Relationship Among Statistical Distributions

Abstract

Samples of clinical data are frequently assessed through 3 variables:

• The mean result of the data.

• The spread or variability of the data.

• The sample size.

Generally, we are primarily interested in the first variable, but mean or proportion does not tell the whole story, and the spread of the data may be more relevant. For example, when studying how two drugs reach various organs, the mean level may be the same for both, but one drug may be more variable than the other. In some cases, too little and, in other cases, dangerously high levels get through. The Chi-square-distribution, unlike the normal distribution, is used for the assessment of such variabilities. Clinical scientists although they are generally familiar with the concept of null-hypothesis-testing of normally distributed data, have difficulties to understand the null-hypothesis testing of Chi-square-distributed data, and do not know how closely Chi-square is related to the normal-distribution or the T-distribution. The Chi-square-distribution has a relatively young history. It has been invented by K. Pearson1 one hundred years ago, three hundred years after the invention of the normal-distribution (A. de Moivre 1667–1754). The Chi-square-distribution and its extensions have become the basis of modern statistics and have provided statisticians with a relatively simple device to analyze complex data, including multiple groups and multivariate analyses. The present paper was written for clinical investigators/scientists in order to better understand the relation between normal and chi-square distribution, and how they are being applied for the purpose of null-hypothesis testing.