scored 55 of 110 shots: that is 50%
- Sam scored 9 of 20 shots: that is only 45%
Hang on! YOU are better!
Sam was better last week and this week ... but you are better over both weeks?
Please explain.
...
Make a table with all the data and do the calculations yourself:
| Sam | You | |
| 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:
- You scored 2 of 10 (20%)
- Sam scored 3 of 10 (30%)
This week:
- You scored 53 of 100 (53%)
- Sam scored 6 of 10 (60%)
For BOTH weeks:
- You scored 55 of 110 (50%)
- Sam scored 9 of 20 (45%)
To be fair, we should really compare the averages when both of you have had about the same number of attempts at goal.
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.
This is an example of "Simpson's paradox" if you want to find out more.
Note: only certain sets of data produce this "paradox", usually everything makes sense.