### Let’s Think About It

William is interested in global warming and climate change. He's noticed that the summers seem to be pretty hot, but there have been some really bad snow storms in the winter as well so he's not sure what global warming is all about. He does some research and finds the following scatter plot.

This scatter plot shows the average temperature for the whole Earth each year since 1880. How should William interpret this scatter plot? Does it show a correlation between year and annual average global temperature?

In this concept, you will learn how to make and read scatter plots of paired real-world data to recognize patterns and make predictions.

### Guidance

A **scatter plot** is a type of graph that shows pairs of data plotted as points.

You can use a scatter plot to analyze trends in your data and to help you to determine whether or not there is a relationship between two variables. A scatter plot can show a positive relationship, a negative relationship, or no relationship.

If the points on the scatter plot seem to form a line that slants up from left to right, there is a **positive relationship** or **positive correlation** between the variables.

If the points on the scatter plot seem to form a line that slants down from left to right, there is a **negative relationship** or **negative correlation** between the variables.

If the points on the scatter plot seem to be scattered randomly, there is **no relationship** or **no correlation** between the variables.

When there is a positive or negative relationship between your variables, you can draw a **line of best fit**. This is a line that will not go through every point on the scatter plot, but will show the general trend of your data.

Let's look at an example.

A teacher wants to know if there is a relationship between the amount of time her students spent working on a social studies report and the grade each student received. She surveyed 10 students and recorded the data below.

Student |
Number of Hours Worked |
Grade |

Ahmad | 5 | 90 |

Becky | 3 | 80 |

Darrell | 3.5 | 80 |

Emma | 1 | 60 |

Guillermo | 4.5 | 90 |

Helene | 1 | 70 |

Kiet | 3 | 75 |

Nykeisha | 4 | 85 |

Ollie | 2 | 70 |

Zivia | 2.5 | 75 |

- Make a scatter plot based on the data.
- Determine if there is a relationship between the number of hours worked and the grade received. If so, describe the relationship.
- Suppose an eleventh student spent 1.5 hours working on her report. Based on the scatter plot, predict the grade you would expect her to receive.

Let's start by looking at part a.

To make the scatter plot you need to first scale your axes. The teacher wants to know if the number of hours worked is related to the grade a student earned, so use the horizontal axis of the scatter plot to show the number of hours worked. Notice that in the table, the number of hours worked ranges from 1 to 5, so draw your horizontal axis so it goes from 0 to 6.

Next, scale the vertical axis. You will use the vertical axis to show the grades students received. The grades in the table range from 60 to 90. Showing every grade from 0 to 100 would make the plot very large, so include a break in the axis between 0 and 60. Use intervals of 10 for the rest of scale.

Now, you can plot a point for each pair of data in the table. For example, Ahmad worked for 5 hours and earned a grade of 90. So, plot a point at (5, 90) on the scatter plot. Do this until you have plotted all 10 data points.

The answer to part a is shown below.

Next let's look at part b.

Notice that when you look at the scatter plot the points seem to form a line going up from left to right. You could sketch in a potential line of best fit that shows this general trend of the data.

The answer is that because the line of best fit slants up from left to right, this scatter plot shows a positive relationship between hours worked and grade on the report. This means that in general, the longer a student spent working on the report, the higher the student's grade on the report.

Last let's look at part c.

Because the scatter plot shows a relationship between hours worked on the report and grade on the report, you can use the scatter plot to make predictions for other values within the original range of your data.

None of the original ten students worked only 1.5 hours on the report; however, you can assume that this student would fit into the trends seen with the rest of the students.

Look at the scatter plot with the line of best fit. Find 1.5 hours on the \begin{align*}x\end{align*}-axis and move up until you hit the line of best fit. You are at a height corresponding to a grade of 65 on the report. This means that if a student only worked 1.5 hours on the report, you can predict that they will get a 65 as a grade on the report.

The answer is that you predict that a student who only worked 1.5 hours on the report will receive a grade of a 65 on the report.

### Guided Practice

Look at the following set of ordered pairs. Would the scatter plot show a positive correlation, a negative correlation or no correlation?

\begin{align*}(1, 2), (3, 5), (7, 10), (9, 15), (4, 8)\end{align*}

First, plot all five points on a graph.

Next, look to see if there is a pattern in the graph. Notice that when you look at the scatter plot the points seem to form a line going up from left to right. This means that there is a positive correlation between the \begin{align*}x\end{align*}-coordinates and the \begin{align*}y\end{align*}-coordinates.

The answer is that the scatter plot shows a positive correlation.

### Examples

#### Example 1

Does a line of best fit on a plot with a positive correlation go up or down as you move from left to right on the scatter plot?

This is something that you have to remember. Positive correlation means that as the first variable increases, the second variable increases as well. This corresponds to points (and a line of best fit) that move up as you go from left to right.

Negative correlation would mean that as one variable increases, the second variable decreases. Negative correlation corresponds to points that move down as you go from left to right on your scatter plot.

The answer is that the line of best fit on a scatter plot with positive correlation will go up as you move from left to right.

#### Example 2

Look at the following set of ordered pairs. Would the scatter plot show a positive correlation, a negative correlation or no correlation?

\begin{align*}(1, 12), (5, 5), (3, 8), (9, 1), (8, 4)\end{align*}

First, plot all five points on a graph.

Next, look to see if there is a pattern in the graph. Notice that when you look at the scatter plot the points seem to form a line going down from left to right. This means that there is a negative correlation between the \begin{align*}x\end{align*}-coordinates and the \begin{align*}y\end{align*}-coordinates.

The answer is that the scatter plot shows a negative correlation.

#### Example 3

Look at the following set of ordered pairs. Would the scatter plot show a positive correlation, a negative correlation or no correlation?

\begin{align*}(1, 12), (8, 9), (3, 10), (2, 4), (5, 2), (6, 6)\end{align*}

First, plot all six points on a graph.

Next, look to see if there is a pattern in the graph. Notice that when you look at the scatter plot there does not seem to be any real pattern to the points. The points don't seem to be trending up or down. This means that there is no correlation between the \begin{align*}x\end{align*}-coordinates and the \begin{align*}y\end{align*}-coordinates.

The answer is that the scatter plot shows no correlation.

### Follow Up

Remember William who was researching global warming? He found a scatter plot that showed the annual average global temperature each year since 1880.

In order to determine if the graph shows any correlation, William should look to see if there is a pattern in the scatter plot. Notice that when you look at the scatter plot overall the points seem to form a line going up from left to right. While there are brief periods where the points head back down, the overall trend is up. This means that there is a positive correlation.

The answer is that the scatter plot shows a positive correlation between year and annual average global temperature.

This means that the average annual global temperature appears to be going up over time. Keep in mind the scale on the vertical axis. The annual average global temperature has only changed \begin{align*}1^\circ \ C\end{align*} over 120+ years so while there is definitely a positive correlation, it's not quite as drastic as it might appear on the scatter plot.

### Explore More

This scatter plot shows the relationship between the last digit of ten students’ phone numbers and their vocabulary quiz scores.

1. Does this scatter plot show a positive relationship, a negative relationship, or no relationship?

2. How many students have a six as the last digit in their phone number?

3. How many students have an eight as the last digit?

4. How many students received a grade of 70%?

5. How many students received a grade of 60%

6. How many students earned 100%?

7. How many students earned a 75%?

8. True or false. No one earned below a 60%.

9. True or false. No one earned an 80%.

10. True or false. No one earned an 85%.

Serena wants to know if there is a relationship between a person’s age and the number of DVDs they purchased in one year. She surveyed a group of people and recorded the data in the table below.

Person |
Age |
Number of DVDs Purchased |

Person 1 | 18 | 14 |

Person 2 | 19 | 13 |

Person 3 | 20 | 13 |

Person 4 | 20 | 12 |

Person 5 | 21 | 11 |

Person 6 | 22 | 12 |

Person 7 | 22 | 11 |

Person 8 | 23 | 10 |

Person 9 | 24 | 9 |

Person 10 | 25 | 9 |

11. Use the grid below to make a scatter plot for the data in the table.

12. True or false. The scatter plot shows a negative correlation.

13. True or false. The scatter plot shows no correlation.

14. True or false. The scatter plot shows a positive correlation.

15. If the trend in the scatter plot continues, predict the number of DVDs you would expect a 27-year-old person to buy in one year.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 4.18.