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“B has the higher conditional probability,
but the marginal probabilities reverse.
The weights we discussed earlier are different.”
B was higher in both groups.
How can A be higher overall?
30-second demo · Statistics
“B can be higher on each device, yet A can be higher overall. This happens because the weights we discussed earlier are different.”
| Device | Page A | Page B |
|---|---|---|
| PC | 90% | 95% |
| Mobile | 10% | 20% |
| Overall | 82% | 35% |
B is higher on desktop and mobile, but the overall result is reversed.
“Of A’s 100 visitors, 90 are on desktop and 10 on mobile. B has 20 desktop and 80 mobile visitors. The overall rate is a weighted average using each device’s share of visitors.”
The lecturer mentioned “weights.”
I missed what those referred to.
“Why is it reversed overall?”
Earlier, the lecturer explained that A had many more desktop visitors, who purchase at a higher rate. A and B do not combine desktop and mobile visitors in the same proportions.
With equal weights, A is 50% and B is 57.5%. The difference was each page’s visitor mix.
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What did “weights” refer to just now?
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Why can groups and their total tell different stories?
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Simpson’s paradox · Conditional probability and weighted averages
“B is higher in both groups. Why is it reversed overall?”
The overall rate combines each group’s rate
in proportion to the size of that group.
First check what is being mixed, and in what amounts.
A: 90% × 0.9 + 10% × 0.1 = 82%
B: 95% × 0.2 + 20% × 0.8 = 35%
With a 50:50 device mix, A is 50% and B is 57.5%. The overall rate alone does not establish the effect of the page itself.
Concept reference: UC Berkeley statistics course
The pages and visitor counts are illustrative examples.
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