Simpson’s paradox is one of the most puzzling phenomena in statistics. It occurs when a trend observed in several distinct groups completely reverses once those groups are combined. In other words, what appears true within each subgroup can become false when the data is analyzed as a whole.
This phenomenon has major implications in medicine, economics, marketing, and the social sciences. Understanding it is essential to avoid incorrect conclusions when analyzing heterogeneous data.
1. A Simple Medical Example#
Imagine a clinical trial involving 160 patients (80 women and 80 men). These patients receive either a drug or a placebo. Treatment effectiveness is measured by the recovery rate.
Results Among Women#
| Women | Recovered | Not Recovered | Total | Recovery Rate |
|---|---|---|---|---|
| Drug | 4 | 16 | 20 | 20% |
| Placebo | 18 | 42 | 60 | 30% |
👉 Among women, the placebo performs better (30% vs 20%).
Results Among Men#
| Men | Recovered | Not Recovered | Total | Recovery Rate |
|---|---|---|---|---|
| Drug | 36 | 24 | 60 | 60% |
| Placebo | 14 | 6 | 20 | 70% |
👉 Among men as well, the placebo is superior (70% vs 60%).
When analyzing the two groups separately, the conclusion seems clear:
the placebo works better.
Overall Results#
| Overall | Recovered | Not Recovered | Total | Recovery Rate |
|---|---|---|---|---|
| Drug | 40 | 40 | 80 | 50% |
| Placebo | 32 | 48 | 80 | 40% |
👉 But when the two groups are combined, the drug becomes more effective (50% vs 40%).
How can a treatment perform worse in every group, yet better overall?
This is exactly what Simpson’s paradox reveals.
2. Why Does the Conclusion Reverse?#
The reversal is caused by a confounding variable — here, the sex of the patients.
In our study:
The drug was mainly given to men (who naturally recover more often: 60–70%), while the placebo was mainly given to women (who naturally recover less often: 20–30%).
As a result, the drug benefits from a favorable distribution, which artificially improves its overall performance.
This composition bias perfectly illustrates the mechanism behind Simpson’s paradox:
A trend can reverse when the proportions of each group differ.
3. Real-World Situations Where the Paradox Appears#
Simpson’s paradox is not just a theoretical curiosity. It appears regularly in real-world data analyses.
Medicine#
A treatment may seem more or less effective depending on age, sex, or disease stage. Without proper stratification, conclusions can be misleading.
Marketing Analytics#
A lower overall conversion rate may hide better performance across individual channels (mobile, desktop, specific campaigns).
Human Resources#
A department may show strong overall performance while each individual team performs worse when analyzed separately.
The Berkeley Case (1973)#
The university appeared to discriminate against women in admissions. In reality, women applied more often to highly selective departments. Department by department, women were slightly favored.
4. How to Avoid This Analytical Trap#
To protect against incorrect conclusions, several best practices are essential:
1. Identify Confounding Variables#
Look for factors that influence both the cause and the outcome.
2. Analyze Data by Subgroups#
Global statistics are often just a summary — and sometimes a misleading one.
3. Use Appropriate Statistical Tools#
Multivariate regressions, adjusted models, or randomization help avoid the trap.
4. Check Group Comparability#
Always ask:
- Are the groups comparable?
- Could an unaccounted variable reverse the interpretation?
5. So Which Analysis Should Be Trusted?#
In our example, the answer depends on the question being asked.
To Measure the Intrinsic Effectiveness of a Treatment:#
➡️ Analyze each group separately.
Here, the placebo is more effective in every subgroup.
To Predict Outcomes in a Population with Similar Composition:#
➡️ Look at the aggregated data.
Here, the drug performs better overall.
In clinical trials, the goal is usually to measure the true effectiveness of a treatment. That makes it crucial to control or balance confounding variables.
Conclusion#
Simpson’s paradox reminds us that aggregated data can hide or even reverse real trends. Meaningful analysis requires understanding the context, identifying structural variables, and choosing the appropriate level of interpretation.
Key Takeaways#
- Global averages can be misleading.
- Data must be analyzed at the correct scale.
- Confounding variables must be identified and controlled.
- A conclusion only makes sense within its analytical context.
Before drawing conclusions from aggregated numbers, always ask:
“Is there a hidden variable that could explain this trend?”
This simple precaution can prevent many interpretation errors — even among experienced analysts.