Correlation

Prerequisites
Values of the Pearson Correlation, Sampling Distribution of Pearson's r, Confidence Intervals

Learning Objectives

  1. State the standard error of z'
  2. Compute a confidence interval on ρ

The computation of a confidence interval on the population value of Pearson's correlation (ρ) is complicated by the fact that the sampling distribution of r is not normally distributed. The solution lies with Fisher's z' transformation described in the section on the sampling distribution of Pearson's r. The steps in computing a confidence interval for ρ are:

    1. Convert r to z'
    2. Compute a confidence interval in terms of z'
    3. Convert the confidence interval back to r.

The conversion of r to z' can be done using a table or calculator. The table contains only positive value of r, but that is not a problem. The value of z' associated with an r of 0.654 is 0.78. Therefore, the z' associated with an r of -0.654 is -0.78.

The sampling distribution of z' is approximately normally distributed and has a standard error of

 

The final step is to convert the endpoints of the interval back to r using a table or calculator. To calculate the 99% confidence interval, you use the Z for a 99% confidence interval of 2.58. Naturally, the 99% confidence interval is wider than the 95% confidence interval.