# Discrete Data – Explanation and Examples

Discrete data is data that only occurs at certain intervals.

Continuous data is the opposite of discrete data since it includes all possible numbers. While the interval for discrete data is often $1$, it can be any real number.

All areas of math and science include discrete data. It is important for young children just learning math and for doctoral candidates studying graph theory. In fact, there is an entire branch of mathematics called discrete data focuses on the study of non-continuous structures.

This section covers:

• What is Discrete Data?
• Discrete Data Definition
• Discrete Data Examples

## What is Discrete Data?

Discrete data is data that only occurs in certain intervals. Essentially it is data that is not continuous.

Think of the integers. The set of integers are discrete because they do not include, for example, the numbers between $0$ and $1$ or between $1$ and $2$.

Sometimes data is discrete because the instruments used to measure it have limited accuracy. For example, a thermometer may only measure to the nearest one-tenth of a degree, so temperatures would only be recorded to the nearest tenth.

Other times, data is discrete because the variable itself is non-continuous. For example, American currency is measured in increments of $0.01$. It would not make sense to say that someone has $\frac{3}{8}$ of a cent or $\sqrt{2}$ cents.

### Discrete Data Definition

Discrete data is data that only occurs at certain points on certain intervals. Specifically, discrete data is countable and non-continuous.

Note that, more often than not, discrete data is defined as non-continuous data.

### Floor and Ceiling Functions

Floor and ceiling functions are non-continuous functions that take a set value for a range of numbers. Ceiling function round up while floor functions round down.

For example, consider a function that takes a student’s age and spits out the student’s grade. Suppose a state says, for example, that all students who turn $5$ between September 1 of the previous year and August 31 of this year can enter kindergarten this year. The age of the students is continuous because a student could, theoretically, be $5.6293$ years old at a given moment. However, the grade that a student is entering is discrete. This function would take all students who are from $5.0$ to $5.9999$ years old on August 31 and map them to $0$ (for grade 0, kindergarten).

In this case, then, a floor or ceiling function maps a continuous variable onto a discrete variable.

## Discrete Data Examples

Suppose Amy goes to the store and buys boxes of cereal. How many boxes can she buy?

$1$ box is fine. $0$ boxes is fine. $5$ boxes or even $50$ boxes makes sense (although that’s a lot of cereal!).

It does not, however, make sense to say Amy bought $9 \frac{3}{4}$ boxes of cereal.

Why?

Cereal is sold by the box. Therefore, one cannot buy fractions or decimals of a box. And certainly no one buys an irrational number of boxes!

No. Amy can only buy a whole number of boxes. The number of boxes she buys is therefore discrete with intervals of $1$.

As mentioned before, although $1$ is a common interval for discrete data, it is not the only one.

Consider, for example, United States shoe sizes. Shoe sizes exist in increments of $\frac{1}{2}$. This is why it makes sense for someone to have a shoe size of $6 \frac{1}{2}$ but not a shoe size of $6 \frac{7}{8}$.

Certainly someone could have a foot that is larger than a $6 \frac{1}{2}$ and smaller than a $7$, but such a person would probably just have to go a size up.

## Common Examples

This section goes over common examples of problems involving discrete data and their step-by-step solutions.

### Example 1

A school is planning a field trip. They want to take students on buses that each carry $54$ people.