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Reece Yang '21
Reece Yang '21
Imagine that it’s the end of the school year. You check your grades and see that your final average GPA is lower than both of your semester average GPAs! Is this an error? Maybe not. You may have fallen victim to Simpson’s Paradox.
Suppose you took two year-long courses and two semester-long courses. Consider the following hypothetical transcript:
Table 1: A hypothetical transcript with two year-long courses and two semester-long courses.
Note that your final grades for your year-long courses are the averages of their semester grades. So when all the final grades are averaged together, your final average GPA is less than the semester GPA averages. This is the essence behind Simpson’s paradox: trends that exist in separate groups of data disappear or reverse when you combine the groups together. The same is true for the other direction: a trend in a dataset disappears when you split the data into groups. In this case, by combining the two semester grades for your year-long courses, the trend of your final average GPA reverses and becomes less than your semester GPA averages.
Simpson’s paradox often appears in statistics when splitting data into groups is involved. Unfortunately, this becomes especially problematic when statistics are used to defend an argument.
For example, in a recent New York Times opinion article, psychiatrist Richard A. Friedman argues, “self-reflection can be intrinsically aversive and that we have a near hysterical dread of boredom.” Friedman cites a study where researchers asked participants to spend 15 minutes in a room and instructed them to “entertain themselves with their own thoughts.” Participants could also choose to administer a small electric shock to themselves. The researchers found that 67 percent of men and 25 percent of women shocked themselves instead of “being alone with their own thoughts” for the duration of the experiment.
Sounds convincing, right? The conclusion of the experiment isn’t so simple, however. The study involved 18 men and 24 women; twelve of the men chose to shock themselves, and six of the women shocked themselves. The following table summarizes the results. Less than half of all participants chose to shock themselves. Although 43% is still a significant amount, the result of the study is definitely not as definitive as Friedman makes it appear.
Table 2: Results of the boredom experiment.
While boredom may not seem like an important problem, situations involving Simpson’s paradox could very likely appear in politics—in data concerning COVID-19 infection rates in children, for example.
The practice of gerrymandering is another example of a phenomenon similar to Simpson’s paradox. Gerrymandering creates an unfair advantage for a political party by drawing electoral district boundaries in a way that favors certain voters. Just like in Simpson’s paradox, splitting an electorate into groups can reverse the outcome of an election, as demonstrated in the figure shown. In this case, the impacts of gerrymandering can have major implications for who has control over the government and how certain groups of people are treated.
The ease at which statistical fallacies may occur show why math education and thinking carefully about the statistics that you encounter are so important. Blindly following the numbers you see might cause you to believe a misleading conclusion.
Image Attribution: M.boli / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0)
Bibliography
Carlson, Bruce W. “Simpson's Paradox.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 7 Jan. 2019, www.britannica.com/topic/Simpsons-paradox.
Friedman, Richard A. “Is the Lockdown Making You Depressed, or Are You Just Bored?” The New York Times, The New York Times, 21 Aug. 2020, www.nytimes.com/2020/08/21/opinion/sunday/covid-depression-boredom.html.
Samarrai, Fariss. “Doing Something Is Better Than Doing Nothing for Most People, Study Shows.” UVA Today, 11 July 2014, news.virginia.edu/content/doing-something-better-doing-nothing-most-people-study-shows.
Wilson, Timothy D, et al. “Social Psychology. Just Think: the Challenges of the Disengaged Mind.” Science (New York, N.Y.), U.S. National Library of Medicine, 4 July 2014, www.ncbi.nlm.nih.gov/pmc/articles/PMC4330241/.
Duignan, Brian. Gerrymandering. 11 Oct. 2019, www.britannica.com/topic/gerrymandering.
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