The table shows the average and maximum longevity of various animals in captivity.

Average |
12 | 25 | 15 | 8 | 35 | 40 | 41 | 20 |

Maximum |
47 | 50 | 40 | 20 | 70 | 77 | 61 | 54 |

Construct a scatter plot for the above data.

Predict the maximum life span in captivity if the average life span was 30 years.

In this concept, you will learn to make a scatterplot to represent data.

### Scatterplots

In the real world, many things are related to each other. Many fields try to find relationships between two variables. One tool that helps you to accomplish this is the **scatterplot**.

A **scatterplot** is a type of graph where corresponding values from a set of data are placed as points on a coordinate plane. A relationship between the points is sometimes shown to be positive, negative, strong, or weak.

Sometimes a scatterplot shows that there is no relationship at all. Aside from finding relationships, scatterplots are useful in predicting values based on the relationship that was revealed.

Let’s look at how a scatterplot can be applied to a situation.

A student had a hypothesis for a science project. He believed that the more students studied math, the better their math scores would be. He took a poll in which he asked students the average number of hours that they studied per week during a given semester. He then found out the overall percent that they received in their math classes. His data is shown in the table below:

Study Time (hours) |
4 | 3.5 | 5 | 2 | 3 | 6.5 | 0.5 | 3.5 | 4.5 | 5 | 1 | 1.5 | 3 | 5.5 |

Math Grade (percent) |
82 | 81 | 90 | 74 | 77 | 97 | 51 | 78 | 86 | 88 | 62 | 75 | 70 | 90 |

In order to understand this data, he decided to make a scatterplot.

The independent variable, or **input data**, is the study time because the hypothesis is that the math grade depends on the study time. That means that the math grade is the dependent variable, or **output data**. The input data is on the \begin{align*}x\end{align*}-axis and the output data is on the \begin{align*}y\end{align*}-axis.

The scales and intervals on the axes will be determined by the data. Since the greatest value on the \begin{align*}y\end{align*}-axis, the greatest value is 97 and, since it is a percent, you can use intervals of 10 until you reach 100.

-axis is 6.5, you can use intervals of 1 until you reach 7. On the

Now you can graph the points on the scatterplot. In order to plot the points, you will show each one as an ordered pair (hours, percent). The first ordered pair, then, is (4, 82). Plot each of the 14 points. Remember, it takes two pieces of data to make a single point.

You can see that there is a relationship between the independent and dependent values of the chart.

Scientists in the real world rarely create scatterplots on a piece of paper and compute equations by hand. They use computer programs that can approximate the trend line much more accurately.

You can make scatterplots on your graphing calculator, if you have one. Then, you can compute the **trend line** called the **linear regression** in some models. Your graphing calculator can put the equation in \begin{align*}y = mx + b\end{align*} form or \begin{align*}Ax + By = C\end{align*} form, depending on the mode you choose. Then, you can choose any input values for which your calculator will tell you the output values.

If you use a graphing calculator for the example above you would get a similar image on your calculator. This image was completed on a TI-84C.

### Examples

#### Example 1

Earlier, you were given a problem about animals and their longevity. You were given data about the average and maximum longevities of various animals. You need to construct a scatterplot of the data and then predict the maximum life span if the average life span was 30 years. The data is below.

Average |
12 | 25 | 15 | 8 | 35 | 40 | 41 | 20 |

Maximum |
47 | 50 | 40 | 20 | 70 | 77 | 61 | 54 |

First, construct the scatterplot. The average life span is the independent variable and the maximum life span is the dependent variable.

Next, use the graph to find the maximum life span if the average life span was 30 years.

Using the trend line, if you draw a line up from 30 and over to the \begin{align*}y\end{align*} axis, you can find the predicted maximum longevity with an average value of 30 years.

The answer is 59 years.

#### Example 2

Create a scatterplot of the data.

After a circus class, the following data was collected. It tracks the number of people who could balance on a tightrope for specific lengths of time.

\begin{align*}1 \ \text{person} = 7 \ \text{minutes} \end{align*}

\begin{align*}3 \ \text{people} = 15 \ \text{minutes}\end{align*}

\begin{align*}7 \ \text{people} = 20 \ \text{minutes}\end{align*}

\begin{align*}9 \ \text{people} = 25 \ \text{minutes}\end{align*}

\begin{align*}14 \ \text{people} = 32 \ \text{minutes}\end{align*}

\begin{align*}18 \ \text{people} = 39 \ \text{minutes}\end{align*}

Here is the scatterplot of the data.

#### Answer the following questions about scatterplots.

#### Example 3

If the points on a scatterplot do not show a pattern, is there a connection between the data?

The answer is no.

When there is no pattern there is no correlation between the independent and dependent values.

#### Example 4

If the points on a scatterplot trend up to the right, is there a connection between the data?

The answer is yes.

If the pattern is a trend to the right it is positive. Therefore there is a positive correlation between the independent and dependent values.

#### Example 5

If the points on a scatterplot trend down and to the left, is there a connection between the data?

The answer is yes.

If the pattern is a trend to the left it is negative. Therefore there is a negative correlation between the independent and dependent values.

### Review

Use what you have learned to answer each question or complete each task.

1. Make a scatterplot to display the data set in the table:

23 | 18 | 30 | 24 | 29 | 45 | 10 | 17 | 27 | 39 | 32 | 40 | 21 | 14 | |

62 | 72 | 54 | 60 | 57 | 30 | 79 | 65 | 55 | 34 | 48 | 41 | 68 | 76 |

What type of relationship would you predict for the following variables, positive, negative, or no relationship?

2. Altitude vs. the amount of oxygen in the atmosphere

3. Number of customers vs. profit

4. Number of siblings vs. grade point average

5. Hours of studying vs. test score

6. Hours of driving vs. distance traveled

7. Speed of a car vs. distance traveled

8. Hours at work vs. amount of money made

9. Age of person vs. intelligence

Use this scatterplot to answer the following questions.

10. True or false. This data shows a positive correlation.

11. True or false. This data shows a negative correlation.

12. True or false. This data shows no correlation.

13. True or false. Data is graphed on a scatterplot by using ordered pairs.

14. True or false. An example of a negative correlation could be the amount of time in school and an increase in intelligence.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.7.