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Chapter 4: Scatterplots and Correlation

Knowledge Prerequisites

Variables

declarative knowledge (definitions)
1. independent variable
2. dependent variable
procedural knowledge
1. how to plot points, $(x, y)$
conditional knowledge
resources

Learning Goals

declarative knowledge (definitions)
1. response variable
2. explanatory variable
3. categorical variable
4. scatterplot
5. positive/negative association
6. correlation
7. correlation coeeficient, $r$
procedural knowledge
1. how to construct a scatterplot
2. how to calculate the correlation coefficient, $r$
3. how to show categorical variables in scatterplots
conditional knowledge
1. know how construct a scatterplot
  1. correctly identify and place the explanatory and response variables
  2. appropriately label the graph
2. know how interpret a scatterplot
  1. identify clusters of data;
  2. the form of the relationship (linear is the only form we will consider in the course);
  3. the direction of the relationship: positively/negatively association;
  4. the strength of the relationship: strong ( $r$ near $\pm 1$ ) or weak ( $r$ near $0$ ).
3. be able to distinguish between types of data.
4. know the facts about correlation listed on p. 91 of text.

Questions to Ponder

What is the best way to study/prepare for the concepts in this section? Are there are other methods to study these concepts?
What questions do you need to ask the instructor?

Purpose of this Section

To study a relationship between two variables. We are often interested in comparisons among several distributions or relationships among several variables. A study of data often leads us to ask whether there is a correlation between two variables that are closely linked in the data.

General Notes

Scatterplots

To study a relationship between two variables, we need to measure both variables on the same individuals. But we need to be cautious of possible lurking variables.

General procedure for studying possible relationships between two variables:

Plot the data and add numerical summaries.
Look for overall pattern and deviations from those patterns.
Use mathematical models to describe regular patterns.

Here is how to construct a scatterplot on the TI-83. Note when drawing a scatterplot make sure both axes have the same scale!

Correlation Coefficient, $r$

Measures the strength and direction of the linear association between two quantitative variables. The correlation coefficient is calculated as follows:

r = \frac{1}{n - 1} \sum (\frac{x_{i} - \bar{x}}{s_{x}}) (\frac{y_{i} - \bar{y}}{s_{y}})

where $x_{i}$ and $y_{i}$ are are the observations of one individual, $\bar{x}$ and $\bar{y}$ are the means of the two variables, and $s_{x}$ and $s_{y}$ are the standard deviations of the two variables. The factor $(\frac{x_{i} - \bar{x}}{s_{x}})$ standardizes all the $x$ observations and $(\frac{y_{i} - \bar{y}}{s_{y}})$ standardizes all the $y$ observations. Here are steps to find the correlation coefficient using the TI-83 calculator.

IMPORTANT: Correlation is not a complete description of twovariable data. You should also include the means and standard deviations of each variable.

Resources:

Summarizing and Presenting Data: Presenting Data for Two Continuous Measurements - http://www.anu.edu.au/nceph/surfstat/surfstat-home/1-4-1.html - This page provides a discussion of scatterplots.
Guessing Correlations - http://www.stat.uiuc.edu/~stat100/java/GCApplet/GCAppletFrame.html - This applet gives you four scatterplots and four correlation coefficients. You match them up.
Correlation/Outliers Applet - http://www.math.tamu.edu/FiniteMath/FinalBuild/Classes/Correlation/ - This applet can provide you with real insight into how to read scatterplots and into how correlation coefficients can be misleading.
Correlation coefficient - http://noppa5.pc.helsinki.fi/koe/corr/cor7.html - This applet allows you to choose the numerical correlation coefficient and they give you a scatterplot with that coefficient.

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