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 transformto a normally distributed quantity. Then, we will discuss how to calculate the probability of obtaining an above a specified value.
Sampling Distribution of Pearson's r
1. State how the shape of the sampling distribution of
2. Transform to '
3. Compute the standard error of '
4. Calculate the probability of obtaining an above a specified value
Assume that the correlation between quantitative and verbal SAT scores in a given population is
The shape of the sampling distribution of 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 cannot take on values greater than and therefore the distribution cannot extend as far in the positive direction as it can in the negative direction. The greater the value of , the more pronounced the skew.
Figure 1. The sampling distribution of
Figure 2 shows the sampling distribution for
Figure 2. The sampling distribution of
Referring back to the SAT example, suppose you wanted to know the probability that in a sample of
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' is normally distributed and has a standard error of
whereis the number of pairs of scores.
Let's return to the question of determining the probability of getting a sample correlation ofor above in a sample of from a population with a correlation of . The first step is to convert both and to their ' values, which are and , respectively. The standard error of ' for is Therefore the question is reduced to the following: given a normal distribution with a mean of and a standard deviation of , what is the probability of obtaining a value of or higher? The answer can be found directly from the applet "Calculate Area for a given " to be . Alternatively, you could use the formula:
and use a table to find that the area above
Source: David M. Lane, https://onlinestatbook.com/2/sampling_distributions/samp_dist_r.html
This work is in the Public Domain.