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Category membership concerns whether an object or event is assigned to a specified category. It is distinct from its judged typicality, how confident a person is about the judgment and how quickly the judgment is made. A category can have a precise membership rule while its examples differ in familiarity.

For example, 3 and 97 are both odd integers: each has the form $2k+1$ for an integer $k$. A participant might nevertheless consider one a more typical example or identify it faster. Those responses concern the categorization process, rather than making one integer mathematically more odd than the other.

Sharon Lee Armstrong and collaborators compared typicality judgments and category verification for everyday categories and well-defined categories including odd and even numbers. Their experiments show why graded responses alone cannot establish that the underlying membership conditions are graded. Distinguishing membership, typicality and reaction time avoids drawing a stronger representational conclusion than the task supports.^[ref]

A developmental psychologist who investigated how children reason about nonobvious properties of living things. Experiments on internal properties and category membership tested whether expectations follow surface appearance or deeper assumptions about what an organism is. This work informs psychological essentialism without treating children’s beliefs as biological facts.^[ref]

Typicality is how representative an example seems of a category. A robin may be judged a more typical bird than a penguin, although both are birds. Such judgments can reflect familiarity, similarity to other members or the purposes for which a category is used; they are not the same measurement as a yes-or-no judgment of category membership.

Even a sharply defined mathematical category can produce graded psychological judgments. People can regard some odd integers as better examples than others, despite every odd integer satisfying the same membership rule. Sharon Lee Armstrong and colleagues used such cases to show that graded typicality does not, by itself, establish that a concept lacks a definition.^[ref]