Simpson's Paradox
What is Simpson's Paradox?
Simpson's paradox is when a trend that appears in separate subgroups of data reverses or disappears once the groups are combined.
A relationship can point one way within every subgroup yet the opposite way in the aggregate, usually because a lurking variable is unevenly distributed across groups. A famous economics case: median U.S. wages rose overall even as wages fell within every education group, because the mix shifted toward more-educated workers. It is a powerful warning against trusting aggregate correlations.
Simpson's Paradox: a worked example
Harlow Analytics runs a Research division and a Field division. In its first year, 20 research staff averaged $80,000 and 80 field staff averaged $40,000, so total pay was $4,800,000 across 100 people, an average of $48,000. The next year the firm cut pay in both divisions, research to $78,000 and field to $38,000, but shifted its mix to 60 research staff and 40 field staff. Total pay is now 60 x $78,000 + 40 x $38,000 = $4,680,000 + $1,520,000 = $6,200,000, an average of $62,000. Every division paid less, yet the firm-wide average wage rose $14,000. Hold the mix at the original 20/80 and the second-year average is 0.2 x $78,000 + 0.8 x $38,000 = $46,000, which is the true $2,000 pay cut showing through.
The mistake students make with simpson's paradox
Students usually assume one of the two figures must be an error, and that the aggregate is the honest one because it uses all the data. Both are arithmetically right. The reversal comes from the weights on each division, not from a mistake, and it does not wash out with a bigger sample. The follow-up error is deciding the subgroup view always wins. It wins when the grouping variable is a confounder you want held constant; if the change in group mix is itself the effect you care about, the aggregate answers the question.
Simpson's Paradox questions
Why does Simpson's paradox happen?
Simpson's paradox happens because subgroups carry unequal weights in the combined total, and those weights shift at the same time as the values being measured. When a high-value group grows and a low-value group shrinks, the combined average is pulled upward even if every group's own average fell. Group membership is the lurking variable: it is correlated with the outcome and its share changes across the periods being compared, so collapsing the groups hides it.
Is Simpson's paradox a real paradox?
Simpson's paradox is not a logical contradiction, only a surprising result. Both the subgroup figures and the combined figure are computed correctly from the same data; the aggregate is a weighted average whose weights changed. It feels like a contradiction because people read an average as a property of the typical member, rather than as a number that depends on how many members sit in each group.
How do you avoid Simpson's paradox in economic data?
Avoiding Simpson's paradox means reporting subgroup figures next to any aggregate and checking whether group shares moved over the same period. Economists also build fixed-weight or composition-adjusted measures, which hold each group's share constant so the headline number reflects only within-group change. If shares are stable, the aggregate and the subgroups move together; if shares moved a lot, expect them to diverge and state which question you are answering.
Related terms
Get AP Econ exam tips in your inbox
Occasional emails with study tips, new interactive graphs, and exam-season reminders. Free, no spam.
No spam. Unsubscribe anytime. Read our privacy policy.
Keep track of what you have studied
A free EconLearn account adds progress tracking, your quiz history, and achievements. Studying here is free either way, and there is nothing to pay for as a student.
Create a free accountAlready have one? Sign in
Last updated