Dependence or Independence of the Sample Mean and Variance In Non-IID or Non-Normal Cases and the Role of Some Tests of Independence

Abstract

Let X₁, …, X_n be independent and identically distributed (iid) random variables. We denote the sample mean $\overline X = n^{ - 1} \sum\nolimits_{i = 1}^n {X_i }$ and the sample variance $S^2 = (n - 1)^{ - 1} \sum\nolimits_{i = 1}^n {(X_i - } \overline X )^2$ for n ≥ 2. Then, it is well-known that if the underlying common probability model for the X’s is N(µ,σ²), the sample mean $\bar X$̄ and the sample variance S² are independently distributed. On the other hand, it is also known that if $\bar X$̄ and S² are independently distributed, then the underlying common probability model for the X’s must be normal (Zinger (1958)). Theorem 1.1 summarizes these. But, what can one expect regarding the status of independence or dependence between $\bar X$̄ and S² when the random variables X’s are allowed to be non-iid or non-normal? In a direct contrast with the message from Theorem 1.1, what we find interesting is that the sample mean $\bar X$̄ and the variance S² may or may not follow independent probability models when the observations X_i’s are not iid or when these follow non-normal probability laws. With the help of examples, we highlight a number of interesting scenarios. These examples point toward an opening for the development of important characterization results and we hope to see some progress on this in the future. Illustrations are provided where we have applied the t-test based on Pearson-sample correlation coefficient, a traditional non-parametric test based on Spearman-rank correlation coefficient, and the Chi-square test to “validate” independence or dependence between the appropriate $\bar x$̄, s data. In a number of occasions, the t-test and the traditional non-parametric test unfortunately arrived at conflicting conclusions based on same data. We raise the potential of a major problem in implementing either a t-test or the nonparametric test as exploratory data analytic (EDA) tools to examine dependence or association for paired data in practice! The Chi-square test, however, correctly validated dependence whenever ($\bar x$̄, s) data were dependent. Also, the Chi-square test never sided against a correct conclusion that the paired data $\bar x$(̄, s) were independent whenever the paired variables were in fact independent. It is safe to say that among three contenders, the Chi-square test stood out as the most reliable EDA tool in validating the true state of nature of dependence (or independence) between $\bar x$̄, S² as evidenced by the observed paired data $\bar x$̄, s whether the observations X₁, …, X_n were assumed iid, not iid or these were non-normal.