# Introduction

A scatterplot is a graph that is used to compare two different data sets.

 City Average January High Temperature (°F) Average July High Temperature (°F) College Station 61 95 Austin 62 96 Longview 58 94 Wichita Falls 54 97 Victoria 65 94 McAllen 71 97 San Angelo 60 95 El Paso 58 95 Amarillo 51 91

A scatterplot is created by rewriting the table values as a set of ordered pairs, and plotting one variable along the x-axis and one variable along the y-axis.

Scatterplots are typically used to determine if there is a relationship between the two variables. Researchers, engineers, and statisticians frequently use scatterplots to look for these relationships since they are a visual representation. In a scatterplot, you may spot trends that you don't easily see in a data table.

In this lesson, you will investigate ways to distinguish between different types of relationships that can be presented in a scatterplot. If the relationship is a linear one, then you will also use a trend line to make predictions.

# Distinguishing Between Linear and Non-Linear Associations

In this section, you will compare linear and non-linear associations in order to distinguish between the two types of association in bivariate data. An association between two data sets occurs when there is a relationship between the values in one data set and the values in the other data set.

Use The ScatterPlot grapher by clicking the image below. The grapher will open in a new tab or window.

1. The table below contains data relating the weight of an alligator in pounds to the length of an alligator in inches.

 Alligator Size Weight of Alligator (pounds) Length of Alligator (inches) 86 83 88 70 72 61 74 54 61 44 90 106 89 84 68 39 76 42 114 197 90 102 78 57 94 130 74 51 147 640 58 28 86 80 94 110 63 33 86 90 69 36 72 38 128 366 85 84 82 80

Copy the numeric portion of the data only (i.e., do not copy the row headers). Paste the data into the Data box of the grapher.

2. In the grapher, use the radio buttons to show the Light Grid Lines and change the plot type to Scatter.

3. Click the Plot/Update button to generate a scatterplot.

See a sample graph.

4. Do the data points appear to follow a linear trend?

5. Do the data points represent a reasonably constant rate of change?

Use The Scatter Plot grapher by clicking the image below. The grapher will open in a new tab or window.

1. The table below contains data relating the length of an alligator in centimeters to the belly width of an alligator in centimeters.
 Alligator Size Length of Alligator (centimeters) Belly Width of Alligator (centimeters) 45 9 48 10 48 8 50 11 55 11 56 13 60 12 62 13 65 14 68 14 70 15 72 17 76 17 80 18 82 17 84 19 88 20 90 21 92 23 94 22 100 23 103 23 105 24 107 26

Copy the numeric portion of the data only (i.e., do not copy the row headers). Paste the data into the Data box of the grapher.

2. In the grapher, use the radio buttons to show the Light Grid Lines and change the plot type to Scatter.
3. Click the Plot/Update button to generate a scatterplot. Use the scatterplot to answer the questions below.

4. Do the data points appear to follow a linear trend?

5. Do the data points represent a reasonably constant rate of change?

Data sets could have a linear association or a non-linear association. But not all data sets have an association. For example, consider the graph below that shows the relationship between the population according to the 2010 U.S. Census of each state and that state’s average 8th grade math score on a national mathematics test in 2013.

Does the data set appear to show a linear association, a non-linear association, or no association?

Pause and Reflect

1. When you look at a scatterplot of data, how can you tell the difference between the appearance of the scatterplot with a linear association or a scatterplot with a non-linear association?
2. How do the rates of change for linear associations and non-linear associations compare?

Practice

For each of the data sets below, decide whether the scatterplot best represents a linear association or non-linear association.

1.

2.

3.

# Distinguishing Between Positive and Negative Linear Associations

In the last section, you used scatterplots to distinguish between linear associations and non-linear associations. In this section, you will use scatterplots to distinguish between positive linear associations and negative linear associations. In a linear association, data will appear to be clustered around a trend line. Data are said to be clustered when the data values seem to be gathered around a particular value.

Describing Characteristics of Positive Trends
In this section, you will practice creating a scatterplot, and then use that scatterplot to analyze a relationship that exhibits a positive trend.

Use The ScatterPlot grapher by clicking the image below.

1. The table below contains data collected twice each month regarding the number of jars of peach preserves sold at a general store in Fredericksburg, Texas, and the number of songs that are downloaded in New York City.
 Semi-Monthly Data Collection Week Number of Jars of Peach Preserves Sold in Fredericksburg, Texas Songs Downloaded in New York City (thousands) January 1 8 16 January 15 15 35 February 1 17 32 February 15 11 28 March 1 19 40 March 15 25 55 April 1 30 60 April 15 31 70 May 1 33 75 May 15 37 80 June 1 35 72 June 15 32 76 July 1 28 55 July 15 15 33 August 1 22 50 August 15 24 50 September 1 28 58 September 15 17 40 October 1 16 36 October 15 8 20 November 1 13 31 November 15 17 35 December 1 11 20 December 15 15 37

Copy the data from the Number of Jars of Peach Preserves column and Songs Downloaded column. Paste the data into the Data box of the grapher.

2.  In the grapher, use the radio buttons to show the Light Grid Lines and change the plot type to Scatter.

3. Click the Plot/Update button to generate a scatterplot. Use the scatterplot to answer the questions below.

4. Do the data points appear to follow a linear association? How can you tell?

5. As you read the graph from left to right, do the points seem to move upward or downward?

6. As the number of jars of peach preserves sold in Fredericksburg, Texas increases, what happens to the number of songs downloaded in New York City?

7. If a greater number of jars of peach preserves are sold in Fredericksburg, what can you predict will happen to the number of songs downloaded in New York City?

8. Do you think that there is a cause-and-effect relationship between the number of jars of peach preserves sold in Fredericksburg, Texas, and the number of songs that is downloaded in New York City? Explain your answer.

9. If a trend line were found, would it have positive or negative slope?

10. Do you think that the relationship between the number of jars of peach preserves sold in Fredericksburg, Texas, and the number of songs that is downloaded in New York City has a positive or negative association? Why or why not?

Describing Characteristics of Negative Trends

Use The ScatterPlot grapher by clicking the image below.

1. The table below contains data describing different U.S. cities’ latitude (in degrees North from the equator) and the city’s average July high temperature.

 City Latitude (°N) Average July High Temperature (°F) Atlanta, Georgia 33.75 89 Austin, Texas 30.25 96 Baltimore, Maryland 39.3 87 Birmingham, Alabama 33.5 91 Boston, Massachusetts 42.6 81 Buffalo, New York 43 80 Charlotte, North Carolina 35.25 89 Chicago, Illinois 41.8 84 Cincinnati, Ohio 39 87 Cleveland, Ohio 41.3 83 Columbus, Ohio 40 85 Dallas, Texas 32.75 96 Denver, Colorado 39.75 88 Detroit, Michigan 42.3 83 Hartford, Connecticut 41.8 85 Houston, Texas 30 94 Indianapolis, Indiana 39.75 85 Jacksonville, Florida 30.2 92 Kansas City, Missouri 39 90 Louisville, Kentucky 38.25 89 Memphis, Tennessee 35.1 92 Milwaukee, Wisconsin 43 80 Minneapolis, Minnesota 45 83 Nashville, Tennessee 36.2 89 New Orleans, Louisiana 30 91 New York, New York 40.8 84 Oklahoma City, Oklahoma 35.5 94 Orlando, Florida 28.5 92 Philadelphia, Pennsylvania 40 87 Pittsburgh, Pennsylvania 40.5 83 Portland, Oregon 45.5 81 Providence, Rhode Island 41.8 83 Raleigh, North Carolina 35.75 90 Richmond, Virginia 37.5 90 Riverside, California 34 95 Rochester, New York 43.2 81 Sacramento, California 38.6 92 Salt Lake City, Utah 40.75 93 San Antonio, Texas 29.5 95 San Jose, California 37.3 82 Seattle, Washington 47.6 76 St. Louis, Missouri 38.6 89 Tampa, Florida 28 90 Virginia Beach, Virginia 36.8 87 Washington, DC 38.8 88

Copy the data from the Latitude (°N) column and Average July High Temperature (°F) column. Paste the data into the Data box of the grapher.

2. In the grapher, use the radio buttons to show the Light Brid Lines and change the plot type to Scatter.

3. Click the Plot/Update button to generate a scatterplot. Use the scatterplot to answer the questions below.

4. Do the data points appear to follow a linear association? How can you tell?

5. As you read the graph from left to right, do the points seem to move upward or downward?

6. As the latitude of the city increases, what happens to the city’s average July high temperature?

7. If a randomly chosen city has greater latitude, what can you predict will be that city’s average July high temperature?

8. Do you think that there is a cause-and-effect relationship between the latitude of a city and that city's average July high temperature? Explain your answer.

9. If a trend line were found, would it have positive or negative slope?

10. Do you think that the relationship between the latitude of a city and that city's average July high temperature has a positive or negative association? Why or why not?

Pause and Reflect

1. How can you tell from a scatterplot whether a set of data shows a positive linear association or a negative linear association? (Hint: think about the slope of the line approximating the data.)
2. How could a trend line help you to determine if the slope is positive or negative?

Practice

Determine whether each of the graphs below shows a positive linear association or a negative linear association.

1.

2.

# Using Trend Lines to Make Predictions

In the last section, you studied the difference between positive linear associations and negative linear associations. Once you know that a data set has a linear association, you can use a trend line to make predictions. In this section, you will practice generating a trend line and using that trend line to make predictions from the data.

The graph below shows the relationship between the length of an alligator (in centimeters) and the belly width of an alligator (in centimeters).

Click and drag the circles below to place a trend line on the graph. Your trend line will not connect every point, but should follow the trend in the data.

Use the trend line you estimated in the graph to answer the questions below.

1. What is the y-intercept, or starting point, of your trend line?
2. What is the approximate slope of your trend line?
3. Use your trend line to estimate the belly width of an alligator that has a length of 30 centimeters.
4. Use your trend line to estimate the length of an alligator that has a belly width of 35 centimeters.

Pause and Reflect

How does a trend line help you to make predictions from a scatterplot?

Practice

1. The graph below shows the relationship between the amount of television watched in one week and a student’s grade point average.

Use a trend line to estimate the grade point average a student would have if they watched 27 hours of television each week.

2. The scatterplot below shows the relationship between the sales at an ice cream store and the outdoor air temperature.

Use a trend line to estimate the temperature required for \$700 in ice cream sales.

# Summary

There are four types of relationships that you analyzed in this lesson.

### Positive Linear Association

A relationship with a positive linear association is one in which both variables increase at the same time at an almost constant rate.

In this example, each point represents the amount of sleep that a student had and the grade that they received on a recent math quiz. As the amount of sleep increases, the math grade increases.

### Negative Linear Association

A relationship with a negative linear association is one in which as one variable increases, the other variable decreases at an almost constant rate.

In this example, each point represents the total points that a player scored and the number of penalties they received during a recent game. As the number of penalties increases, the total points scored decreases. Likewise, as the total points scored increases, the number of penalties received decreases.

### Non-Linear Association

A relationship with a non-linear association is one in which as one variable increases, the other variable changes in a way that is not constant but is predictable.

In this example, as the number of days increases, the number of bacteria in a Petri dish increases. However, the number of bacteria does not increase at a constant rate, as it would for a positive linear association. Instead, the data appear to follow a curve, which is a non-linear relationship.

### No Association

Sometimes, a relationship shows no trend. In this case, there is no detectable pattern in the data that allows you to say that as one variable changes, the second variable changes in a particular way. In this example, each point represents a student's shoe size and his or her recent social studies exam score. There does not appear to be a relationship between the shoe size and the exam score.