Simpson’s Paradox
What is Simpson’s Paradox?
Sometimes numbers can play tricks on us. They might tell us a story that seems to make sense, but when we look closer, we find out that story isn’t quite right. This is what Simpson’s Paradox is all about. It shows up when we have a bunch of information that we squeeze together into a summary and then, the summary doesn’t quite match what’s going on in all the little parts.
Let’s say there’s a lemonade stand that is open every day of one week. If we look at the total number of lemonades sold over the whole week, it might look like they sold more lemonade on sunny days. But, if we look at it day by day, we’d find out they actually sold more on cloudy days. The total isn’t wrong; it’s just not showing us the full picture. When we squish all those individual days together, we lose some key details about each day. That’s a sort of magic trick that numbers can do without us even realizing it.
Definitions
Simple Definition 1
Imagine you have a puzzle with lots of tiny pieces. When you look at each piece by itself, you can see small details like a bit of sky or part of a tree. But if you only look at the big picture on the box without focusing on the pieces, you might miss those details. Simpson’s Paradox is like that. It happens when the big picture of data, which is made up of lots of tiny pieces, seems to tell a different story from what’s actually happening in those tiny pieces.
Simple Definition 2
Think of it like a movie with a plot twist. The whole movie might lead you to think one character is the hero. But at the end, you find out it was really someone else. With data, Simpson’s Paradox is like a plot twist where the summary tells you one thing, but when you look at each part of the data, like each scene of the movie, you discover the truth is different.
Examples
College Admissions
A university has two colleges, the College of Technology and the College of Humanities. Looking at the overall admission data, it looks like the university favors men over women. However, within each college, they actually admit a higher rate of women than men. This is Simpson’s Paradox because combining the data from both colleges together hides the fact that each college actually admits women at a higher rate.
Player Batting Averages
Two baseball players, Mia and Joey, play during two seasons. Mia has a higher batting average than Joey in both seasons when looked at individually. But when we combine the data from both seasons, Joey ends up with a higher overall average. This happens because the second season had a lot more games where Joey improved a lot while Mia did not have that many opportunities, twisting the overall picture through Simpson’s Paradox.
Kidney Stone Treatments
A study compares two treatments for kidney stones. Treatment A appears to be more effective when looking at all patients together. But if you look closer, breaking down the data by the size of the kidney stones, Treatment B is actually more effective for both small and large stones. This is an example of Simpson’s Paradox because taking all the data together hides the fact that Treatment B works better regardless of the stone size.
Related Topics

Confounding Variables: A confounding variable is an outside influence that changes the effect of a dependent and independent variable. This can often play a role in creating Simpson’s Paradox, as it might be the hidden factor that changes how the groups are compared.

Data Stratification: This is a method used to separate data into different layers or strata to highlight differences that might not be apparent in the aggregated data. It’s a technique that can be used to overcome Simpson’s Paradox.

Causal Inference: Causal inference is about determining what causes what. It’s a complex process that is often made more difficult by paradoxes like Simpson’s. To make good causal inferences, carefully analyzing data without assuming that correlation means causation is essential.
Why is it Important?
Understanding Simpson’s Paradox is crucial because it teaches us to think critically about the information we’re given. Not everything is as it seems on the surface, especially with numbers. For the average person, knowing about this paradox can be super helpful when making decisions based on statistics. It could be something as simple as choosing which school to go to, based on graduation rates, or as serious as deciding between different health treatments when looking at success rates.
Data is everywhere in our lives, from school grades to sports stats to election results, and mistakes in interpreting data can lead to wrong conclusions. That can affect what job you think is the best, which neighborhood you believe is the safest, or what diet you feel is the healthiest based on what the numbers say. So it’s not just important—it’s part of being a smart thinker in today’s world.
Conclusion
Numbers are powerful, and Simpson’s Paradox shows how they can sometimes be tricky. It serves as a reminder to all of us that we should never stop asking questions about the world around us, especially when it comes to making decisions based on statistics. Looking beyond the surface, understanding the context, and analyzing the details is key. This paradox isn’t about doubt or confusion but a lesson in being thorough and careful, making it a valuable piece of wisdom in our datadriven world.