Venn Diagram Calculator

A Venn diagram represents sets as overlapping shapes, and the regions between them hold the items that belong to one set, to two, or to all three. With three sets the diagram has seven disjoint regions, and every figure people quote about the diagram can be recovered from them.

The practical difficulty is that the seven regions are hidden while the set sizes and pairwise overlaps are the numbers actually measured. Solving in that direction is the job of inclusion–exclusion, and getting it wrong is the usual source of impossible overlap figures.

Start from the seven regions, which partition the union. Each set size is then the sum of the four regions touching it: the A-only region, the two pairwise regions involving A, and the triple intersection.

The union is simply the sum of all seven regions. This is why a union is never as large as the sum of the set sizes — that sum counts every doubly or triply shared item more than once.

Going the other way, subtract the triple intersection from each pairwise intersection to isolate the pairwise-only regions, then subtract those from each set size to isolate the only-regions. Each step can produce a negative number, which is the signal that the inputs cannot all be true.

For comparing pairs, the Jaccard index divides the shared part by the union of the pair, while the overlap coefficient divides it by the smaller set. The first answers "how much of the pair do these have in common", the second "how much of the smaller set is contained here".

Set sizes overlap each other while regions never do. A is 10 and B is 8 in the worked example, yet only 19 distinct items are involved, because 5 counts appear in both sizes.

An empty or impossible combination is rejected rather than corrected. If the pairwise overlaps with A exceed the size of A, no diagram exists, and silently clamping the region to zero would hide the mistake in the source data.

The largest set is not necessarily the most overlapping. Comparing Jaccard indices across the three pairs shows which relationship is strongest, which is usually the point of the exercise.

Percentages are shares of the union unless stated otherwise. A set covering 52.6% of the union holds 10 of its 19 items, not 10 of the population.

• Whether the intersections are quoted inclusively. Mixing an inclusive |A ∩ B| with an exclusive A∩B-only region is the most common source of figures that cannot be reconciled.
• The size of the overlap relative to the smaller set. The overlap coefficient is insensitive to a large set and highly sensitive to a small one.
• The total population. It changes nothing about the diagram itself but determines how many items sit outside all three sets.
• Rounding in survey data. Percentages converted to counts can produce overlaps that are impossible by a fraction, which is why the check is worth reading before trusting a result.
• The number of sets. Three is the practical limit for this kind of arithmetic; beyond it the regions multiply and the figures stop determining a unique answer.

• Checking whether a set of survey or database figures describes a possible Venn diagram at all.
• Finding how many distinct items three overlapping groups cover, once the sizes and pairwise overlaps are known.
• Comparing the strength of relationships between pairs of groups with Jaccard and overlap coefficients.
• Reporting membership shares: how much of a population each group covers, and how many belong to none of them.
• Teaching inclusion–exclusion, where seeing the seven regions and the six figures line up is the whole point.

• Items are counted once per set, so multi-valued attributes such as "buys both brands" are fine but duplicated records are not.
• Sets must be finite counts. Percentages alone are ambiguous unless the sample size is known.
• The three sets are treated symmetrically. There is no notion of nesting beyond what the counts themselves imply.
• Only the six standard figures and the triple intersection can be given as inputs; a full four-set diagram needs fifteen regions.
• Nothing here measures statistical significance. Two regions of size 1 in a survey of 20 mean something quite different from two in a survey of 20,000.