# Does correlation (PMCC) require variables to be normal

When calculating product moment correlation coefficient between two variables, say height and weight, is it necessary that the variables are normally distributed for the PMCC to be valid/relevant.

I've heard that the data should be bivariate. What does this mean? I've read what is on wikipedia. Does it simply mean any two data sets? Or is the definition more strict.

Edit 1

The original question I had in mind was to do with an A-Level statistics question involving Hypothesis testing. Testing for a correlation between BMR (Metabolic Rate) and BMI (Body Mass Index). We are given some data and asked to carry out a hypothesis test for H0:rho=0, H1:rho<>0

One of the follow up questions is

"What assumption must be made if the product moment correlation coefficient is to be preferred to the Spearman’s rank correlation coefficient, for the investigation of an association between BMR and BMI?"

Is it necessary in this context that BMR and BMI are normally distributed?

Thank You

## 1 Answer

There's no such thing as validity or relevance without some further context. If you have a sample consisting of pairs $(x_1,y_1),\ldots,(x_n,y_n)$ you can find a correlation between $\pm 1$ (inclusive). "Bivariate" in this context simply means the data consists of a bunch of ordered pairs. In regard to probability distributions, a bivariate distribution is the probability distribution of a random ordered pair. A bivariate probability distribution has a correlation associated with it provided the variance of each component of the pair is finite.

Now suppose for example, you have a null hypothesis that in the population from which the sample was taken, the correlation between the two variables is zero. The validity of the hypothesis test might then depend on an assumption that the population has a bivariate normal distribution, and then one could consider the question of how robust the test is, i.e. how much deviation from normality would not substantial disturb the validity of the test.

For example, there is this highly publicized study of the correlation between chocolate consumption and Nobel prizes. (I don't think it's intended to be taken literally; but perhaps it is intended to be taken seriously as giving us certain things to think about.) The claimed p-value in that study is not valid because the data make it obvious that it's not from anything close to a bivariate normal population. I would replace the $x$- and $y$-variables with their logarithms and re-do the test. In that case, one does get something like looks approximately normal, and only after taking logarithms does one see that perhaps there's something different about China and Brazil.

PS:

"What assumption must be made if the product moment correlation coefficient is to be preferred to the Spearman’s rank correlation coefficient, for the investigation of an association between BMR and BMI?"

Here I suspect they have in mind bivariate normality. The word "must" may be a bit exaggerated.

• Thanks for your answer. I've added a bit more context to the question. It was an A-Level statistics question to do with hypothesis testing. – NumberCruncher May 21 '15 at 20:14