On the Interrelation Between the Sample Mean and the Sample Variance

The linearity (or lack thereof) of association between sample mean and sample variance is explored in this note with the intent of providing new insights. Of particular interest is a well-known inequality involving the measures of skewness and kurtosis that is derived as a consequence of an identity involving the correlation between sample mean and sample variance. The nature of association between the two is explored further by means of the conditional expectation of the sample variance given the mean. We present several characterization results where the specific relationship of this conditional expectation and sample mean uniquely determines the parent population. The note is presented at a level accessible to graduate or upper-level undergraduate students with several illustrative examples included as teaching aid.