II.  The Pearson Correlation Coefficient

   A. Correlation measures:

            1.  Direction of relationship

        2.  Form of relationship

        3.  Degree of relationship

Conceptually, Pearson computes

Hours Studied (X)            Errors on Test (Y)

            0                                            19

            1                                             6

            2                                             2

            4                                             1

            4                                             4

            5                                             0

            3                                             3

            5                                             5

   B. Calculations

C.  Hypothesis testing with Pearson correlation

Ho: ρ = 0 (no population correlation) OR
In the population, there is no correlation between x and y.

H1 : ρ ≠ 0   OR
In the population, there is a correlation between x and y.

 

Figure 16-12  (p. 539)
A scatterplot of a population of X and Y values with a near-zero correlation. However, a small sample of n = 3 data points from this population shows a relatively strong, positive correlation. Data points in the sample are circled.

 

Degrees of Freedom

df = n - 2

Look up the critical values in Table B.6 (p. 709)

III.  The Spearman correlation

Measures the relationship between variables when both are measured on an ordinal scale (rank-order data)

Rank the data for x and y separately—SMALLEST score is ranked 1, etc...can rank highest score as 1..just be consistent for the two variables

To rank ties, compute the mean of their rank positions

Then use the formula from the Pearson correlation coefficient

To look up the critical value, use table B. 7 (G&W p. 710)

IV. coefficient of determination     

Indicates the strength of the relationship between x and y.  Measures the proportion of y variability that is associated with the x variable.