3. Algebraic Relations & Functions

Lesson

Bivariate data is the technical name for numerical data consisting of to 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*. *

Types of variables

Independent variable - can be changed freely, does not depend on any other variables

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

45 | 12 |

13 | 4 |

67 | 13 |

35 | 10 |

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

Consider the following variables:

- Number of ice cream cones sold
- Temperature (°C)

Which of the following statements makes sense?

The temperature affects the number of ice cream cones sold.

AThe number of ice cream cones sold affects the temperature.

BThe temperature affects the number of ice cream cones sold.

AThe number of ice cream cones sold affects the temperature.

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

The independent variable is the temperature and the dependent variable is the number of ice cream cones sold.

AThe independent variable is the number of ice cream cones sold and the dependent variable is temperature.

BThe independent variable is the temperature and the dependent variable is the number of ice cream cones sold.

AThe independent variable is the number of ice cream cones sold and the dependent variable is temperature.

B

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

- ABCDEABCDE

The scatter plot shows the relationship between sea temperature and the amount of healthy coral.

Which variable is the dependent variable?

Sea temperature

ALevel of healthy coral

BSea temperature

ALevel of healthy coral

BWhich variable is the independent variable?

Sea temperature

ALevel of healthy coral

BSea temperature

ALevel of healthy coral

B

Determine the independent and dependent variable, given a practical situation modeled by a linear function