Math 243: Lecture File 3
N. Christopher Phillips
7 April 2009
N. Christopher Phillips () Math 243: Lecture File 3 7 April 2009 1 / 46
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Scatterplots and Correlation: Describing the Relationship between Quantitative Variables -, Study notes of Probability and Statistics
A lecture file from math 243, discussing the use of scatterplots and correlation to describe the relationship between two quantitative variables. The file includes examples of scatterplots and their analysis, as well as an explanation of the concept of correlation and its significance.
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Math 243: Lecture File 3
N. Christopher Phillips
7 April 2009
Scatterplots and correlation
How do we describe the relation between two quantitative variables?
We begin with several examples, to give the idea before giving details of definitions.
Scatterplots
Scatterplots are a graphical way to compare two variables.
Example: Heights and weights of Math 243 students. (Heights are in inches and weights are in pounds.) What kind of relationship do you expect?
Heights and weights of my Math 243 students in Fall 2003
Histograms showing heights (left) and weights (right).
60 65 70 75 80
2
4
6
8
10
12
100 150 200 250 300
5
10
15
Scatterplot: height is x, weight is y.
60 65 70 75 80
50
100
150
200
250
300
There is clearly a relationship, even though the points are scattered.
Heights and UO GPAs of my Math 243 students in Fall
What kind of relationship do you expect?
Scatterplot: height is x, UO GPA is y.
60 65 70 75 80
1
2
3
4
- Correlation r ≈
- Correlation r ≈ −
Correlation r ≈ − 0. 0896136
r 2 ≈ 0 .0080306: The change in height explains about 0.8% of the change in GPA.
Which variable is x and which is y?
Let x be the variable on the horizontal axis, and let y be the variable on the vertical axis.
Explanatory variable x and response variable y : You hope changes in x cause, or at least explain, (part of) the corresponding changes in y.
Example 1: Age and height of Douglas fir trees.
You want to know how age is related to height for Douglas fir trees.
Which is the explanatory variable and which is the response variable?
Example 2: Two IQ tests.
You have constructed a new IQ test, and you want to know whether scores on it can be used to predict scores on an older intelligence test.
Which is the explanatory variable and which is the response variable?
Example 2: Two IQ tests.
You have constructed a new IQ test, and you want to know whether scores on it can be used to predict scores on an older intelligence test.
Which is the explanatory variable and which is the response variable?
The score on the new test is the explanatory variable, even though a change in it does not cause a change in the score on the older test (especially if the older test was taken first).
So put the score on the new test on the horizontal axis and the score on the older test on the vertical axis.