For a sample correlation that uses a sample from a bivariate normal distribution with correlation , the statistic

has a Student’s distribution with () degrees of freedom.
With the monotone transformation of the correlation (Fisher, 1921)

the statistic has an approximate normal distribution with mean and variance


where .
For the transformed , the approximate variance is independent of the correlation . Furthermore, even the distribution of is not strictly normal, it tends to normality rapidly as the sample size increases for any values of (Fisher, 1973, pp. 200–201).
For the null hypothesis , the values are computed by treating

as a normal random variable with mean zero and variance , where (Fisher 1973, p. 207; Anderson 1984, p. 123).
Note that the bias adjustment, , is always used when computing values under the null hypothesis in the CORR procedure.
The ALPHA= option in the FISHER option specifies the value for the confidence level , the RHO0= option specifies the value in the hypothesis , and the BIASADJ= option specifies whether the bias adjustment is to be used for the confidence limits.
The TYPE= option specifies the type of confidence limits. The TYPE=TWOSIDED option requests twosided confidence limits and a value under the hypothesis . For a onesided confidence limit, the TYPE=LOWER option requests a lower confidence limit and a value under the hypothesis , and the TYPE=UPPER option requests an upper confidence limit and a value under the hypothesis .
The confidence limits for the correlation are derived through the confidence limits for the parameter , with or without the bias adjustment.
Without a bias adjustment, confidence limits for are computed by treating

as having a normal distribution with mean zero and variance .
That is, the twosided confidence limits for are computed as


where is the percentage point of the standard normal distribution.
With a bias adjustment, confidence limits for are computed by treating

as having a normal distribution with mean zero and variance , where the bias adjustment function (Keeping, 1962, p. 308) is

That is, the twosided confidence limits for are computed as


These computed confidence limits of and are then transformed back to derive the confidence limits for the correlation :


Note that with a bias adjustment, the CORR procedure also displays the following correlation estimate:

Fisher (1973, p. 199) describes the following practical applications of the transformation:
testing whether a population correlation is equal to a given value
testing for equality of two population correlations
combining correlation estimates from different samples
To test if a population correlation from a sample of observations with sample correlation is equal to a given , first apply the transformation to and : and .
The value is then computed by treating

as a normal random variable with mean zero and variance .
Assume that sample correlations and are computed from two independent samples of and observations, respectively. To test whether the two corresponding population correlations, and , are equal, first apply the transformation to the two sample correlations: and .
The value is derived under the null hypothesis of equal correlation. That is, the difference is distributed as a normal random variable with mean zero and variance .
Assuming further that the two samples are from populations with identical correlation, a combined correlation estimate can be computed. The weighted average of the corresponding values is

where the weights are inversely proportional to their variances.
Thus, a combined correlation estimate is and . See Example 2.4 for further illustrations of these applications.
Note that this approach can be extended to include more than two samples.