Linear Regression and Correlation: Scatter Plots

Module by: Susan Dean, Dr. Barbara Illowsky

Summary: This module provides an overview of Linear Regression and Correlation: Scatter Plots as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

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

Subfigure 1.1: Table showing the number of m-commerce users (in millions) by year. Subfigure 1.2: 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 shows the direction and strength of a relationship between the variables. A clear direction happens when there is either:

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 Subfigure 2.1 Subfigure 2.2

Figure 3

Negative Linear Pattern (Strong) Negative Linear Pattern (Weak) Subfigure 3.1 Subfigure 3.2

Figure 4

Exponential Growth Pattern No Pattern Subfigure 4.1 Subfigure 4.2

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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Last edited by James Cooper on Oct 27, 2008 6:31 pm GMT-5.