Correlation Coefficients

Introduce the concepts of positive, zero, and negative correlations with examples (eating and weight, the number of notes they take and the price of tea in China, number of suicidal thoughts and happiness). You can also have them learn about correlation coefficients by using the authors' tutorial called " The Correlator." Alternatively, you could one or more of the following online tutorials:
  • 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. )
  • A nice interactive scatterplot applet Students plot points on a graph and get to see how the correlation coefficient changes with each point they plot.
  • Another interactive scatterplot applet (geared for high school teachers; has lesson plan, including student exercises) 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). From this point, go on to 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); whether r-squared might be more informative than p values, or multiple regression.
    3. Multiple regression. Two useful references are
    Cohen, J. & Cohen, P. (1983). Applied multiple regression/correlation analysis for
    the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.

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

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