15.4 Review of Concepts

The Meaning of Correlation

Correlation is an association between two variables and is quantified by a correlation coefficient. A positive correlation indicates that a participant who has a high score on one variable is likely to have a high score on the other, and someone with a low score on one variable is likely to have a low score on the other. A negative correlation indicates that someone with a high score on one variable is likely to have a low score on the other. All correlation coefficients must fall between −1.00 and 1.00. The strength of the correlation is independent of its sign.

Correlation coefficients are useful, but they can be misleading. When interpreting a correlation coefficient, we must be certain not to confuse correlation with causation. We cannot know the causal direction in which two variables are related from a correlation coefficient, nor can we know if there is a hidden third variable that causes the apparent relation.

The Pearson Correlation Coefficient

The Pearson correlation coefficient is used when two scale variables are linearly related, as determined from a scatterplot. Calculating a correlation coefficient involves three steps. (1) We calculate the deviation of each score from its mean, multiply the deviations on each variable for each participant, and sum the products of the deviations. (2) We multiply the sums of squares for each variable, then take the square root of the product. (3) We divide the sum of the products of the deviations (from step 1) by the square root of the product of the sums of squares (from step 2). We use the six steps of hypothesis testing to determine whether the correlation coefficient is statistically significantly different from 0 on the r distribution.

Applying Correlation in Psychometrics

Psychometrics is the statistics of the development of tests and measures. Psychometricians assess the reliability and validity of a test. Reliability is sometimes measured by test–retest reliability, whereby participants’ scores on the same measure at two different points in time are correlated. With coefficient alpha, a computer essentially calculates the average of all possible split-half correlations (e.g., odd and even items, first and second halves of items). Validity is sometimes assessed by correlating a new measure with existing measures that have been shown to be valid.

With partial correlation, researchers quantify the degree of association between two variables that remains when the correlations of these two variables with a third variable are mathematically eliminated. Researchers use partial correlation when a third variable may be influencing the co-relation of the first two variables.