Combined Rate Trap Calculator
How can both groups improve while the total gets worse?
Because the mix between the groups changed. On the figures here, group A goes from 90% to 95% and group B from 20% to 25% — both up — while the combined rate falls from 55% to 32%. Group A was half of the total and is now a tenth, so the weaker group now dominates the average. This is Simpson's paradox.
Every department improved. The company total got worse. Nobody is lying.
The mix moved, and the mix is what the combined number is really measuring.
Built by Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology, who also writes the paid version of this tool. When a headline rate and every underlying rate disagree, the mix is almost always the answer.
Two groups, before and after
Put in how many succeeded and out of how many, for each group in each period.
How a total can contradict every part of it
A combined rate is a weighted average, and the weights are the group sizes. If the sizes stay put, the combined rate follows the group rates. If the sizes move, it follows the sizes.
On the figures this page opens with, group A succeeds at 90% and group B at 20%. Both improve by five points. But group A shrinks from half the total to a tenth of it, so the combined rate is now mostly measuring group B - and group B, even improved, is far below where the combined rate used to sit.
The result is a headline that fell twenty-three points while every underlying number rose.
Where this actually happens
A hiring process where the easy roles were filled and the remaining vacancies are the hard ones. A conversion rate after a campaign brought in a lot of colder traffic. A recovery rate after a clinic took on more difficult cases. A pass rate after entry requirements were widened.
In every one of those, the people doing the work improved and the number they are judged on got worse. If you have ever had to explain a falling metric while insisting the team is doing better, this is very often the arithmetic behind it.
It runs the other way too: a rate can improve because the mix shifted toward the easy cases, and somebody will take the credit.
What to report instead
Give the group rates. A combined figure that reverses its own components is not a summary of them, and presenting it as one is how a true statement becomes a misleading report.
If a single number is unavoidable, say what the mix did in the same breath. The mix shift is not a caveat; on these figures it is the entire explanation.
What this will not tell you
This shows whether a reversal is present and how far the mix moved. It cannot tell you why the mix moved, whether the change was good, or which figure you ought to report - those depend on what the groups are and what the number is for.
Questions people actually ask
What is Simpson's paradox?
A reversal in which every subgroup moves one way while the combined figure moves the other. It happens when the relative sizes of the subgroups change, because a combined rate is a weighted average and the weights are the group sizes.
How can both groups improve but the total get worse?
Because the group with the lower rate became a larger share of the total. The combined figure is then mostly measuring that group, so it falls even though both groups individually improved.
Is Simpson's paradox a statistical error?
No. Every number in it is correct. The error is in treating a combined rate as a summary of its parts when the mix has shifted.
What should I report when this happens?
The group rates, and what the mix did. A single combined figure that contradicts all of its components misleads even when it is arithmetically true.
Cite this tool
Free to use and free to quote. If a figure here is useful in something you are writing, a link back is all that is asked.
Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology. Combined Rate Trap Calculator (Simpson's Paradox). Shahzeb Speaks. https://www.shahzebspeaks.com/combined-rate-trap-calculator/
The other free tools
Each one answers a single question and refuses the ones it cannot. All free, none asks for an email address.
Need to strip out what was never comparable?
The paid Combined Rate Trap Calculator sits with a like-for-like tool that takes out one-offs, trading days and price rises one at a time, so a comparison can be argued about a step at a time rather than accepted or rejected whole.
See the full tool