Bivariate data is the name for numerical data consisting of pairs of values. We generate these pairs to find out whether there is a simple relation between the numbers in each pair.

For example, we may conduct an experiment on a group of people where each person’s bone density is measured against their age. Their age is the input quantity and this could be any value. Their bone density is the level of response that is recorded against their age.

Then each person’s age and bone density make a pair of values in the bivariate data set.

Univariate data consists of only one numerical variable. A data set collecting just the heights of people, or the number of cats people own, are univariate as there is only one variable. Even when comparing the heights from two different classes this is univariate data as this is the same variable just for two different groups of people.

The paired values in a bivariate data set are called the independent variable and the dependent variable. In the above context, the independent variable is the person’s age and the dependent variable is their bone density. The dependent variable is the one that should change based on the independent variable. We could then check whether age is a good predictor for bone density. In other words, we could determine whether bone density depends on a person’s age.

A single data point in a bivariate data set is written in the form (x,\,y), with the first number x being the independent variable and the second number y being the dependent variable. We display bivariate data graphically by plotting the data points with the value of the independent variable on the horizontal axis and the value of the dependent variable on the vertical axis. This is known as a scatterplot.

Examples

Example 1

Scientists were looking for a relationship between the number of hours of sleep we receive and the effect it has on our motor and process skills. Some subjects were asked to sleep for different amounts of time, and were all asked to undergo the same driving challenge in which their reaction time was measured. The table shows the results, which are to be presented as a scatter plot.

Amount of sleep (hours)	Reaction time (seconds)
9	3
6	3.3
4	3.5
10	3
3	3.7
7	3.2
2	3.85
5	3.55

Create a scatter plot for the observations in the table.

Worked Solution

Create a strategy

Plot the points on a cartesian plane.

Apply the idea

\text{Sleep (hours)}

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

\text{Reaction time (seconds)}

The scatter plot of the data is shown.

Reaction time is dependent on how much sleep the person gets. So the values on the horizontal axis represent the number of hours of sleep and the values on the vertical axis represent the reaction time.

According to the results, describe the relationship between amount of sleep and reaction time.

Worked Solution

Create a strategy

Describe what happens to the dependent variable (reaction time) as the independent variable (sleep) increases.

Apply the idea

The sleep hours increases from left to right. We can see from the graph that the reaction time decreases (falls) from left to right.

So as the hours of sleep increases, the reaction time decreases.

Idea summary

Bivariate data

Data consisting of ordered pairs of two variables

Univariate data

Data with only one variable

Independent variable

A variable that is not determined by another variable.

Dependent variable

A variable that is determined by some other variable.

Data point

A value or ordered pair taken from a data set

Scatterplot

A visualisation of bivariate data where ordered pairs are plotted on a number plane

Outcomes

VCMSP352

Use scatter plots to investigate and comment on relationships between two numerical variables

11.06 Bivariate data

Ideas

Bivariate data

Examples

Example 1

Outcomes

VCMSP352

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