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This Science Saturday will be breaking new ground. For the first time in it’s history, we’re actually going to talk about football.
Shocking, I know.
Let’s say we have two teams, well call them the Leafs and the Dams, and we want to see who has the better defense. How would we do this?
One way to measure this is to talk about yards per play. And since there’s two types of plays, passing and rushing, we can see who does better in each category. For rushing we’ll use yards per carry, and for passing we’ll use net yards per attempt (a net passing yard is just like a regular passing yard except it counts sacks as pass plays).
|NY/A (Passing)||Y/C (Running)|
Clearly the Leafs had the better defense. They were better in both passing and rushing.
But just for fun let’s see what the combined yardage looks like. How much better did the Leafs do in yards per play?
Wait, now the Dams have the better defense? How can it be possible for them to have a better overall defense when the Leafs are better against the pass and the run?
This is called Simpon’s Paradox, and has to do with some funny things that can happen with averages. Since these per-play stats are averages, they can run into this issue.
Here’s how the Leafs and Dams example worked.
|Team||NY/A||Pass Attempts||Y/C||Run Attempts||Y/P (Total)|
Since passing plays gained a lot more yards than rushing plays, having more passing plays is going to raise the total average. In this example the Leafs face 200 more passing attempts and 150 fewer running attempts, so even though their average on both were better, their total average was worse.
Or think of it this way if you want a more intuitive example. Would you rather have 10 quarters and 2 dimes or 2 quarters and 10 dimes? Its the same number of coins, and the quarters and dimes individually have the same value, but having more quarters makes the total higher.
You’ve probably heard a saying about how you can lie with statistics, and while I don’t totally agree with that the Simpson’s Paradox is a way people can use statistics to be deceptive.
Consider a private school versus a public school. The public school has to accept all students, while the private school can pick and choose. So even if the public school teaches both low performing students and high performing students better the private school can look better by simply accepting only the high performers.
Another example is a hospital. A hospital can make their statistics look better by only treating those patients who have a milder form of whatever disease they want to promote. They may not actually be any better, but because they reduce the number of high risk patients and increase the number of low risk ones their average fatality rate will be much lower than the hospital who doesn’t discriminate.
This is one science lesson that everybody should know. It’s important to keep it in mind when trying to determine who’s really as good as they say they are.