13 - Distribution of Means

Abstract

This chapter argues that the calculation of standard deviation involves the ratio of two other quantities—(1) the sum of squares, which goes into the numerator and (2) the degrees of freedom, which go into the denominator. As more and more data are included in the standard deviation formula, both of these quantities increase numerically, becoming better and statistically better defined. Conversely, as fewer and fewer data points are included, both the quantities decrease and become more and more poorly defined. This chapter illustrates the relationship between the distribution of a population and the distribution of means taken from that population. For convenience it is assumed that the parent population is Normally distributed, although this need not be so. The distribution of averages, on the other hand, must be Normal, from the central limit theorem. For the distributions, the standard deviation and the 95% confidence intervals are indicated. Since the distributions are Normal, each 95% confidence interval extends to 1.96 times the standard deviation of its corresponding distribution. For the hypothetical data shown, the 95% confidence interval for the distribution of means is approximately the same as the standard deviation of the parent population.