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Chapter 4: Describing the Relationship Between Two Variables4.1 Scatter Diagrams and Correlation4.2 Least-Squares Regression4.3 Diagnostics on the Least-Squares Regression Line4.4 Contingency Tables and Association
In Chapter 3, we looked at numerically summarizing data from one variable (univariate data), but newspaperarticles and studies frequently describe the relationship between two variables (bivariate data). It's this secondclass that we'll be focusing on in Chapter 4.
There are plenty of variables which seem to be related. The links below are articles from various news sources, alldiscussing relationships between two variables.
Do SAT Scores Really Predict Success?Range of Variables Affect How SAT Correlates to College GPA
Proximity to highways affects newborns' health: study
Study: Weight-loss surgery cuts cancer risk in women
Our goal in this chapter will be to find ways to describe relationships like the one between a student's SAT scoreand his/her GPA, and to describe the strength of that relationship.
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Objectives
Example 1
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Section 4.1: Scatter Diagrams and Correlation4.1 Scatter Diagrams and Correlation4.2 Least-Squares Regression4.3 Diagnostics on the Least-Squares Regression Line4.4 Contingency Tables and Association
By the end of this lesson, you will be able to...
1. draw and interpret scatter diagrams2. describe the properties of the linear correlation coefficient (LCC)3. estimate the LCC based on a scatter diagram4. compute and interpret the LCC5. explain the difference between correlation and causation
In Chapter 3, we looked at numerically summarizing data from one variable (univariate data), but newspaperarticles and studies frequently describe the relationship between two variables (bivariate data). It's this secondclass that we'll be focusing on in Chapter 4.
There are plenty of variables which seem to be related. The links below are articles from various news sources, alldiscussing relationships between two variables.
Do SAT Scores Really Predict Success?Range of Variables Affect How SAT Correlates to College GPA
Proximity to highways affects newborns' health: study
Study: Weight-loss surgery cuts cancer risk in women
In each case, there's a response variable (GPA, newborn's health, cancer levels) whose value can be explainedat least in part by a predictor variable (SAT score, proximity to highways, weight-loss pill consumption).
Remember, unless we perform a designed experiment, we can only claim an association betweenthe predictor and response variables, not a causation.
Our goal in this chapter will be to find ways to describe relationships like the one between a student's SAT scoreand his/her GPA, and to describe the strength of that relationship.
First, we need a new type of graph.
Scatter Diagrams
Scatter diagrams are the easiest way to graphically represent the relationship between two quantitative variables.They're just x-y plots, with the predictor variable as the x and the response variable as the y.
The data below are heart rates of students from a Statistics I class at ECC
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during the Spring semester of 2008. Students measured their heart rates(in beats per minute), then took a brisk walk and measured their heartrates again.
before after
before after
before after86 98 58 128 60 7062 70 64 74 80 9252 56 74 106 66 7090 110 76 84 80 9266 76 56 96 78 11680 96 72 82 74 11478 86 72 78 90 11674 84 68 90 76 94
We can see that the heart rate before going on the walk is the predictor(x), and the heart rate after the walk is the response (y).
Here's an excellent video showing a scatter diagram on steroids created by the BBC:
Technology
Here's a quick overview of the steps for creating scatter diagrams in StatCrunch.
1. Select Graphics > Scatter plot2. Select quantitative variables for the X & Y axes.
You can also go to the video page for links to see videos in either Quicktime or iPod format.
Types of Relationships
Not all relationships have to be linear, like the before/after heart rate data. The images below show some of thepossibilities for the relationship (or lack thereof) between two variables.
Linear Linear
Nonlinear No relation
The price of a manufactured item and the profit the company gains from it, for example, do not have a linearrelationship. When prices
