Correlation has significance in all sciences. Various attempts have been made over two centuries by many authors to explore its usefulness. In this research an attempt has been made to derive a distribution of the mean of k-independent sample correlation coefficients each of which is based on "n" pairs of observations. This distribution has been developed by expanding the rth power of the modified Bessel function by using the Taylor series. The contribution of this research is to derive, 1)the distribution of mean of two independent sample correlation coefficients and their weighted means, 2)the distribution of mean of k-independent samples correlation coefficients 3)various properties of the distribution of mean of two and k- independent sample correlation coefficients.