A **categorical distribution** is a discrete probability distribution that describes the probability that a random variable will take on a value that belongs to one of *K* categories, where each category has a probability associated with it.

For a distribution to be classified as a categorical distribution, it must meet the following criteria:

- The categories are discrete.
- There are two or more potential categories.
- The probability that the random variable takes on a value in each category must be between 0 and 1.
- The sum of the probabilities for all categories must sum to 1.

The most obvious example of a categorical distribution is the distribution of outcomes associated with rolling a dice. There are *K* = 6 potential outcomes and the probability for each outcome is 1/6:

This distribution satisfies all of the criteria to be classified as a categorical distribution:

- The categories are discrete (e.g. the random variable can only take on discrete values – 1, 2, 3, 4, 5, 6)
- There are two or more potential categories.
- The probability of each category is between 0 and 1.
- The sum of the probabilities add up to 1: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1.

Rule of Thumb:

If you can

countthe number of outcomes, then you are working with a discrete random variable – e.g. counting the number of times a coin lands on heads.

But if you can

measurethe outcome, you are working with a continuous random variable – e.g. measuring height, weight, time, etc.

**Other Examples of Categorical Distributions**

There are plenty of categorical distributions in the real world, including:

**Example 1: Flipping a Coin.**

When we flip a coin there are 2 potential discrete outcomes, the probability of each outcome is between 0 and 1, and the sum of the probabilities is equal to 1:

**Example 2: Selecting Marbles from an Urn.**

Suppose an urn contains 5 red marbles, 3 green marbles, and 2 purple marbles. If we randomly select one marble from the urn, there are 3 potential discrete outcomes, the probability of each outcome is between 0 and 1, and the sum of the probabilities is equal to 1:

**Example 3: Selecting a Card from a Deck.**

If we randomly select a card from a standard 52-card deck, there are 13 potential discrete outcomes, the probability of each outcome is between 0 and 1, and the sum of the probabilities is equal to 1:

**Relation to Other Distributions**

For a distribution to be classified as a **categorical distribution**, it must have *K* ≥ 2 potential outcomes and *n* = 1 trial.

Using this terminology, a categorical distribution is similar to the following distributions:

**Bernoulli distribution:** *K* = 2 outcomes, *n* = 1 trial

**Binomial distribution:** *K* = 2 outcomes, n ≥ 1 trial

**Multinomial distribution:** *K* ≥ 2 outcomes, n ≥ trial

**Additional Resources**

What Are Random Variables?

An Introduction to the Binomial Distribution

An Introduction to the Multinomial Distribution