# Why isn't the sample distribution of r normal?

The sample distribution of $r$ is positively or negatively skewed if $\rho \neq 0$. However, according to the central limit theorem - the more samples you take from a population, no matter what shape the distribution is, the more normal your sampling distribution becomes. So why, despite the central limit theorem, is the sample distribution of $r$ not normally distributed?

• Which $r$? What is $\rho$? You talk about skew - so variance is finite ... if it were not then CLT in its basic form does not hold.
– Ric
Apr 24 '15 at 8:08
• The sampling distribution of $r$ is necessarily bounded by its limits $[-1,1]$. This bites, and will bite hardest when the underlying correlation is near $\pm 1$. The Fisher $z$ transformation atanh$(r)$ takes most of the the bite out of the bounds. Apr 24 '15 at 8:45
• You have a highly imprecise idea about the CLT. Apr 24 '15 at 8:49
• @Richard: I'm not exactly sure which r and ρ, since I have so little knowledge in statistics. Let's say we wanted to reject the null hypothesis ρ =0. We may take many samples of two scores X and Y, and compute the Pearson correlation r of them. The problem here is that when plotting the sampling distribution of r, we get skew if ρ =/= 0. Hence the Fisher z transformation. But we don't get skew when plotting the sampling distribution of the mean, explained in part because of the CLT. So what makes these different? Apr 24 '15 at 9:20
• I get an idea what you mean after reading your remark. But shouldn't this be formulated more clearly in the question ..beginning with $X,Y$ and their true correlation $\rho$ (if this is what you mean) ...
– Ric
Apr 24 '15 at 11:31

according to the central limit theorem - the more samples you take from a population, no matter what shape the distribution is, the more normal your sampling distribution becomes.

This is incorrect in several respects. (I expect you've been told something close to this, but it's not the case.)

• "no matter what shape the distribution is" ... not so. There are distributions for which the CLT doesn't hold (other limit theorems may apply).

• "the more samples you take [...] the more normal your sampling distribution becomes" ... the central limit theorem doesn't actually assert this. It does happen (under certain conditions), but it's not really the CLT that says that. An example of a result that does say something (more or less) along these lines would be the Berry-Esseen theorem, which gives a bound on how far the distribution of the standardized mean can be from the standard normal distribution, and which bound decreases as $1/\sqrt{n}$.

So why, despite the central limit theorem, is the sample distribution of r not normally distributed?

The sample correlation statistic $r$ can be construed as a kind of (scaled) mean (especially if you define it with an $n$ denominator), and the CLT should apply to it. But the CLT is a theorem about the limit as $n\to\infty$; you have finite $n$. At finite sample sizes the distribution won't be normal -- indeed this should be obvious, since $r$ is bounded. The bounds will tend to make it skew as you move toward one or the other.

Indeed, even the effect you thought was CLT -- the one where increasing the sample size tends to make the sampling distribution of means more normal -- applies to $r$ too. As you take larger samples, $r$ does tend to be more nearly normally distributed in a very wide variety of circumstances.

You can investigate this fairly easily using simulation.

However, while it becomes more nearly normal as $n$ increases for a given population $\rho$, it is still generally skew at every finite sample size (when $\rho\neq 0$). The same applies when taking sample means from skewed distributions --- the skewness decreases with larger samples, but it doesn't go away.

But we don't get skew when plotting the sampling distribution of the mean

Not so! If the distribution you draw samples from is skewed, the sample means will also be skewed (but typically substantially less so):

As the sample size gets larger, the skewness in the sample mean will tend to reduce.

If we drew samples from a suitable distribution on $[-1,1]$, say a scaled-shifted beta, then as the population mean approached those bounds for a given $n$, the sample mean would tend to be more skewed (left skew for a population mean near $1$, right skew for a population mean near $-1$). But as the sample size increased for a given mean, the distribution would be less skewed. The situation is similar for the sample correlation.