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Measures of Association/Correlation Coefficient

In many situations, researchers are interested in evaluating the relationship between variables of interest. Such associations are important for testing theories and hypotheses in which changes in one variable are tied to changes in another. In other words, is an increase in one variable associated with a systematic increase or decrease in the other? The most frequently reported measure of association within industrial and organizational psychology is the correlation coefficient (r). Correlation is a standardized index of the extent to which two sets of scores vary together. As an index, correlation can vary between −1.00 (i.e., a perfect negative relationship) and +1.00 (i.e., a perfect positive relationship). Correlations near zero indicate the absence of a linear relationship between the variables of interest. Squaring the correlation (i.e., r2) provides an indication of the percentage of variance in one variable that can be explained by the other variable. For example, if the correlation between height and weight is .5, then 25% of the variance in height can be explained by weight, or vice versa.

Numerical Representation of Correlations

The correlation between two variables can be described in one of two ways: numerically or graphically. The following example illustrates how correlation is computed and what the numerical value indicates. First, assume that five individuals respond totwomeasures (x and y). Scores for the five individuals on x are 2, 4, 8, 6, and 5, and scores on y are 5, 7, 6, 8, and 4. Thus, the mean of x is equal to 5 (i.e., 25/5) and the mean of y is 6 (i.e., 30/5).

As a second step, deviation scores can be computed for each person on each variable by subtracting the mean of each distribution from each raw score. As a result, the deviation scores for x are −3, −1, 3, 1, and 0, and the deviation scores for y are −1, 1, 0, 2, and −2. These deviation scores can then be used to compute the standard deviation and variances for these measures.

With the current data, these values are as follows:

None

In addition to computing standard deviations and variances, deviation scores can also be used to construct a matrix of these data. Because there are only two variables, it will be a simple 2 × 2 matrix that describes all possible relationships among the data. To create the matrix, take the sums of the cross-products that are produced by cross-multiplying the deviation scores—that is x2, y2, and xy.

None

In matrix form, these values can be represented as shown in Table 1.

Table 1 Cross-Products Matrix
xy
x204
y410

The matrix in Table 1 can then be transformed into a variance-covariance matrix (see Table 2) by dividing the elements by the number of cases (e.g., 20/5, 4/5, 10/5).

Table 2 Variance-Covariance Matrix
xy
x4.00.8
y0.82.0

Importantly, the covariance in this example (0.8) is a measure of association between the two variables of interest. An important consideration in using covariance as a measure of association, however, is that it is fundamentally related to the scales of measurement for both x and y and therefore unstandardized. To get the correlation matrix, divide each of the elements by the product of the standard deviations of the variables involved. For

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