Discrete vs continuous variables explained

By Jude Wallis · Published July 28, 2026

A discrete variable takes countable, separate values (like the number of pets), while a continuous variable can take any value in an interval (like height), so it is measured rather than counted.

AP Statistics: Unit 1 (topics 1.2 Variables, 2.10 The Binomial Distribution, 2.11 The Normal Distribution). In the Fall 2026 AP Statistics course, discrete and continuous are defined for quantitative variables in Unit 1 topic 1.2, and the same split separates the discrete binomial model in topic 2.10 from the continuous normal model in topic 2.11.

What a discrete variable is

A discrete variable is a quantitative variable that can take on only a countable number of values. Countable means you could list the possible values one by one, even if the list never ends, as with the whole numbers $0, 1, 2, 3, \ldots$ . The number of text messages you send in a day and the number of students in a class are discrete, because the results land on separate, isolated values with gaps between them. You count a discrete variable rather than measure it.

What a continuous variable is

A continuous variable is a quantitative variable that can take on any value within an interval. Between any two possible values there is always another possible value, so the results form an unbroken scale with no gaps. Height, weight, temperature, and elapsed time are continuous, because a measurement can in principle be refined to more and more decimal places. You measure a continuous variable rather than count it.

Discrete vs continuous at a glance

Here is the split on the features that decide which type you have.

Feature	Discrete	Continuous
Possible values	Countable, with gaps	Any value in an interval
How you get it	Counting	Measuring
Example	Number of pets	Body weight
Probability attaches to	Individual values	Intervals of values
Distribution graph	Probability histogram	Density curve
AP model distribution	Binomial	Normal

How to tell which one you have

Ask whether you could, in principle, list every possible outcome. If the answer is yes, even for an unending list like $0, 1, 2, \ldots$ , the variable is discrete. If instead the outcomes fill an interval so that no two are truly next to each other, the variable is continuous. A quick tell is the verb: counted quantities are discrete, and measured quantities are continuous.

The classic mix-up and how to avoid it

The tricky cases are quantities that are truly continuous but get recorded to a fixed precision. Age is continuous, yet a form that asks for whole years makes the recorded data look discrete; money is measured to the cent, but the underlying amount is continuous. Judge the variable by the quantity it represents, not by how precisely your instrument happens to report it. When in doubt, ask whether a value halfway between two readings is meaningful: half a person is not, so headcount is discrete, but 70.5 kilograms is, so weight is continuous.

A discrete distribution: heads in three coin flips

Flip a fair coin 3 times and let $X$ be the number of heads. The possible values are the countable set $\{0, 1, 2, 3\}$ , with probabilities $P(0) = \frac{1}{8}$ , $P(1) = \frac{3}{8}$ , $P(2) = \frac{3}{8}$ , and $P(3) = \frac{1}{8}$ . Confirm the probabilities sum to 1, then find the expected value $E(X)$ .

Check the total probability, which a discrete distribution requires to equal 1: $\frac{1}{8} + \frac{3}{8} + \frac{3}{8} + \frac{1}{8} = \frac{8}{8} = 1$ .
Set up the expected value as the sum of each value times its probability: $E(X) = \sum x P(x)$ .
Multiply term by term: $0 \times \frac{1}{8} = 0$ ; $1 \times \frac{3}{8} = \frac{3}{8}$ ; $2 \times \frac{3}{8} = \frac{6}{8}$ ; $3 \times \frac{1}{8} = \frac{3}{8}$ .
Add the pieces: $0 + \frac{3}{8} + \frac{6}{8} + \frac{3}{8} = \frac{12}{8} = 1.5$ .
Notice what made this work: probability sat on individual values and summed to 1. For a continuous variable the probability of any single exact value is 0, and you would work with an interval instead.

The probabilities sum to 1 and $E(X) = 1.5$ heads; because $X$ is discrete, probability attaches to each separate value, unlike a continuous variable where it attaches to intervals.

Frequently asked questions

Is money discrete or continuous?

Money is quantized to the nearest cent, so strictly it is discrete, and many texts classify it that way. It is often modeled as continuous only as a convention, because the cent is so small next to typical amounts that a continuous curve fits the values well. If a question asks, say which model you are using and why rather than assuming one answer, while a count of coins stays clearly discrete.

Can a variable be neither discrete nor continuous?

Those two types cover quantitative variables, which are numerical. A variable that records a category, such as eye color or brand, is categorical rather than quantitative, so the discrete-versus-continuous split does not apply; see qualitative vs quantitative data.