Chapter 19. Positive and Negative Correlations

Learning Objectives

correlation
a way of measuring the relationship between two variables
correlation coefficient (r)
a statistic that indicates the precise numerical relationship between two variables; r can range from –1.0 to +1.0
negative correlation
a relationship between two variables, such that people or things with high scores on one variable tend to have low scores on the other variable
no correlation
no apparent relationship between two variables; people or things with high scores on one variable are equally likely to have low OR high scores on the other variable
positive correlation
a relationship between two variables, such that people or things with high scores on one variable tend to have high scores on the other variable
scatterplot
a graph showing the relationship between two variables; each point on the graph represents one person or thing, with the location of the point determined by the score on each of the variables
statistic
a calculated number that summarizes important information about a distribution of scores
variable
anything that can vary, or take different values
Positive and Negative Correlations
true
true
true
Learning Objectives:

Identify scatterplots associated with the three types of relationships between variables.

Understand that correlation does not imply causation.

Review

concept_review

Review

Select the NEXT button to continue with the Review.

The graph is a scatterplot. The horizontal X axis shows Grade Point Average (GPA). The vertical Y axis shows hours of study. The graph has approximately 30 points scattered in a pattern beginning in the lower-left corner where GPA and hours of study are the lowest. The pattern extends toward the upper-right corner where GPA and hours of study are highest.

1. Correlation is a statistical measure of the strength of the relationship between two variables—that is, the extent to which scores on the two variables go up or down together. A scatterplot provides a visual representation of the relationship. This scatterplot shows the relationship between study hours per week and GPA.

Review

concept_review

Review

Select the NEXT button to continue with the Review.

The graph is a scatterplot. The horizontal X axis shows Height in inches ranging from 60 to 75 in increments of 5. The vertical Y axis shows Weight in pounds ranging from 90 to 200 in increments of 40. The graph has 10 points tightly scattered in a pattern beginning in the lower-left corner where height and weight are the lowest. The pattern extends toward the upper-right corner where height and weight are highest.

2. Two variables are positively correlated if they systematically vary in the same direction, increasing or decreasing together. In this example, a person with a high score on the height variable would also tend to have a high score on the weight variable.

Review

concept_review

Review

Select the NEXT button to continue with the Review.

The graph is a scatterplot. The horizontal X axis shows Number of Drinks ranging from 0 to 10 in increments of 2. The vertical Y axis shows Dexterity Test Score ranging from 0 to 100 in increments of 20. The graph has 11 points tightly scattered in a pattern beginning in the upper-left corner where number of drinks is the lowest but dexterity test score is the highest. The pattern extends toward the lower-right corner where number of drinks is the highest but dexterity test score is the lowest.

3. Two variables are negatively correlated if they vary systematically in the opposite direction, with one increasing while the other decreases. In this example of drinking and dexterity (physical coordination), a person with a high score on one variable would tend to have a low score on the other variable.

Review

concept_review

Review

Select the NEXT button to continue with the Review.

The graph is a scatterplot. The horizontal X axis shows Height in inches ranging from 40 to 80 in increments of 10. The vertical Y axis shows Grade point average (GPA) ranging from 0.0 to 4.0 in increments of 1. The graph has approximately 50 points loosely scattered throughout the entire graph with no discernable pattern.

4. If two variables have no correlation (zero correlation), a person with a high score on one variable (such as height) is equally likely to have either a high or a low score on the other variable (such as GPA). Knowing one score tells you nothing about the other score.

Review

concept_review

Review

Select the NEXT button to continue with the Review.

The graph is a scatterplot. The horizontal X axis shows Height in inches ranging from 140 to 190 in increments of 10. The vertical Y axis shows Arm Span in inches ranging from 140 to 190 in increments of 10. The graph has 15 points  scattered in a pattern beginning just above the lower-left corner where height and arm span is around 150 inches. The pattern extends toward the upper-right corner where height and arm span are the highest. A dotted line runs through the points from the Y axis, where the arm span is at 150 inches and the height is 140 inches, to the point where both the arm span and height is 190 inches. A vertical arrow extends from the X axis at 157 inches to the dotted line, where a horizontal arrow extends to the Y axis at 163 inches.

5. If two variables are correlated (either positively or negatively), then a person's score on one variable (such as a person’s height) can be used to predict or estimate the likely score on the other variable (such as a person’s arm span).

Review

concept_review

Review

Select the NEXT button to continue with the Review.

The graph is a scatterplot. The horizontal X axis shows Computer Science Ph.D. degrees awarded ranging from 700 to 1,900 in increments of 200. The vertical Y axis shows Traffic Fatalities in the United States ranging from thirty thousand to forty-six thousand in increments of two thousand. The graph has 9 points scattered in a pattern beginning in the upper-left corner where degrees awarded is the lowest but traffic fatalities are the highest. The pattern extends toward the lower-right corner where degrees awarded is the highest but traffic fatalities are the lowest. Each point has a year above or below it. The years increase from left to right in increments of 1 year from 2002 to 2010.

6. A correlation between two variables does not indicate that one variable influences the other variable. In other words, correlation does not imply causation. For example, between the years 2002 and 2010, U.S. traffic fatalities fell steadily as the number of doctoral degrees in computer science increased (a negative correlation), but no one believes that the number of computer scientists had anything to do with this reduction in traffic accidents!

Practice 1: Constructing a Scatterplot

line_drawing
true

Practice 1: Constructing a Scatterplot

Select each of the student names to plot that student’s location on the graph. Then, select the NEXT button and move to Practice 2.

A correlation describes a relationship between two variables. Correlations are usually shown on a graph called a scatterplot. Each point on a scatterplot represents a single person or thing.

We can construct a scatterplot from the scores in this table. As you select each student name, that student's score for each variable is shown as a dotted line, with a point for that student added to the graph at the intersection of the two lines.

An empty scatterplot appears on the right side of the screen. The horizontal X axis shows Variable 1 ranging from 0 to 70 in increments of 10. The vertical Y axis shows Variable 2 ranging from 0 to 70 in increments of 10. A table with 10 students' names and their scores on Variables 1 and 2 appears on the left side of the screen. The data are summarized in the following table. As you click on a student in the table, a red dot appears in the graph where two dotted lines intersect. The first line runs vertically from the x-axis to the dot, while the second line runs horizontally from the y-axis to the dot. This intersection is the point where the scores on the two variables meet. When all students have been selected the graph has 10 points tightly scattered in a pattern beginning in the lower-left corner where Varaibles 1 and 2 are the lowest. The pattern extends toward the upper-right corner where Variables 1 and 2 are highest.

Practice 2: The Correlation Coefficient

practice_2
true

Practice 2: The Correlation Coefficient

Select each button to see examples of the three types of scatterplots. Then, select the NEXT button and move to Quiz 1.

The correlation coefficient (r) is a precise measure of the strength of the relationship between the two variables. The relationship can take three forms: positive correlation (r up to +1.0), negative correlation (r as low as –1.0) or no correlation (r near zero).

An empty scatterplot appears in the middle of the screen. The horizontal X axis shows Variable 1 ranging from 0 to 70 in increments of 10. The vertical Y axis shows Variable 2 ranging from 0 to 70 in increments of 10. The following boxes appear on the left side of the screen: positive correlation, negative correlation, and no correlation. When a box is selected points appear in the graph as a scatterplot representing that correlation. Details about the correlation appear to the right of the graph. The following describes each graph and the details provided. When positive correlation is selected: 10 points are tightly scattered in a pattern beginning in the lower-left corner where Varaibles 1 and 2 are the lowest. The pattern extends toward the upper-right corner where Variables 1 and 2 are highest. This scatterplot shows a strong positive correlation. People who scored high on Variable 1 tended to also have high scores on Variable 2, and vice versa. The correlation, represented by the letter r, is positive 0.91. When negative correlation is selected: 10 points are tightly scattered in a pattern beginning in the upper-left corner where Variable 1 is the lowest but Variable 2 is the highest. The pattern extends toward the lower-right corner where Variable 1 is the highest but Variable 2 is the lowest. This scatterplot shows a strong negative correlation. People who scored high on Variable 1 tended to have low scores on Variable 2. The correlation, represented by the letter r, is negative 0.96. When no correlation is selected: 10 points are loosely scattered throughout the graph with no discernable pattern. This scatterplot shows no correlation between the two variables. The correlation, represented by the letter r, is positive 0.03, which is near zero.

Quiz 1

dnd_test

Quiz 1

Drag each term at the top of the screen to the gray area above the appropriate scatterplot. When all the terms have been placed, select the CHECK ANSWER button.

Select the NEXT button and move to Quiz 2.
Perhaps you should go back to review types of scatterplots.
positive correlation
no correlation
negative correlation

Quiz 2

matching_test

Quiz 2

Match the terms to the descriptions by dragging each colored circle to the appropriate gray circle. When all the circles have been placed, select the CHECK ANSWER button.

Select the NEXT button and move to the Conclusion.
Perhaps you should go back to review types of scatterplots.
positive correlation
negative correlation
no correlation
People with high scores on one variable tend to have low scores on the other variable.
People with high scores on one variable are equally likely to have low OR high scores on the other variable.
People with high scores on one variable tend to have high scores on the other variable.

Conclusion

end_slide
Congratulations!

You have completed the activity Positive and Negative Correlations.