How do I estimate a pooled correlation using multiple scores from a set of subjects?

In general from Boniface (1995) a Pearson correlation, r, from n pairs of observations can be tested using a t distribution on n-2 degrees of freedom and a test statistic equal to

r sqrt[(n-2) / (1-r²) ].

Equivalently the square of this test statistic may be compared to a F distribution having 1, n-2 degrees of freedom.

The above, however, assumes that the observations are independent. If this is not the case, for example combining more than one score on the same subject, then it is suggested that either an analysis of covariance is performed to combine within subject correlations into a R-squared measure (see here) or a subject summary score measure such as a mean score is used so that variation is only between subject (see here) . A general article on the dangers of combining scores to obtain correlations without taking into account within and between subject variation is given here.

Reference

Boniface DR (1995) Experiment design and statistical methods for behavioural and social research. Chapman and Hall:London.