# Sampling Distribution of r

 Site: Saylor Academy Course: MA121: Introduction to Statistics Book: Sampling Distribution of r
 Printed by: Guest user Date: Tuesday, April 16, 2024, 5:00 PM

## Description

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

### Learning Objectives

1. State how the shape of the sampling distribution of $r$ deviates from normality
2. Transform $\mathrm{r}$ to $\mathrm{z}$'
3. Compute the standard error of $z$'
4. Calculate the probability of obtaining an $\mathrm{r}$ above a specified value

Assume that the correlation between quantitative and verbal SAT scores in a given population is $0.60$. In other words, $\rho=0.60$. If $12$ students were sampled randomly, the sample correlation, $r$, would not be exactly equal to $0.60$. Naturally different samples of $12$ students would yield different values of $\mathrm{r}$. The distribution of values of $r$ after repeated samples of $12$ students is the sampling distribution of $r$.

The shape of the sampling distribution of $r$ for the above example is shown in Figure 1. You can see that the sampling distribution is not symmetric: it is negatively skewed. The reason for the skew is that $r$ cannot take on values greater than $1.0$ and therefore the distribution cannot extend as far in the positive direction as it can in the negative direction. The greater the value of $\rho$, the more pronounced the skew.

Figure 1. The sampling distribution of $r$ for $N=12$ and $\rho=0.60$.

Figure 2 shows the sampling distribution for $\rho=0.90$. This distribution has a very short positive tail and a long negative tail.

Figure 2. The sampling distribution of $r$ for $N=12$ and $\rho=0.90$.

Referring back to the SAT example, suppose you wanted to know the probability that in a sample of $12$ students, the sample value of $r$ would be $0.75$ or higher. You might think that all you would need to know to compute this probability is the mean and standard error of the sampling distribution of $r$. However, since the sampling distribution is not normal, you would still not be able to solve the problem. Fortunately, the statistician Fisher developed a way to transform $\mathrm{r}$ to a variable that is normally distributed with a known standard error. The variable is called $z$' and the formula for the transformation is given below.

$z^{\prime}=0.5 \ln [(1+r) /(1-r)]$

The details of the formula are not important here since normally you will use either a table or calculator to do the transformation. What is important is that $z$' is normally distributed and has a standard error of

$\frac{1}{\sqrt{N-3}}$

where $N$ is the number of pairs of scores.

Let's return to the question of determining the probability of getting a sample correlation of $0.75$ or above in a sample of $12$ from a population with a correlation of $0.60$. The first step is to convert both $0.60$ and $0.75$ to their $z$' values, which are $0.693$ and $0.973$, respectively. The standard error of $z$' for $N=12$ is $0.333$ Therefore the question is reduced to the following: given a normal distribution with a mean of $0.693$ and a standard deviation of $0.333$, what is the probability of obtaining a value of $0.973$ or higher? The answer can be found directly from the applet "Calculate Area for a given $X$" to be $0.20$. Alternatively, you could use the formula:

$z=(x-\mu) / \sigma=(0.973-0.693) / 0.333=0.841$

and use a table to find that the area above $0.841$ is $0.20$.

Source: David M. Lane, https://onlinestatbook.com/2/sampling_distributions/samp_dist_r.html
This work is in the Public Domain.

### Questions

Question 1 out of 4.
What is the shape of the sampling distribution of $r$?

Normal

Bimodal

Skewed

Question 2 out of 4.
What is the corresponding $z$' for $r = -.65$?

Question 3 out of 4.
Which of these r values is the most different from its corresponding $z$'?

$r = 0$

$r = .6$

$r = .2$

$r = -.8$

Question 4 out of 4.
The population has a correlation of $.60$. What is the probability that you will get a sample correlation of at least $r = .50$ if you sample $19$ pairs?

### Answers

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 $.718$.