## Sampling Distribution of r

Now, we'll talk about how the shape of the sampling distribution of Pearson correlation deviates from normality and then discusses how to transform $r$ to a normally distributed quantity. Then, we will discuss how to calculate the probability of obtaining an $r$ above a specified value.

### Sampling Distribution of Pearson's r

1. Skewed- Unless $r = 0$, the sampling distribution is skewed. The reason for the skew is that $r$ cannot take on values greater than $1.0$ or less than $-1.0$, and therefore the distribution cannot extend as far in one direction as it can in the other.
2. $-.775$ Use a calculator or table to transform $r$ to $z$'. You get $-.775$.
3. $r = -.8$ Although you could plug all of these values into the $r$ to $z$' calculator, you don't need to do that. You know the $r$ with the biggest absolute value has the most skewed sampling distribution, so it is the most different from its corresponding $z$'.
4. $.718$ Population correlation = $.6$ to $z$' = $.693$, SE = $1/sqrt(N-3)$ = $1/sqrt(19-3)$ = $.25$, $r$ = $.50$ to $z$' = $.549$, Use the "Calculate area for a given $X$" applet. Plug in $.693$ for the mean and $.25$ for the SD, and then calculate area above $.549$. You get $.718$.