A prospectus might report a strong employment percentage and a strong satisfaction percentage. A reader may want to know how many students experienced both. Two separate percentages do not normally answer that question. The overlap depends on how the outcomes occur together within the same population.

OpenStax's probability formula review states the addition rule connecting two events, their union and their intersection. The calculations here are an independent CollegeVerse illustration using invented student counts. They identify what missing joint information permits, rather than claiming a particular university has produced misleading statistics.

Establish one shared population first

Imagine exactly 100 fictional students with complete observations on two outcomes. Seventy are employed and eighty report satisfaction. The definitions and observation period are stipulated to be identical for every student. We know each total, but not which students belong to both groups.

At most seventy can meet both outcomes, because the overlap cannot exceed the smaller group. At least fifty must meet both: seventy plus eighty is 150 memberships, and only 100 people can be in the union. The extra fifty memberships must be shared.

Feasible arrangement Both Employed only Satisfied only Neither
Smallest possible overlap 50 20 30 0
Largest possible overlap 70 0 10 20

Both rows reproduce seventy employed, eighty satisfied and 100 students in total. Yet the joint outcome ranges from 50% to 70%. The published marginal totals alone do not identify which row, or which intermediate arrangement, describes the group.

For counts a and b out of the same n, the lower bound is the greater of zero and a plus b minus n. The upper bound is the smaller of a and b. These are logical limits, not confidence intervals and not forecasts about a future student.

Do not multiply without a justified assumption

Multiplying 70% by 80% gives 56%. That would be the overlap under independence, but independence has not been established. Employment and satisfaction might be associated, positively or negatively. The existence of a neat multiplication result does not make that assumption true.

An overlap of 56 students is feasible: it leaves fourteen employed only, twenty-four satisfied only and six in neither group. Feasibility is weaker than evidence. Many other tables are equally compatible with the supplied totals, so 56 must not be reported as an observed result.

If the producer publishes a cross-tabulation, use it with its own definitions and disclosure rules. If joint data are suppressed or unavailable, report the bounds or state that the exact overlap cannot be determined. Do not reverse-engineer protected counts from rounded percentages and present them as recovered facts.

Stop when the denominators do not match

Real publications may measure employment among responding graduates and satisfaction among current students. Those are different populations, so the 100-person worksheet cannot simply be applied. Even two graduate surveys may use different cohorts, response windows or exclusions.

Record the base, reference period and missing-data treatment for each measure before any overlap calculation. Rounded percentages can also widen the possible range. If weighting is used, distinguish the weighted population estimate from the actual number of respondents rather than turning a weighted total into a headcount.

The overlapping category guide explains why categories need not sum to a partition. This worksheet addresses a different question: what can be inferred when the joint table is absent. A useful shortlist note separates the two marginal results, the available joint evidence and any explicit assumption. That lets two attractive headlines remain informative without silently converting them into a stronger combined outcome claim.