It is well-known that a necessary and sufficient condition that a probability distribution be normal is that its mean and standard deviation be independent. Beyond this, little seems to be known about the relationship between the mean and standard deviations s of samples drawn from non normal populations. The exponential distribution and more generally the gamma distribution adequately fit many distributions arising in technological application of statistics, particularly in the fields of reliability studies, life testing, queuing theory, wheatear analysis, and medicine. Inference for parameters of more than two gamma distributions is quite rare in the literature. Tripathi et al (1993) proposed a test for parameters of m ≥ 2 gamma distributions based on a generalized minimum Chi-square procedure. For gamma populations with integer shape parameters m ≥ 2, this paper derives, among other things, the regression function and the conditional distribution function of s given XnF(s | ). The latter is particularly important in determining whether the correct gamma distribution (as identified by its shape parameter) is used in connection with its (frequent) application to real world problems in diverse fields. For example, the gamma density function is used both as a p.d.f. and as a Bayesian prior density function in reliability analysis (Mann et al (1974), p. 127, p. 379). Mooley (1973) discusses the gamma model for summer monsoon rainfall in millimeters (See Bowman and Shenton (1988), p. 90.). Bordi et al (2001) discussed using gamma distribution for drought monitoring in the Mediterranean area. Masyma and Kuroiwa (1951) give data on the sedimentation rate during the period of normal pregnancy, and fit the gamma distribution to the data. (See Bowman and Shenton (1988), p. 90.) Amorosa (1925) used the gamma distribution in analyzing the distribution of income. And of course there are many other applications. But whatever the area or nature of application, the proper use of the gamma distribution depends heavily upon using the correct value of the shape parameter, which can be simply and easily tested by using the conditional distribution function of s given to compare the value of s relative to that of . Furthermore, when using analytical models involving the mean and standard deviation of gamma distribution. It is necessary to know whether the correlation between them is sufficiently high that it cannot be required. Other interesting results obtained in this paper from the derived distribution g(, s), are: (1) the regression function is linear. (2) the scedastic function is quadratic. (3) for any integer value of the shape parameter m, the coefficient of variation for density function of is 1/(2m)1/2. (4) the correlation ratios for the shape parameters m = 2, 3 are, respectively, 0.70076 and 0.48564. The author conjectures that values of the correlation ratios decrease with increasing values of m, but this has not been proven.