Correlation Coefficients

• Download IBM version of " The Correlator."
• Download Mac version of " The Correlator."

• A 12-page scatterplot and correlation tutorial from our publisher. (The last page is an interactive scatterplot: Students type data into a table, click, and then get a correlation coefficient. )
• An interactive scatterplot applet (Students plot points on a graph and get to see how the correlation coefficient changes with each point they plot. Site is geared for high school teachers. Indeed, it has a lesson plan that includes student exercises.)
• Another interactive applet (Students can just estimate correlations with this one. However, they can also use it to get a sense of regression.)
• An interactive applet that allows you to see what happens as you change the size of the correlation and the sample size. Tip: To manipulate "r" and "n," you need to use the "sliders" under their respective boxes. 

Next, have every student state a relationship between the two variables and then tell us whether this relationship is positive, zero, or negative. Then, graphically demonstrate the concepts with scatterplots. Then, give them positively, zero, and negatively related data sets and have them compute Spearman's rho (because it's simpler to calculate than Pearson's r). Finally, show how Pearson's r makes better use of the data than rho when there is continuous interval data.

Once students understand the Pearson r, you could discuss the following three topics:

1. The different types of correlation coefficients. These are summarized in table 7-4 (p. 191).
2. The coefficient of determination. Begin by showing students that +1 and -1 relationships are perfect, so r-squared for each of them should be (and is) 1.00. Zero correlations indicate no relationship, so knowing one variable shouldn't help at all in knowing the other variable (and zero squared is zero). Having thus reviewed two essential points about correlation coefficients (#1 negative correlations are not weaker than positive correlations and #2 zero correlations indicate no relationship), go on to discuss the r-squared values commonly found in psychology (.09 to .25). Explain that although these r-square's indicate that we fail to account for a great deal of the variability in human behavior, that's to be expected because (1) we have measurement error, and (2) we wouldn't expect a single variable (e.g., IQ) to account for all the variance in another variable (college performance) because people are simple rather than complex. Emphasize that because r-squared values will be below 1, there will be exceptions to all rules. Thus, exceptions and "for examples" are not proof. That is, as Bradley Bushman has pointed out, the argument that video games do not promote violence because "I'm not a killer and I play video games" is not a good argument.  After the class understands r-squared, you could go on to debate--or have the class debate--
   •  the value of the SATs,
   the utility or futility of using r-squared to address the nature-nurture controversy (emphasizing the notions of correlation and causality and apparent size of effect and restriction of range); or
   • whether r-squared might be more informative than p values.
   
   
3. Multiple regression. Two useful references are

the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.

Kachigan, S. K. (1982). Multivariate statistical analysis: A conceptual introduction.