Sampling Distribution of Pearson's r


David M. Lane


Values of the Pearson Correlation, Introduction to Sampling Distributions

Learning Objectives
  1. State how the shape of the sampling distribution of r deviates from normality
  2. Transform r to z'
  3. Compute the standard error of z'
  4. 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 error of

where N is the number of pairs of scores.

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