Is homoscedasticity an assumption for Pearson's correlation? I'm running correlation analysis in SPSS between my variables and I'm starting by checking the assumptions to run Pearson's correlation (r). I'm confused as to whether or not homoscedasticity is one of the assumptions for Pearson's correlation. In some places online it says that homoscedasticity isn't an assumption for Pearson's correlation, but in other places it says that it is an assumption.
Also, in the case that it is an assumption, is there an easier way to test for homoscedasticity in SPSS than having to observe every single scatterplot between all my variables (I have a ton)? Is there some sort of test that tests for homoscedasticity? For example, when testing for normality there is the Shapiro-wilk test. Is there something like that for homoscedasticity?
Thanks,
FBH

Please forgive me for being a tad pedantic here at the start of this answer.  But, Pearson's correlation is a defined quantity.  As such, it is what it is, and the value itself does not have any assumptions.  In particular, this is just the scaled covariance of any 2 variables.
However, if you wish to test if the correlation (of a given population) is some value based on a sample, then there are assumptions that this test would require.  Such a test would assume a bivariate normal distribution among the variables.
So, homoscedasticity is indirectly an assumption for such distributions.  But, the real reason people might argue that this is an assumption is because the correlation is often used as a measure of the linear relationship.  And, borrowing from regression, it would be an assumption for regression analyses (and then by extension, might be assumed for correlation analyses).
