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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 scatter plot. The following example illustrates a scatter plot.

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 scatter plot. Let xx = the year and let yy = the number of m-commerce users, in millions.

Figure 1
(a) Table showing the number of m-commerce users (in millions) by year.(b) Scatter plot showing the number of m-commerce users (in millions) by year.
xx (year) yy (# of users)
2000 0.5
2002 20.0
2003 33.0
2004 47.0
A scatter plot with the x-axis representing the year and the y-axis representing the number of m-commerce users in millions. There are four points plotted, at (2000, 0.5), (2002, 20.0), (2003, 33.0), (2004, 47.0).

A scatter plot 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 scatter plot 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)
Scatterplot of 6 points in a straight ascending line from lower left to upper right.Scatterplot of 6 points in a straight ascending line from lower left to upper right with one additional point in the upper left corner.
Figure 3
Negative Linear Pattern (Strong) Negative Linear Pattern (Weak)
(a) (b)
Scatterplot of 6 points in a straight descending line from upper left to lower right.Scatterplot of 8 points in a wobbly descending line from upper left to lower right.
Figure 4
Exponential Growth Pattern No Pattern
(a) (b)
Scatterplot of 7 points in a exponential curve from along the x-axis on the left to slowly ascending up the graph in the upper right.Scatterplot of many points scattered everywhere.

In this chapter, we are interested in scatter plots 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 scatter plot. 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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