Categorical Distribution

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This pie chart shows a distribution with different categories. Image credit: AdrianPeng2 via Wikimedia Commons

The term “categorical distribution” has come to mean two separate things:

  • Informally, it’s any distribution with categories;
  • Alternatively (and more precisely), it’s a generalization of the Bernoulli distribution for a categorical random variable.

While a random variable in a Bernoulli distribution has two possible outcomes, a categorical random variable has multiple possibilities.

The sample space for a Bernoulli distribution is {0, 1} and for a categorical distribution, it’s {0,1…n}. For example, a dice roll, where there are six outcomes {1,2,3,4,5,6} is a categorical distribution. When there is a single trial, the categorical distribution is equal to a multinomial distribution.

As this distribution only deals with discrete outcomes, it is sometimes called a discrete categorical distribution.

Examples of a categorical distribution

  • Throw a six-sided dice fifty times and observe the outcomes. The possible outcomes (the sample space) are 1,2,3,4,5,6. Each outcome has a probability of 1/6. The number of trials, “n” is 10.
  • Count the number of times a word appears in a book. The possible outcomes are any words in the text (e.g. the, and, to…). The probability depend on the number of words in the text.
  • Count the number of times a player on a hockey team scores a goal. The possible number of outcomes (i.e. goals) depend on the length of the game but will likely be somewhere between 0 and 10. Different probabilities will be assigned to the players, depending on their positions and their ability. For example, a forward has a higher chance of scoring a goal than a goalkeeper. And a forward with an excellent scoring record has a higher probability than a player with a poor record.


Image: AdrianPeng2, CC BY-SA 4.0, via Wikimedia Commons