Comparing two Correlation Coefficients

More about this z-test for comparing two sample correlation coefficients so you can better use the results delivered by this solver.

Can you compare two correlation coefficients?

Yes, indeed. A z-test for comparing sample correlation coefficients allow you to assess whether or not a significant difference between the two sample correlation coefficients \(r_1\) and \(r_2\) exists, or in other words, that the sample correlation correspond to population correlation coefficients \(\rho_1\) \(\rho_2\) that are different from each other.

The null and alternative hypotheses to be tested in this case are:

\[H_0: \rho_1 = \rho_2\] \[H_a: \rho_1 \ne \rho_2\]

You will reject the null hypothesis when there is enough evidence to claim that the sample correlations come from population with different population correlations.

Z-test for comparing correlation coefficients

The formula for a z-statistic for two population means is:

\[z = \frac{z_1 - z_2}{\sqrt{\frac{1}{n_1-3} +\frac{1}{n_2-3} }} \]

where

\[z_1 = \frac{1}{2} \ln\left(\frac{1+r_1}{1-r_1}\right)\] \[z_2 = \frac{1}{2} \ln\left(\frac{1+r_2}{1-r_2}\right)\]

The null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed). You can also use our correlation coefficient calculator if you have sample data and you want to compute the actual correlation coefficients.