Notice that the 95 percent confidence interval goes from

to

, a range that includes the

value zero. This means that the true r value could indeed be zero, which is consistent with the non-

significant p value of 0.098 that you obtained from the significance test of r in the preceding section.

Determining whether two r values are statistically significantly different

Suppose that you have two correlation coefficients and you want to test whether they are statistically

significantly different. It doesn’t matter whether the two r values are based on the same variables or

are from the same group of participants. Imagine that a significance test for comparing two correlation

coefficient values (which we will call

and

) that were obtained from

and

participants,

respectively. You can utilize the Fisher z transformation to get

and

. The difference (

) has a

standard error (SE) of

. You obtain the test statistic for the

comparison by dividing the difference by its SE. You can convert this to a p value by referring to a

table (or web page) of the normal distribution.

For example, if you want to compare an r1 value of 0.4 based on an N1 of 100 participants with an r2

value of 0.6 based on an N2 of 150 participants, you perform the following steps:

1. Calculate the Fisher z transformation of each observed r value:

2. Calculate the (

) difference:

3. Calculate the SE of the (

) difference:

4. Calculate the test statistic:

5. Look up 2.05 in a normal distribution table or web page such as

https://statpages.info/pdfs.html(or edit and run the R code provided earlier in “Testing

whether r is statistically significantly different from zero”), and observe that the p value is

0.039 for a two-sided test.

A two-sided test is used when you’re interested in knowing whether either r is larger than the

other. The p value of 0.039 is less than 0.05, meaning that the two correlation coefficients are

statistically significantly different from each other at α = 0.05.

Determining the required sample size for a correlation test

If you are planning to conduct a study where the outcome is a correlation between two

variables designated X and Y, you need to be sure to enroll a large enough sample so that if the

correlation is indeed statistically significant, you have enough sample for r to show it. As

described in Chapter 11 with the t test and the ANOVA, the sample size can be estimated through