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# Continuous Variable in Statistics

Joao Amadeu, Elizabeth Weaver, Kathryn Boddie
• Author

Joao Amadeu has more than 10 years of experience in teaching physics and mathematics at different educational levels. Joao earned two degrees at Londrina State University: B.S. in Physics and M.S. in Science and Mathematics Education. He is currently working on his PhD in Science Education at Western Michigan University.

• Instructor
Elizabeth Weaver

Elizabeth has taught college Mathematics and has a master's degree in Mathematics.

• Expert Contributor
Kathryn Boddie

Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.

What is a continuous variable in statistics? Learn what constitutes a continuous variable by looking at the definition, seeing examples, and comparing it to discrete and quantitative variables. Updated: 09/24/2021

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## Identifying Continuous Variables: Additional Practice

In the following problems, students will identify if a variable is continuous, or not, and explain the reasoning behind the identification.

## Problems

1. Out of the following, which one is NOT a continuous variable. Why not?

• Length of hair
• Number of children
• Weight of a box
• Height of a tree

2. Out of the following, which is a continuous variable? Why?

• Number of pets
• Color of shirt
• Temperature
• Outcomes from flipping a coin

3. A student was giving a presentation on continuous variables in statistics class and received a failing grade because they described the number of people in a household as a continuous variable, using the statistic that the mean household size is 4.2 people. Explain why the number of people in a household is not continuous, even though the mean contains a decimal.

### Solutions

1. The number of children is not a continuous variable. It is not continuous because we cannot have a fraction of a child - only whole numbers.

2. Temperature is a continuous variable because it can take on any value in an interval of values - we can have decimals in a temperature measurement.

3. The number of people in a household is not continuous because you cannot have a fraction of a person - only whole people. This does not mean that the mean household size cannot have a fraction or decimal, just that each individual household can only have a whole number of people. For example, the mean household size for a sample of households of size 3, 4, 2, and 5 is (3 + 4 + 2 + 5) / 4 = 14 / 4 = 3.5, even though we cannot have a fraction of a person.

#### How do you know if a variable is continuous?

At a first glance, any variable that can be measured in decimals or fractions can be considered continuous. On the other hand, variables that can only be presented as whole numbers are called discrete.

#### What are two examples of continuous variables?

Two examples of continuous variables are:

1. Body weight (e.g., 34.879 Kg)

2. Height (e.g.: 1.7589 m)

In both examples the value could present an unlimited number of digits after the decimal point.

#### What are continuous variables in math?

In mathematics, a continuous variable can assume a value within a range that includes infinite possible values. The are commonly presented as decimals or fractions.

#### What is the difference between continuous and discrete variables?

Continuous variable can assume an infinite number of values within a range. They are usually represented by decimals or fractions. Discrete variables are whole number values that represent units that cannot be presented as decimals or fractions.

#### What is considered a continuous variable?

A continuous variable is any value that can be measured as decimals or fractions. These are infinitely situated between two values of reference. For example, there are infinite values between 1 and 2.

Continuous variables can be described as numbers that may assume one of infinite values between any two values of reference. For example, using the values 1 and 2 as reference, there is an infinite number of decimals between them. Using decimals, one may try to list all values between 1 and 2, such as:

1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2

The interval between the values above is 0.1. However, there is no limit to the numbers that can appear after a decimal point. Thus, even between 1 and 1.1, there is an infinite possibility of values. In the interval between 1 and 1.1

1, 1.0000000000000 ~... ~1, ~...1.1

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## Definition of a Continuous Variable

My niece once did a research project on popcorn and measured the time it took for bags of popcorn to stop popping. If you've ever made popcorn, you probably remember the time given on the package is just an estimate of the time it should take, but to really gauge when the popcorn is done, you listen for pops to slow to only one every couple of seconds. Each bag of popcorn is different, even if they're made by the same company.

By measuring the exact time it took for pops to slow down to the point of having two seconds in between each pop, my niece was able to collect several values of x, where x was the continuous variable that measured the time it took for a bag of popcorn to pop.

A continuous variable is a specific kind a quantitative variable used in statistics to describe data that is measurable in some way. If your data deals with measuring a height, weight, or time, then you have a continuous variable.

Let's further define a couple of the terms used in our definition. A variable in statistics is not quite the same as a variable in algebra. In statistics, a variable is something that gives us data. Some examples of variables in statistics might include age, eye color, height, number of siblings, gender, or number of pets. Our definition of a continuous variable also mentions that it's quantitative. Quantitative data involves quantities or numbers. In the examples of variables listed earlier, your age, height, number of siblings, and number of pets are all quantitative variables.

The definition also mentions that the data is measurable in some way. To understand this, you need to understand discrete variables. Data is considered to be discrete if the data is a count. When we count things, we use whole numbers like 0, 1, 2, and 3. My favorite example of a discrete variable is how many eggs a chicken lays. Each day a hen may or may not lay an egg, but there are two things that can never happen. There can never be a negative number of eggs, and there can never be a fraction or a portion of an egg.

Continuous variables are variables that measure something. My favorite example of a continuous variable is how many gallons of milk a cow gives. To the best of my knowledge, cows don't know how to stop producing or giving milk after exactly 4 gallons! Bessie may give 4.17 gallons on Monday, 3.89 gallons on Tuesday, and 4.2 gallons on Wednesday. Notice continuous variables allow us to have decimals, or fractions. An error occurred trying to load this video.

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Continuous variables are better understood in contrast with discrete variables, which bring another type of measurement for different categories. Discrete variables come in the form of whole numbers and, sometimes, can be counted 'by fingers.' Number of students in a class, number of cars parked in a street, and number of children in a family are examples of discrete variables. One can never claim that a family has 1.7 children.

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It is imperative for a researcher to choose the right type of variable that better serves a specific context. The variable 'age,' for example, when described as a continuous variable may become an infeasible count. There would be always a more precise value to be added.

Age of a person as a continuous variable:

27 years, 5 months, 2 days, 2 hours, 4 minutes, 37 second, 22 milliseconds, ... and so on.

Age, perhaps, can be better measured as a discrete variable by choosing a specific unit of measurement (e.g., years).

Age of a person as a discrete variable:

27 years

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This section brings real world examples of quantitative variables displayed in different forms, such as datasheets and graphs.

### Discrete vs Continuous: Datasheet

The image below displays two datasheets. The first one (to the left) presents test scores of ten students as discrete variables. The second one (to the right) presents the heights, measured as continuous variables, of a group of ten high school soccer players. ### Discrete vs Continuous: Graphing

Different types of graphs may be chosen to visually display continuous and discrete variables. The images below present a dot plot and a line graph representing a variable related to soda consumption.

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This lesson presented the concept of continuous variables in statistics. These were described as variables that can be measured within a range that includes infinite values. The values can be displayed as decimals (or fractions) and have unlimited degrees of precision. A researcher, for example, can measure the weekly distance covered by joggers for a week. The mileage for a jogger can result in a data that looks like 18.87639735390375 miles. Continuous variables have the freedom to be limited to a specific number of digits after the decimal point. Therefore, the aforementioned mileage can be reduced to 18.976 miles in a data set.

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Video Transcript

## Definition of a Continuous Variable

My niece once did a research project on popcorn and measured the time it took for bags of popcorn to stop popping. If you've ever made popcorn, you probably remember the time given on the package is just an estimate of the time it should take, but to really gauge when the popcorn is done, you listen for pops to slow to only one every couple of seconds. Each bag of popcorn is different, even if they're made by the same company.

By measuring the exact time it took for pops to slow down to the point of having two seconds in between each pop, my niece was able to collect several values of x, where x was the continuous variable that measured the time it took for a bag of popcorn to pop.

A continuous variable is a specific kind a quantitative variable used in statistics to describe data that is measurable in some way. If your data deals with measuring a height, weight, or time, then you have a continuous variable.

Let's further define a couple of the terms used in our definition. A variable in statistics is not quite the same as a variable in algebra. In statistics, a variable is something that gives us data. Some examples of variables in statistics might include age, eye color, height, number of siblings, gender, or number of pets. Our definition of a continuous variable also mentions that it's quantitative. Quantitative data involves quantities or numbers. In the examples of variables listed earlier, your age, height, number of siblings, and number of pets are all quantitative variables.

The definition also mentions that the data is measurable in some way. To understand this, you need to understand discrete variables. Data is considered to be discrete if the data is a count. When we count things, we use whole numbers like 0, 1, 2, and 3. My favorite example of a discrete variable is how many eggs a chicken lays. Each day a hen may or may not lay an egg, but there are two things that can never happen. There can never be a negative number of eggs, and there can never be a fraction or a portion of an egg.

Continuous variables are variables that measure something. My favorite example of a continuous variable is how many gallons of milk a cow gives. To the best of my knowledge, cows don't know how to stop producing or giving milk after exactly 4 gallons! Bessie may give 4.17 gallons on Monday, 3.89 gallons on Tuesday, and 4.2 gallons on Wednesday. Notice continuous variables allow us to have decimals, or fractions.

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