Final of the 2012 FIFA Club World Cup. The two teams qualified for the final that will take place in Japan are: Sport Club Corinthians from Sao Paulo, winner of the Copa Libertadores and Chelsea F.C. from London, winner of the UEFA Champions League.
It's minute 89, the actual score is 2-2, and the referee has just awarded a penalty for Chelsea. Coach Roberto Di Matteo has to decide which player will kick the penalty. For this election, he consults his technical assistant, Steve Holland, on which player has better stats in penalties shoot out.
Steve brings the table below, according to which this year in the Premier League, Frank Lampard has shot 18 penalties and has scored 12 goals, while Fernando Torres has shot 10 penalties and has scored 6 goals (a 66.67% success rate for Lampard vs. a 60% success rate for Fernando Torres). In international competitions, Lampard has shot 9 penalties and has scored 5 goals (55,58%) while Fernando Torres has kicked 2 and has scored 1 goal (50%).
And if we look at the other seasons statistics, numbers are always favorable to Frank Lampard (91,67% > 90%, 75% > 72,73%).
Therefore, based solely on information provided by his assistant, Di Matteo decides Lampard to be the player who will shoot the penalty.
Do you think he's made the right decision?
Apparently numbers are conclusive. Lampard is better penalty-shooter than Fernando Torres. Or at least that's what it seems.
We can even add all the penalties kicked this season (those of the Premier League plus those scored in international competitions): Lampard has shot 27 and has scored 17 (62,96% success rate), while Torres has shot 12 and has scored 7 (58,33%). Numbers remain favorable to Lampard.
Let's try to add penalties kicked in other seasons: Lampard kicked 16 and scored 14 goals (87,50% success rate) while Torres shot 31 and scored 26 goals (83,87%). Clearly Di Matteo was right to choose Lampard. Or may be not.
Consider the data from another point of view. We add all the penalties that each player have shot in the Premier League in all seasons. It turns out that both players have kicked 30 penalties, and that while Lampard scored only 23 goals, Torres scored 24! (76,67% < 80%).
Now add the penalties kicked by both players in international competitions at all seasons: both have shot 13 penalties, and while Lampard scored only 8 goals, Torres scored 9! (61,54% < 69,23%).
And finally, add all the penalties kicked by both players in all seasons and in all competitions: Lampard kicked 43 penalties and scored 31 goals while Torres kicked the same number of penalties (43), but scored 2 more goals (33 goles)!
So, contrary to what the data indicated in the beginning, Torres would have been a better option than Lampard to shoot this crucial penalty.
In mathematical therms:
That way we find an example of the Simpson's paradox, also called Yule-Simpson effect, which appears when in a comparison between two variables we disaggregate the data by groups. It was first described by Edward H. Simpson, but similar effects were studied earlier by statiscians Karl Pearson and George Udni Yule.
In practice, this paradox is often used to present biased conclusions on some given statistical data, by disaggregating them arbitrarily to get partial results which match the theory that we want to prove.
And sometimes it has been applied in politics, in the process of setting electoral districts, through the interesting phenomenon known as 'Gerrymandering', that consists in manipulating geographic boundaries in order to achieve a desired electoral result.
At the end, just an advice: be very careful with the conclusions based on a statistical basis, and always look back the original data. You can meet with many surprises, such as this example.
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