11.01 Identify dependent and independent 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.

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 (x) axis and the value of the dependent variable on the vertical (y) axis. In our example the point (45, 12) would represent a person who lifts weights for 45 minutes and is able to do 12 pull ups.

An experiment or observational study is typically designed to investigate a relationship between two variables, but just because there appears to be a relationship (or correlation) does not mean that a change in one variable causes a change in the other. For instance, it may seem like a person is able to do more pull ups because they spent more time lifting weights but it could actually be because they are smaller, and therefore weigh less. We can say that there is a correlation but not necessarily a causation between the variables.

Examples

Example 1

Consider the following variables:

• Number of ice cream cones sold
• Temperature (\degree \text{F})

a

Which of the following statements makes sense?

A

The temperature affects the number of ice cream cones sold.

B

The number of ice cream cones sold affects the temperature.

Worked Solution

Create a strategy

Determine which variable has an effect on the other.

Apply the idea

When hot weather occurs, more ice cream cones will be sold. Less ice cream cones will be sold in cold weather. So, the correct answer is option A.

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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 temperature and the dependent variable is the number of ice cream cones sold, since the temperature affects the number of ice creams sold.

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Example 2

For the following sets of axes, which have the variables placed in the correct position? Select all the correct options.

A

B

C

D

E

Worked Solution

Create a strategy

For each graph, determine whether the variable on the horizontal axis affects the variable on the vertical axis.

Apply the idea

Among the options, the graph with horizontal axis of time spent exercising and vertical axis of fitness level is the only one with variables are correctly placed. So, the correct answer is option D.

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Example 3

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

1

2

3

4

5

\text{Months}

10

20

30

40

50

60

70

80

90

100

\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.