An institution can have a higher outcome rate within each of two groups and still have a lower combined rate. That pattern is mathematically possible when the institutions have different proportions of students in the groups. Before treating an aggregate as a ranking, inspect the counts that determine its weights.

The Office for National Statistics discusses population composition in its explanation of Simpson's paradox. Its published illustration concerns earnings; the college example below is CollegeVerse's own invented rate calculation. It does not reproduce ONS data or establish a finding about any real university.

Build a complete fictional table

Imagine two colleges, A and B, each with 100 students in a precisely defined fictional cohort. Every student belongs to exactly one of two hypothetical entry groups, X and Y. The same completion definition and observation window apply to all four cells. There are no missing outcomes, exclusions or overlapping groups in this exercise.

College and entry group Completed Cohort count Completion rate
A, X 20 25 80%
A, Y 30 75 40%
B, X 60 80 75%
B, Y 5 20 25%

Within X, A's rate exceeds B's by five percentage points. Within Y, A's rate exceeds B's by fifteen points. Nevertheless, A has 50 completers out of 100, while B has 65 out of 100. The overall rates are 50% and 65%, putting B ahead in the unadjusted comparison.

There is no arithmetic inconsistency. A's actual cohort consists mostly of Y, the lower-rate group in this example. B's consists mostly of X. The aggregate rates answer what happened to each college's own mix, rather than what would result from giving both colleges the same mix.

Make the weighting visible

A's aggregate is 0.25 multiplied by 80%, plus 0.75 multiplied by 40%, giving 50%. B's is 0.80 multiplied by 75%, plus 0.20 multiplied by 25%, giving 65%. Averaging the two group percentages equally would not reproduce either actual aggregate because those are not the actual group shares.

Now perform a separate illustrative standardization: give X and Y equal weight at both colleges. A's standardized rate becomes 60%, the average of 80% and 40%. B's becomes 50%, the average of 75% and 25%. The comparison reverses under this explicitly chosen reference mix.

These standardized figures are calculated scenarios. They are not observed overall completion rates and must not replace the reported 50% and 65% without explanation. A different reference population would answer a different standardized question. Naming the weights is part of presenting the result, not a technical detail to hide in a footnote.

A reversal does not identify a cause

Nothing in the table establishes why the group rates differ, why students entered either group, or what would happen if a student changed college. There may be important differences within X and Y that this simple classification misses. Assigning identical group labels does not make students otherwise interchangeable.

Likewise, the calculation cannot separate teaching effects from prior preparation, course selection, resources or other factors. It demonstrates the sensitivity of an aggregate to composition. It does not establish that A provides better teaching, or predict a personal chance of graduating. Keep that boundary alongside the admission-rate interpretation guide, which addresses a related personal-probability mistake.

Audit a real comparison without reconstructing protected data

Request published subgroup definitions, denominators, outcome windows and coverage notes. Confirm that the groups are mutually exclusive and cover the intended population. If counts are suppressed, retain the limitation rather than attempting to recover protected observations. Rounded counts may also prevent an exact reconstruction.

Preserve the unadjusted total, any subgroup comparisons and any explicitly documented standardized result as separate outputs. Do not combine rates across inconsistent cohorts; the retention and graduation guide explains why the clock and population matter.

A useful conclusion is conditional: the aggregate comparison depends on these observed group shares, and a stated reference mix yields this different calculation. That tells a reader what the evidence supports without turning a composition effect into a causal college ranking.