Scatterplots and Correlation

214343

Scatterplots and Correlation

Diana Mindrila, Ph.D.

Phoebe Balentyne, M.Ed.

Based on Chapter 4 of The Basic Practice of Statistics (6th ed.)

Concepts:

ƒ Displaying Relationships: Scatterplots

ƒ Interpreting Scatterplots

ƒ Adding Categorical Variables to Scatterplots

ƒ Measuring Linear Association: Correlation

ƒ Facts About Correlation

Objectives:

¾ Construct and interpret scatterplots.

¾ Add categorical variables to scatterplots.

¾ Calculate and interpret correlation.

¾ Describe facts about correlation.

References:

Moore, D. S., Notz, W. I, & Flinger, M. A. (2013). The basic practice of statistics (6th ed.). New York, NY: W. H. Freeman and Company.

Scatterplot

ƒ The most useful graph for displaying the relationship between two quantitative variables is a scatterplot. ƒ Many research projects are correlational studies because they investigate the relationships that may exist between variables. Prior to investigating the relationship between two quantitative variables, it is always helpful to create a graphical representation that includes both of these variables. Such a graphical representation is called a scatterplot.

StudentStudentGPAMotivation

Joe2.050

Lisa2.048

Mary2.0100

Sam2.012

Deana2.334

Sarah2.630

Jennifer2.678

Gregory3.087

Thomas3.184

Cindy3.275

Martha3.683

Steve3.890

Jamell3.890

Tammie4.098

Scatterplot Example

and their GPA is being investigated. ƒ The table on the left includes a small group of individuals for whom GPA and scores on a motivation scale have been recorded. GPAs can range from 0 to 4 and motivation scores in this example range from 0 to 100. Individuals in this table were ordered based on their GPA. ƒ Simply looking at the table shows that, in general, as GPA increases, motivation scores also increase. ƒ However, with a real set of data, which may have hundreds or even thousands of individuals, a pattern cannot be detected by simply looking at the numbers. Therefore, a very useful strategy is to represent the two variables graphically to illustrate the relationship between them. ƒ A graphical representation of individual scores on two variables is called a scatterplot. ƒ The image on the right is an example of a scatterplot and displays the data from the table on the left. GPA scores are displayed on the horizontal axis and motivation scores are displayed on the vertical axis. ƒ Each dot on the scatterplot represents one individual from the data set. The location of each point on the graph depends on both the GPA and motivation scores. Individuals with higher GPAs are located further to the right and individuals with higher motivation scores are located higher up on the graph. ƒ Sam, for example, has a GPA of 2 so his point is located at 2 on the right. He also has a motivation score of 12, so his point is located at 12 going up. ƒ Scatterplots are not meant to be used in great detail because there are usually hundreds of individuals in a data set. ƒ The purpose of a scatterplot is to provide a general illustration of the relationship between the two variables. motivation score. ƒ One of the students in this example does not seem to follow the general pattern: Mary. She is one of the students with the lowest GPA, but she has the maximum score on the motivation scale. This makes her an exception or an outlier.

Interpreting Scatterplots

How to Examine a Scatterplot

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Interpreting Scatterplots: Direction

ƒ One important component to a scatterplot is the direction of the relationship between the two variables.

This example compares

motivation and their GPA.

These two variables have a

positive association because as GPA increases, so does motivation.

This example compares

of absences. These two variables have a negative association because, in general, absences decreases, their GPA increases.

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Interpreting Scatterplots: Form

ƒ Another important component to a scatterplot is the form of the relationship between the two variables.

This example illustrates a linear

relationship. This means that the points on the scatterplot closely resemble a straight line. A relationship is linear if one variable increases by approximately the same rate as the other variables changes by one unit.

This example illustrates a

relationship that has the form of a curve, rather than a straight line.

This is due to the fact that one

variable does not increase at a constant rate and may even start decreasing after a certain point.

This example describes a

curvilinear relationship between the variable Dzagedz and the variable

Dzworking memory.dz  -Š‹•

example, working memory increases throughout childhood, remains steady in adulthood, and begins decreasing around age 50.

Interpreting Scatterplots: Strength

ƒ Another important component to a scatterplot is the strength of the relationship between the two variables. ƒ The slope provides information on the strength of the relationship. ƒ The strongest linear relationship occurs when the slope is 1. This means that when one variable increases by one, the other variable also increases by the same amount. This line is at a 45 degree angle. ƒ The strength of the relationship between two variables is a crucial piece of information. Relying on the interpretation of a scatterplot is too subjective. More precise evidence is needed, and this evidence is obtained by computing a coefficient that measures the strength of the relationship under investigation.

Measuring Linear Association

ƒ A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. ƒ A correlation coefficient measures the strength of that relationship. ƒ Calculating a Pearson correlation coefficient requires the assumption that the relationship between the two variables is linear. ƒ There is a rule of thumb for interpreting the strength of a relationship based on its r value (use the absolute value of the r value to make all values positive):

Absolute Value of r Strength of Relationship

r < 0.3 None or very weak

0.3 < r <0.5 Weak

0.5 < r < 0.7 Moderate

r > 0.7 Strong ƒ The relationship between two variables is generally considered strong when their r value is larger than 0.7.

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Correlations

Example: There is a moderate, positive, linear relationship between GPA and achievement motivation. r = 0.62 ƒ Based on the criteria listed on the previous page, the value of r in this case (r = 0.62) indicates that there is a positive, linear relationship of moderate

Scatterplots and Correlation

Diana Mindrila, Ph.D.

Phoebe Balentyne, M.Ed.

Based on Chapter 4 of The Basic Practice of Statistics (6th ed.)

Concepts:

ƒ Displaying Relationships: Scatterplots

ƒ Interpreting Scatterplots

ƒ Adding Categorical Variables to Scatterplots

ƒ Measuring Linear Association: Correlation

ƒ Facts About Correlation

Objectives:

¾ Construct and interpret scatterplots.

¾ Add categorical variables to scatterplots.

¾ Calculate and interpret correlation.

¾ Describe facts about correlation.

References:

Moore, D. S., Notz, W. I, & Flinger, M. A. (2013). The basic practice of statistics (6th ed.). New York, NY: W. H. Freeman and Company.

Scatterplot

ƒ The most useful graph for displaying the relationship between two quantitative variables is a scatterplot. ƒ Many research projects are correlational studies because they investigate the relationships that may exist between variables. Prior to investigating the relationship between two quantitative variables, it is always helpful to create a graphical representation that includes both of these variables. Such a graphical representation is called a scatterplot.

StudentStudentGPAMotivation

Joe2.050

Lisa2.048

Mary2.0100

Sam2.012

Deana2.334

Sarah2.630

Jennifer2.678

Gregory3.087

Thomas3.184

Cindy3.275

Martha3.683

Steve3.890

Jamell3.890

Tammie4.098

Scatterplot Example

and their GPA is being investigated. ƒ The table on the left includes a small group of individuals for whom GPA and scores on a motivation scale have been recorded. GPAs can range from 0 to 4 and motivation scores in this example range from 0 to 100. Individuals in this table were ordered based on their GPA. ƒ Simply looking at the table shows that, in general, as GPA increases, motivation scores also increase. ƒ However, with a real set of data, which may have hundreds or even thousands of individuals, a pattern cannot be detected by simply looking at the numbers. Therefore, a very useful strategy is to represent the two variables graphically to illustrate the relationship between them. ƒ A graphical representation of individual scores on two variables is called a scatterplot. ƒ The image on the right is an example of a scatterplot and displays the data from the table on the left. GPA scores are displayed on the horizontal axis and motivation scores are displayed on the vertical axis. ƒ Each dot on the scatterplot represents one individual from the data set. The location of each point on the graph depends on both the GPA and motivation scores. Individuals with higher GPAs are located further to the right and individuals with higher motivation scores are located higher up on the graph. ƒ Sam, for example, has a GPA of 2 so his point is located at 2 on the right. He also has a motivation score of 12, so his point is located at 12 going up. ƒ Scatterplots are not meant to be used in great detail because there are usually hundreds of individuals in a data set. ƒ The purpose of a scatterplot is to provide a general illustration of the relationship between the two variables. motivation score. ƒ One of the students in this example does not seem to follow the general pattern: Mary. She is one of the students with the lowest GPA, but she has the maximum score on the motivation scale. This makes her an exception or an outlier.

Interpreting Scatterplots

How to Examine a Scatterplot

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Interpreting Scatterplots: Direction

ƒ One important component to a scatterplot is the direction of the relationship between the two variables.

This example compares

motivation and their GPA.

These two variables have a

positive association because as GPA increases, so does motivation.

This example compares

of absences. These two variables have a negative association because, in general, absences decreases, their GPA increases.

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Interpreting Scatterplots: Form

ƒ Another important component to a scatterplot is the form of the relationship between the two variables.

This example illustrates a linear

relationship. This means that the points on the scatterplot closely resemble a straight line. A relationship is linear if one variable increases by approximately the same rate as the other variables changes by one unit.

This example illustrates a

relationship that has the form of a curve, rather than a straight line.

This is due to the fact that one

variable does not increase at a constant rate and may even start decreasing after a certain point.

This example describes a

curvilinear relationship between the variable Dzagedz and the variable

Dzworking memory.dz  -Š‹•

example, working memory increases throughout childhood, remains steady in adulthood, and begins decreasing around age 50.

Interpreting Scatterplots: Strength

ƒ Another important component to a scatterplot is the strength of the relationship between the two variables. ƒ The slope provides information on the strength of the relationship. ƒ The strongest linear relationship occurs when the slope is 1. This means that when one variable increases by one, the other variable also increases by the same amount. This line is at a 45 degree angle. ƒ The strength of the relationship between two variables is a crucial piece of information. Relying on the interpretation of a scatterplot is too subjective. More precise evidence is needed, and this evidence is obtained by computing a coefficient that measures the strength of the relationship under investigation.

Measuring Linear Association

ƒ A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. ƒ A correlation coefficient measures the strength of that relationship. ƒ Calculating a Pearson correlation coefficient requires the assumption that the relationship between the two variables is linear. ƒ There is a rule of thumb for interpreting the strength of a relationship based on its r value (use the absolute value of the r value to make all values positive):

Absolute Value of r Strength of Relationship

r < 0.3 None or very weak

0.3 < r <0.5 Weak

0.5 < r < 0.7 Moderate

r > 0.7 Strong ƒ The relationship between two variables is generally considered strong when their r value is larger than 0.7.

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Correlations

Example: There is a moderate, positive, linear relationship between GPA and achievement motivation. r = 0.62 ƒ Based on the criteria listed on the previous page, the value of r in this case (r = 0.62) indicates that there is a positive, linear relationship of moderate