Sampling Distribution of Pearson's r
of the Pearson Correlation, Introduction
to Sampling Distributions
- State how the shape of the sampling distribution of r deviates from
- Transform r to z'
- Compute the standard error of z'
- Calculate the probability of obtaining an r above a specified value
The shape of the sampling distribution of r for
an 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 ρ, the more pronounced the skew.
|Figure 1. The sampling distribution of
r for N = 12 and ρ = 0.60.
The statistician Fisher developed a way to transform
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' = 0.5 ln[(1+r)/(1-r)]
z' is normally distributed and has a standard
where N is the number of pairs of scores.