# Alternatives to Pearson's correlation given small sample size and normality hypothesis rejected?

I would like to use Pearson's $$r$$ to determine the correlation strengths among around $$20$$ variables. Unfortunately, I had to reject the hypotheses that the data from some (around $$5$$) comes from normal distributions with Shapiro–Wilk test and $$n \in [10, 20]$$. What are some good alternatives of correlation measurement to the $$r$$ for these not-normally-distributed data? Thanks.

P.S. am using SPSS 25.

Suppose $$x = 1, 2, 3, 4, 5$$ and $$y = 2x$$ and so Pearson's $$r$$ is $$1$$.
What's being premised here is that there is a lurking property "correlation strength" and that if one measure is not ideal, we should use another. It is the other way round. Each measure has its own idea of what a strong correlation is. If it is Pearson $$r$$, it is a linear relationship. If it is Spearman's or Kendall's, it is a monotonic relationship. And there are other measures that purport to find any kind of relationship (departure from independence).
• Eh my $x$ is fixed and the Shapiro–Wilk test doesn't reject the normality hypothesis, but rejects for some of my $y$s. I look for normality of the data a lot because I want to take independent t-tests with them. Btw I'm not very familiar with bootstrapping or permutation test (esp. in SPSS). Do you have any sources about these? Jan 9, 2020 at 11:21