Simpson's Paradox: Can Every Segment Improve While the Total Gets Worse?
Aggregate rates move with both segment performance and segment weights.
Precise claimAn aggregate rate is a weighted result: every segment can improve while the total falls when the population mix shifts toward a lower-rate segment.
Applies
A synthetic two-period, two-device completion-rate fixture with exact integer counts, fixed segment definitions and a controlled new-period traffic-share slider.
Does not prove
The page does not determine causality, prescribe which variables to condition on, claim segments always outrank aggregates, or generalize the synthetic rates to a real product.
Portable rule
If an aggregate rate changes, then inspect both segment rates and segment weights before attributing the movement to performance.
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aggregation-illusion puzzle
Simpson's Paradox: Can Every Segment Improve While the Total Gets Worse?
Judge an exact product-metric table before the totals are combined, then keep all segment rates fixed and change only the traffic mix to inspect what the aggregate can support.
Audit ledger · one product, two periods; successes and sessions are fully visible.
Segment
Old
New
Change
Desktop
480 / 60080%
340 / 40085%
+5 pp ↑
Mobile
160 / 40040%
270 / 60045%
+5 pp ↑
All sessions
Hidden until you commit — no rows or counts are missing.
Now combine the exact counts
Every row rose. The total fell.
This is a weighting reversal, not an arithmetic contradiction.
An aggregate rate is a weighted result: every segment can improve while the total falls when the population mix shifts toward a lower-rate segment.
This proves a composition reversal in this exact table. It does not decide which segmentation is causally appropriate for a real product.
The rebuilt model
You assumed:
If desktop and mobile completion rates both improve, the overall completion rate must improve too.
The actual model:
A total rate is not an unweighted vote among segment directions. It equals the sum of each segment rate multiplied by that segment's population weight, so performance and composition can move the total in opposing directions.
The variable that failed you:
Segment weight — desktop shrank from 60% to 40% of traffic while the lower-rate mobile segment grew from 40% to 60%.
Change one variable
Freeze every rate. Move only the mix.
Use the new rates — desktop 85%, mobile 45% — throughout. The old 64% total stays fixed. Start with 60% desktop traffic, then move to the observed 40%.
Desktop rate 85%Mobile rate 45%Old baseline 64%Changed new traffic mix
Reference mode opens the same arithmetic control without manufacturing a prediction.
Equation strip · rates fixed, mix adjustable
Desktop rate85%fixed
×
Desktop share60.0%adjustable
+
Mobile rate45%fixed
×
Mobile share40.0%complement
=
New aggregate69.0%live result
Weighted contribution +
0%47.5% crossover100%
Desktop 60.0%Mobile 40.0%
Use the range for continuous adjustment. Presets are exact shortcuts.
The original contingency-table analysis associated with Simpson's paradox demonstrates why relationships in component tables and combined tables require careful interpretation.
Association reversal can occur because conditional and aggregate rates use different weights; the appropriate causal interpretation depends on the question and data-generating structure.
A consequential historical case in which aggregate admissions data showed a misleading pattern and disaggregated departmental analysis changed the interpretation.
paper — checked 2026-07-21.
Scope: A synthetic two-period, two-device completion-rate fixture with exact integer counts, fixed segment definitions and a controlled new-period traffic-share slider.
Does not prove: The page does not determine causality, prescribe which variables to condition on, claim segments always outrank aggregates, or generalize the synthetic rates to a real product.