chapter 7 scatterplots, association, and correlation © 2010 pearson education 1

Chapter 7Scatterplots, Association,

and Correlation

© 2010 Pearson Education

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A scatterplot, which plots one quantitative variable against another, can be an effective display for data.

Scatterplots are the ideal way to picture associations between two quantitative variables.


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7.1 Looking at Scatterplots

The direction of the association is important.

A pattern that runs from the upper left to the lower right is said to be negative.

A pattern running from the lower left to the upper right is called positive.


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The second thing to look for in a scatterplot is its form.

If there is a straight line relationship, it will appear as a cloud or swarm of points stretched out in a generallyconsistent, straight form. This is called linear form.

Sometimes the relationship curves gently, while still increasing or decreasing steadily; sometimes it curves sharply up then down.


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The third feature to look for in a scatterplot is the strength of the relationship.

Do the points appear tightly clustered in a single stream or do the points seem to be so variable and spread out that we can barely discern any trend or pattern?


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Finally, always look for the unexpected.

An outlier is an unusual observation, standing away from the overall pattern of the scatterplot.


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7.2 Assigning Roles to Variables in Scatterplots

To make a scatterplot of two quantitative variables, assign one to the y-axis and the other to the x-axis.

Be sure to label the axes clearly, and indicate the scales of the axes with numbers.

Each variable has units, and these should appear with the display—usually near each axis.


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Each point is placed on a scatterplot at a position that corresponds to values of the two variables.

The point’s horizontal location is specified by its x-value, and its vertical location is specified by its y-value variable.

Together, these variables are known as coordinates and written (x, y).


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One variable plays the role of the explanatory or predictor variable, while the other takes on the role of the response variable.

We place the explanatory variable on the x-axis and the response variable on the y-axis.

The x- and y-variables are sometimes referred to as the independent and dependent variables, respectively.


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7.3 Understanding Correlation

The ratio of the sum of the product zxzy for every point in the scatterplot to n – 1is called the correlation coefficient.

1x yz z

rn

Two of the more common alternative formulas for correlation are:

2 2 1 x y

x x y y x x y yr

n s sx x y y


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Correlation Conditions

Correlation measures the strength of the linear association between two quantitative variables.


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Correlation Conditions

Before you use correlation, you must check three conditions:

• Quantitative Variables Condition: Correlationapplies only to quantitative variables.

• Linearity Condition: Correlation measures the strength only of the linear association.

• Outlier Condition: Unusual observations can distortthe correlation.


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Correlation Properties

• The sign of a correlation coefficient gives the direction of the association.

• Correlation is always between –1 and +1.

• Correlation treats x and y symmetrically.

• Correlation has no units.


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Correlation Properties

• Correlation is not affected by changes in the center or scale of either variable.

• Correlation measures the strength of the linear association between the two variables.

• Correlation is sensitive to unusual observations.


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Correlation Tables

Sometimes the correlations between each pair of variables in a data set are arranged in a table like the one below.


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*7.4 Straightening Scatterplots

Example: After the Dow Jones Industrial Average, the S&P 500 is the most widely-watched index of U.S. stocks.

The time series plot of the data does not seem to indicate a strong linear association:


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However, if we look at the logarithm of the S&P 500 over Time:

the plot looks straighter, so the correlation is now a more appropriate measure of association.


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Simple transformations such as the logarithm, square root, or reciprocal can sometimes straighten a scatterplot’s form.


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7.5 Lurking Variables and Causation

There is no way to conclude from a high correlation alonethat one variable causes the other.

There’s always the possibility that some third variable—a lurking variable—is simultaneously affecting both of the variables you have observed.


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What Can Go Wrong?

• Don’t say “correlation” when you mean “association.”

• Don’t correlate categorical variables.

• Make sure the association is linear.

• Beware of outliers.

• Don’t confuse correlation with causation.

• Watch out for lurking variables.


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What Have We Learned?

• Begin our investigation by looking at a scatterplot.

• The sign of the correlation tells us the direction of the association.

• The magnitude of the correlation tells us of the strength of a linear association.

• Correlation has no units, so shifting or scaling the data, standardizing, or even swapping the variables has no effect on the numerical value.


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What Have We Learned?

To use correlation we have to check certain conditions forthe analysis to be valid:

• Check the Linearity Condition.

• Watch out for unusual observations.

We’ve learned not to make the mistake of assuming that a high correlation or strong association is evidence of a cause-and-effect relationship.

chapter 7 scatterplots, association, and correlation © 2010 pearson education 1

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dependent variables

response variable

explanatory variable

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predictor variable

yvalue variable

assigning roles