A significance test for Pearson's r is described in the section inferential statistics for b and r. The significance test described in that section assumes normality. This section describes a method for testing the significance of r that makes no distributional assumptions.

Table 1. Example data.

X

Y

1.0

1.0

2.4

2.0

3.8

2.3

4.0

3.7

11.0

2.5

The approach is to consider the X variable fixed and compare the correlation obtained in the actual data to the correlations that could be obtained by rearranging the Y variable. For the data shown in Table 1, the correlation between X and Y is 0.385. There is only one arrangement of Y that would produce a higher correlation. This arrangement is shown in Table 2 and the r is 0.945. Therefore, there are two arrangements of Y that lead to correlations as high or higher than the actual data.

Table 2. The example data arranged to give the highest r.

X

Y

1.0

1.0

2.4

2.0

3.8

2.3

4.0

2.5

11.0

3.7

The next step is to calculate the number of possible arrangements of Y. The number is simply N!, where N is the number of pairs of scores. Here, the number of arrangements is 5! = 120. Therefore, the probability value is 2/120 = 0.017. Note that this is a one-tailed probability since it is the proportion of arrangements that give an r as large or larger. For the two-tailed probability, you would also count arrangements for which the value of r were less than or equal to -0.385. In randomization tests, the two-tailed probability is not necessarily double the one-tailed probability.