Can someone explain its significance and maybe a simple example as well?
In: 3585
I’m going to steal [Skafi’s example](https://www.reddit.com/r/nba/comments/4p63ku/ysk_simpsons_paradox_and_basketball/):
Before getting to Simpson’s paradox, I’m going to define some basketball terms for anyone who is not familiar. In basketball, there are two types of field goal attempts. 2-pointers and 3-pointers. You can calculate their percentages individually or together as an overall field goal percentage. For example, let’s say that a player attempted 40 2-point field goals, making 30 of them, and attempted 10 3-point field goals, making 3 of them.
Her 2-point% is 30/40 = 75%.
Her 3-point% is 3/10 = 30%.
You can also look at overall field goal % by treating both types of shots the same and disregarding whether they were 2-point or 3-point attempts.
She attempted 50 total field goals (40 2-point + 10 3-point) and made a total of 33 (30 2-point + 3 3-point).
Her overall field goal % is then 33/50 = 66%.
An example of Simpson’s Paradox is the following. Say that you are told the 2-point% and 3-point% for two different players:
Player | 2-Point% | 3-Point% |
—————-|——-|——-|
Larry Bird | 50.9% | 37.6% |
Reggie Miller | 51.6% | 39.5% |
Reggie Miller’s % is higher than Larry Bird’s in both categories. The logical assumption would be that Reggie Miller’s combined field goal% would be higher than Larry Bird’s as well because that Reggie’s percentage is higher in both components of field goal%.
However, the actual values:
Player | 2-Point% | 3-Point% | Overall FG% |
—————-|——-|——-|————-|
Larry Bird | 50.9% | 37.6% | 49.6%
Reggie Miller | 51.6% | 39.5% | 47.1%
How can Larry Bird have a higher overall field goal % when he had a lower percentage for every component of the calculation? It’s because there was another factor not considered.
37% of Reggie Miller’s career field goal attempts were 3-Pointers, while only 10% of Larry Bird’s career field goal attempts were 3-Pointers. Because 3-point field goal attempts have a lower chance of success, Reggie’s 3-point % dragged his 2-point % further down than Larry’s 3-point % dragged his 2-Point % down.
The specific overall field goal% calculations:
Reggie Miller: 51.6%*63% + 39.5%*37% = 47.1%
Larry Bird: 50.9%*90% + 37.6%*10% = 49.6%
Again, you can see that Reggie’s overall field goal% was much more influenced by the relatively less likely 3-pointers than Larry’s was.
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