Introduction

Simpson’s Paradox is a statistical phenomenon that occurs when a result or trend in disaggregated data either reverses or disappears when the data are aggregated (Grigg, 2018). It can occur when a lurking variable is present in the data. A lurking variable is a hidden variable that changes the relationship between two variables. When data are aggregated, the lurking variable is not taken into account in the relationship. Splitting data into groups can change the relationships between the variables by taking the lurking variable into account.

A famous example of Simpson’s Paradox is admissions data from UC Berkeley’s graduate school programs. UC Berkeley was accused of having a bias against women in graduate admissions. The overall admissions data appeared to support such a claim, but when the data were disaggregated by individual graduate programs, the results were different. Some programs actually had higher rates of admission for women, and in others there was no significant bias. Introducing graduate department into the analysis changed the result because department was a lurking variable in the data. The following discussion will demonstrate Simpson’s Paradox using sample UC Berkeley data.

Overall Graduate Admissions

The sample dataset contains admissions data for six graduate departments at UC Berkeley and includes admission status and gender. The table below shows the overall numbers of male and female students admitted and rejected.

The graph below further shows that the number of males admitted is more than twice the number of females admitted.

In the next graph, the proportions of males and females admitted are compared. Not only are more males being admitted to UC Berkeley, the proportion of males admitted is higher than the proportion of females.

To check for Simpson’s Paradox, we should consider other variables that may be lurking and affecting the result. In the case of the UC Berkeley data, statistician Peter Bickel analyzed the graduate admission data by department, and found that disaggregating the data by departments changed the results (Grigg, 2018).

Graduate Admissions by Department

In our example data, we see what happens when the data are disaggregated and analyzed by department. The graph below shows the numbers of males and females admitted by the six graduate departments, A through F.

The next graph shows the proportion of male and female students admitted to each department.

The proportions of female students admitted to Departments A, B, and D are higher than the proportions of males admitted to those departments. Further, in departments A and B, very few females apply compared to males. While the admit rates in the other three departments, C, E, and F are higher for males, there are no large gaps in the rates compared to females. This sample data demonstrates Simpson’s Paradox in that the overall result of males having higher admission rates does not hold when the data are disaggregated by department. Department is a lurking variable that should be taken into account in the data.

Works Cited

Grigg, Tom. “Simpson’s Paradox and Interpreting Data.” Towards Data Science, 2018, https://towardsdatascience.com/simpsons-paradox-and-interpreting-data-6a0443516765.