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.

This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Use The ScatterPlot grapher by clicking the image below.

Click for additional directions on how to use the grapher.

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  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


  2. Interactive popup. Assistance may be required. Click to see a popup table.
    Number of Jars of Peach Preserves Sold in Fredericksburg, Texas Songs Downloaded in New York City (thousands)
    8
    16
    15
    35
    17
    32
    11
    28
    19
    40
    25
    55
    30
    60
    31

    70

    33
    75
    37
    80
    35
    72
    32
    76
    28
    55
    15
    33
    22
    50
    24
    50
    28
    58
    17
    40
    16
    36
    8
    20
    13
    31
    17
    35
    11
    20
    15
    37
    Close Pop Up


    From the table in the popup, 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.


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

    Interactive popup. Assistance may be required.

    See a sample graph.

    Close Pop Up
  5. Do the data points appear to follow a linear association? How can you tell?
    Interactive popup. Assistance may be required.

    Check Your Answer

    Yes. Even though you cannot connect all of the data points with one line, all of the data points lie clustered near a line.Close Pop Up Interactive popup. Assistance may be required.
  6. As you read the graph from left to right, do the points seem to move upward or downward? Interactive popup. Assistance may be required.

    Check Your Answer

    The points move upward as you read the graph to the right.Close Pop Up
  7. researcher examining data
  8. 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?

    Interactive popup. Assistance may be required.

    Need a hint?

    The number of jars of peach preserves sold is along the horizontal axis, and the number of songs downloaded is along the vertical axis. The values along the horizontal axis increase from left to right, and the values along the vertical axis increase from bottom to top. Close Pop Up Interactive popup. Assistance may be required.

    Check Your Answer

    As the number of jars of peach preserves sold in Fredericksburg, Texas, increases, the number of songs downloaded in New York City also increases. Close Pop Up
  9. 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? Interactive popup. Assistance may be required.

    Need a hint?

    Use your answer from the previous question to help you. A greater number of jars of peach preserves is indicated by a point farther to the right along the horizontal axis. Close Pop Up Interactive popup. Assistance may be required.

    Check Your Answer

    If a greater number of jars of peach preserves is sold in Fredericksburg, Texas, then there should be a greater number of songs downloaded in New York City. Close Pop Up
  10. 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.

    Interactive popup. Assistance may be required.

    Check Your Answer

    No. There may be an association between the two events, but it is unlikely that the number of songs downloaded in New York City is caused by the number of jars of peach preserves sold in Fredericksburg, Texas. Close Pop Up
  11. If a trend line were found, would it have positive or negative slope?

    Interactive popup. Assistance may be required.

    Check Your Answer

    The trend line would have a positive slope.Close Pop Up

     

  12. 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?

    Interactive popup. Assistance may be required.

    Check Your Answer

    Possible answer: The relationship has a positive association because the data values are clustered along a trend line with a positive slope.Close Pop Up

     

Describing Characteristics of Negative Trends

This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Use The ScatterPlot grapher by clicking the image below.

screen shot from the Shodor interactive scatterplot grapher

Click for additional directions on how to use the grapher.

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  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


  2. Click to see a popup table.

    From the table in the popup, 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.

    Latitude (°N)

    Average July High Temperature (°F)

    33.75

    89

    30.25

    96

    39.3

    87

    33.5

    91

    42.6

    81

    43

    80

    35.25

    89

    41.8

    84

    39

    87

    41.3

    83

    40

    85

    32.75

    96

    39.75

    88

    42.3

    83

    41.8

    85

    30

    94

    39.75

    85

    30.2

    92

    39

    90

    38.25

    89

    35.1

    92

    43

    80

    45

    83

    36.2

    89

    30

    91

    40.8

    84

    35.5

    94

    28.5

    92

    40

    87

    40.5

    83

    45.5

    81

    41.8

    83

    35.75

    90

    37.5

    90

    34

    95

    43.2

    81

    38.6

    92

    40.75

    93

    29.5

    95

    37.3

    82

    47.6

    76

    38.6

    89

    28

    90

    36.8

    87

    38.8

    88

    Close Pop Up
  3. In the grapher, use the radio buttons to show the Light Brid Lines and change the plot type to Scatter.

    Image showing Light Grid Lines radio button and Scatter radio button selected
  4. Click the Plot/Update button to generate a scatterplot. Use the scatterplot to answer the questions below.

    Interactive popup. Assistance may be required.

    See a sample graph.

    Close Pop Up
  5. Do the data points appear to follow a linear association? How can you tell?
    Interactive popup. Assistance may be required.

    Check Your Answer

    Yes. Even though you cannot connect all of the data points with one line, all of the data points lie clustered near a line.Close Pop Up Interactive popup. Assistance may be required.
  6. As you read the graph from left to right, do the points seem to move upward or downward? Interactive popup. Assistance may be required.

    Check Your Answer

    The points move downward as you read the graph to the right.Close Pop Up
  7. researcher examining data
  8. As the latitude of the city increases, what happens to the city’s average July high temperature?

    Interactive popup. Assistance may be required.

    Need a hint?

    The latitude of each city is plotted along the horizontal axis, and the average July high temperature of each city is plotted along the vertical axis. The values along the horizontal axis increase from left to right and the values along the vertical axis increase from bottom to top. Close Pop Up Interactive popup. Assistance may be required.

    Check Your Answer

    As the latitude of the city increases, the city’s average July high temperature decreases. Close Pop Up
  9. If a randomly chosen city has greater latitude, what can you predict will be that city’s average July high temperature? Interactive popup. Assistance may be required.

    Need a hint?

    Use your answer from the previous question to help you. A city with greater latitude is indicated by a point farther to the right along the horizontal axis. Close Pop Up Interactive popup. Assistance may be required.

    Check Your Answer

    If a randomly chosen city has greater latitude, then that city's average July high temperature will be lower. Close Pop Up
  10. 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.

    Interactive popup. Assistance may be required.

    Check Your Answer

    Yes, there could be. A city with a greater (higher) latitude is farther north, and cities that are farther north are typically cooler than cities that are closer to the equator. Close Pop Up
  11. If a trend line were found, would it have positive or negative slope?

    Interactive popup. Assistance may be required.

    Check Your Answer

    The trend line would have a negative slope.Close Pop Up

     

  12. 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?

    Interactive popup. Assistance may be required.

    Check Your Answer

    Possible answer: The relationship has a negative association because the data values are clustered along a trend line with a negative slope.Close Pop Up

     

Pause and Reflect

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.)

Interactive popup. Assistance may be required.

Check Your Answer

A positive linear association shows data with both variables increasing. The line approximating the data will have a positive slope. A negative linear association shows data with one variable increasing and the other variable decreasing. The line approximating the data will have a negative slope.Close Pop Up

How could a trend line help you to determine if the slope is positive or negative?

Interactive popup. Assistance may be required.

Check Your Answer

A trend line shows how the data values tend to be clustered. If the slope of the trend line is positive, then the data set shows a positive linear association. If the slope of the trend line is negative, then the data set shows a negative linear association. Close Pop Up

Practice

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


  1. screen shot from the Shodor interactive scatterplot grapher
    Interactive popup. Assistance may be required.

    Need a hint?

    If you were to connect most of the points in the scatterplot, would the line have a positive or a negative slope?Close Pop Up
    Interactive popup. Assistance may be required.

    Check Your Answer

    Positive linear associationClose Pop Up

  2. screen shot from the Shodor interactive scatterplot grapher
    Interactive popup. Assistance may be required.

    Need a hint?

    If you were to connect most of the points in the scatterplot, would the line have a positive or a negative slope? Close Pop Up
    Interactive popup. Assistance may be required.

    Check Your Answer

    Negative linear associationClose Pop Up