Topics
3. Relations & Functions
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United States of AmericaVirginia
Grade 8

3.04 Independent and dependent variables

Lesson
Practice

Independent and dependent variables

Bivariate data is the technical name for numerical data consisting of two variables organized into pairs of values. When we are analyzing bivariate data we are interested in determining whether there is a relationship between the two variables.

We may, for example, conduct an experiment where we measure how much time a person spends lifting weights in a week and how many pull ups they can do in a row. The two variables here are time spent lifting weights and number of pull ups. We can describe these as the independent variable and the dependent variable.

In our example a person can freely choose how much time they will spend lifting weights, so that is the independent variable. The number of pull ups they are able to do depends on how much weight lifting they do so the number of pull ups is the dependent variable. See the table below for some data collected on these variables.

Time lifting weights (in minutes)Number of pull ups
4512
134
6713
3510

Notice from the table above, it seems like the more time spent lifting weights, the more pull ups a person is able to do, but it is often useful to display data like this graphically to see if that is really what is happening. Each data pair represents a point on the graph. We plot the data points with the value of the independent variable on the horizontal \left(x\right) axis and the value of the dependent variable on the vertical \left(y\right) axis. In our example the point \left(45, 12\right) would represent a person who lifts weights for 45 minutes and is able to do 12 pull ups.

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\text{Time lifting weights (in minutes)}
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\text{Number of pull ups}

Examples

Example 1

Consider the following variables:

  • Number of people at the beach

  • Season

a

Which of the following statements makes sense?

A
The season affects the number of people at the beach.
B
The number of people at the beach affects the season.
Worked Solution
Create a strategy

Determine which variable has an effect on the other.

Apply the idea

During the summer season, more people tend to go to the beach. There are typically fewer people at the beach during other seasons, and it is much less llikely that people will visit during winter. Therefore, the correct answer is option A.

b

Which is the dependent variable and which is the independent variable?

Worked Solution
Create a strategy

Recall that the independent variable is not affected by the other variable. In contrast, the dependent variable is affected or changed by the other variable.

Apply the idea

Based on the previous problem, we can see that the independent variable is the season and the dependent variable is the number of people at the beach, since the season affects the number of people at the beach, but changing the number of people at the beach will not change the season.

Example 2

The linear graph shows the relationship between Lesley's savings (in dollars) over a few months.

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\text{Months}
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\text{Savings}
a

Which variable is the dependent variable?

A
Number of months
B
Savings
Worked Solution
Create a strategy

The dependent variable is placed on the vertical axis and is affected by the independent variable.

Apply the idea

The amount of savings is determined by number of months and is on the vertical axis, making it the dependent variable. So, the correct answer is B.

b

Which variable is the independent variable?

A
Number of months
B
Savings
Worked Solution
Create a strategy

An independent variable is a variable that stands alone and is not changed by the other variables you are measuring.

Apply the idea

From the previous problem, we know that the amount of savings is the dependent variable, so this means the number of months is the independent variable. The correct answer is A.

Idea summary

Independent variable - the input, it can be changed freely, does not depend on any other variables.

Dependent variable - the output, it changes as a result of the independent variable, it depends on the value of the independent variable

We plot the data points with the value of the independent variable on the horizontal \left(x\right) axis and the value of the dependent variable on the vertical \left(y\right) axis.

Outcomes

8.PFA.3

The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

8.PFA.3b

Describe key characteristics of linear functions including slope (m), y-intercept (b), and independent and dependent variables.

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