This study is concerned with a mathematical entity known as the Kolmogoroff-Smirnoff statistic. Its purpose was to provide rigorous demonstrations of the properties, applications, and exact probability distribution of the statistic. Its main result is a general technique by which the distribution of the statistic may be computed for any sample size, n, and an explicit representation of this distribution for n=1, 2, 3, and 4. In Chapter I, the intuitive notion of a statistic is defined and a discussion is given regarding the purpose, importance, and results of the study. Chapter II is a brief outline of the theory of sets, functions, and probability. In particular, the distinction is made between interpretive and mathematical probability. The Kolmogoroff axioms for mathematical probability are presented and explained with regard to motivation and interpretation. Also, a proof is given of the equivalence of the existence of a probability measure on the Borel sets and a "distribution function," an idea of importance in the definition of the Kolmogoroff-Smirnoff statistic. The chapter concludes with the mathematical definitions of statistic, random vector, and stochastic independence. Chapter III is a detailed exposition of the properties and applications of the Kolmogoroff-Smirnoff statistic. Following its mathematical definition there is a discussion of related statistics and a proof that the Kolmogoroff-Smirnoff statistic is "distribution-free." The mathematical structure of the statistic is then discussed and the chapter ends with an example relating to the probability distribution of automobile traffic. In the final chapter, Chapter IV, the main result of the study is presented: a technique for the computation of the exact distribution of the Kolmogoroff-Smirnoff statistic for any sample size, n. The technique is illustrated for n = 4 and the distributions for n=1, 2, 3, and 4 are presented both as distribution functions in equation form and as densities plotted in Figures 4.1-4.4.