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Linear Regression and Correlation: Scatter Plots

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables xx and yy. The most common and easiest way is a scatterplot. The following example illustrates a scatterplot.

Example 1

From an article in the Wall Street Journal: In Europe and Asia, m-commerce is becoming more popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. In the next few years, will there be a relationship between the year and the number of m-commerce users? Construct a scatterplot. Let xx = the year and let yy = the number of m-commerce users, in millions.

Table 1
x y
2000 0.5
2002 20.0
2003 33.0
2004 47.0

Figure 1
Figure 1 (linrgs_scater1.png)

A scatterplot shows the direction and strength of a relationship between the variables. A clear direction happens when there is either

  • high values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.
  • high values of one variable occurring with low values of the other variable

You can determine the strength of the relationship by looking at the scatterplot and seeing how close the points are to a line, a power function, an exponential function, or to some other type of function.

When you look at a scatterplot, you want to notice the overall pattern and any deviations from the pattern. The following scatterplot examples illustrate these concepts.

Figure 2
Positive Linear Pattern (Strong) Linear Pattern w/ One Deviation
(a) (b)
 Positive Linear Pattern (Strong) (linrgs_scater7.png) Linear Pattern w/ One Deviation (linrgs_scater6.png)
Figure 3
Negative Linear Pattern (Strong) Negative Linear Pattern (Weak)
(a) (b)
 Negative Linear Pattern (Strong) (linrgs_scater2.png) Negative Linear Pattern (Weak) (linrgs_scater5.png)
Figure 4
Exponential Growth Pattern No Pattern
(a) (b)
 Exponential Growth Pattern (linrgs_scater3.png) No Pattern (linrgs_scater4.png)

In this chapter, we are interested in scatterplots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line. If we think that the points show a linear relationship, we would like to draw a line on the scatterplot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If xx is the independent variable and yy the dependent variable, then we can use a regression line to predict yy for a given value of xx.

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