When Are the Sample Mean and Standard Deviation Independent?
Perhaps one of the most important and least intuitive theorems of classical statistics states that, for data from a normal population, the sample meanand standard deviation s are independent.2 One reason this fact is important is that the derivation of Students t-distribution depends on it. In practice, if and s are not independent, then tests of hypothesis and confidence intervals based on the t-distribution may give misleading results. However, it seems strange that and s would be independent. Each of these statistics arises from somewhat different manipulations of the same data X1, X2, …, Xn. The definition and “computation formula” for s even involve : . Sometimes proofs satisfy the intuition as well as the intellect, but the standard proofs of this independence use such methods as multivariate transformations, matrix manipulations, and moment generating functions, which have little intuitive appeal for many students. Intuitive suspicions about the claimed independence of and s a
Perhaps one of the most important and least intuitive theorems of classical statistics states that, for data from a normal population, the sample meanand standard deviation s are independent.2 One reason this fact is important is that the derivation of Students t-distribution depends on it. In practice, if and s are not independent, then tests of hypothesis and confidence intervals based on the t-distribution may give misleading results. However, it seems strange that and s would be independent. Each of these statistics arises from somewhat different manipulations of the same data X1, X2, …, Xn. The definition and “computation formula” for s even involve : . Sometimes proofs satisfy the intuition as well as the intellect, but the standard proofs of this independence use such methods as multivariate transformations, matrix manipulations, and moment generating functions, which have little intuitive appeal for many students. Intuitive suspicions about the claimed independence of and s a
