In health sciences many variables may exhibit half-normal distribution. For example an inflammation process regarding one cell type or population in a tissue sample. The lack of the inflammatory cells describes the lack of disease and with increasing cell count in a sample the disease process becomes more severe and finally terminal. Moreover, a mild disease (small cell count) is more common than severe disease (high cell count) and thus the distribution of cell count is half-normal.

Assuming I have a variable with half-normal distribution and another variable which is normally distributed. Is it valid to use Pearson correlation coefficient with these two variables?

  • 3
    $\begingroup$ What is intended by "valid" there? It's possible for there to be a linear relationship (in that if $X$ is half normal and $Y$ be conditionally normal, its possible to have $E(Y|X)$ be linear - easily so, indeed I just generated some data like that now, though it looks as though $E(X|Y)$ will be curved). However, I wouldn't necessarily expect that you'd be able to get the marginal distribution of $Y$ to be normal with a linear relationship. If the relationship is not linear, the Pearson will only capture part of the relationship. $\endgroup$
    – Glen_b
    Jul 27, 2015 at 8:11
  • $\begingroup$ @Glen_b thanks! I guess with 'valid' I intend to mean I am not violating any critical assumptions needed to meet. My data has its llimitations (half-normal) and I guess I must deal with some amount of uncertainty. I was unsure whether to choose pearson or non-parametric methods. What would be the consequence of non-normal marginal distribution for Y. $\endgroup$
    – arkiaamu
    Jul 29, 2015 at 8:42
  • 1
    $\begingroup$ It's not that the marginal for Y being non-normal is a problem; it's that if you're saying that for you the marginal distribution for $Y$ is normal, and that for $X$ is half-normal, the conditional distribution for $Y$ won't be normal (and that's not necessarily a problem either; the bigger issue is whether the relationship you're trying to measure is really linear, or (perhaps) whether you're trying to measure the linear part of a non-linear relationship, or indeed whether you want to measure something else, such as monotonic relationship. $\endgroup$
    – Glen_b
    Jul 29, 2015 at 9:29

1 Answer 1


The Pearson correlation coefficient is valid for comparing two samples drawn from distributions with finite means and variances, in that it estimates a linear relationship between them.

If the two samples are drawn from a bivariate normal distribution then the Pearson correlation coefficient has nice properties, otherwise there may be some bias in the estimator (unless the sample sizes are very large).

  • $\begingroup$ Note that a biased estimate is not necessarily bad. In fact, the usual estimate of standard deviation, $s$, is biased, and we’re usually content to use it. $\endgroup$
    – Dave
    Sep 8, 2020 at 11:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.