Aitken A. Statistical Mathematics

Edinburgh: Oliver & Boyd, 1947. — 160 p.The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s. They were sold at a price that students could afford and tended to cover the right amount of material for a lecture coure. One of the books in the series was Statistical Mathematics by A C Aitken."The concern of the present book will for the most part be with statistics (in the singular) as a science. The typical order of development of the "exact" sciences (as they are somewhat loosely called) has been along the following lines. First of all, the examination of data collected in a particular field of inquiry is found to disclose elements of regularity, suggesting a law or laws. This is the stage of inductive synthesis. These laws are expressed, if possible, in the form of logical or numerical axioms, resembling those of Euclidean geometry. The methods of logic and mathematics are then brought into play to develop the consequences of the axioms, producing an assemblage of theorems or propositions. This department of the science, namely the posing of axioms and the deduction of theorems, is usually called the pure branch of the science. Even if future observations should invalidate the axioms extrinsically, the discrepancies between theory and fact being too great to be explained away, these axioms and the deductions based on them would still have an abstract validity, as a logical structure of propositions exempt from self-contradiction; but for the description and explanation of the phenomena a new set of axioms would have to be found. On the other side, the corroborative part of the science consists in interpreting the abstract functions, formulae, equations, constants, invariants and the like, which occur in the pure formulation, as measures and measurable relations of actual phenomena, or numbers constructed from those measures in a definite way. This interpretative discipline constitutes the applied branch of the science.
Such a division or dichotomy into pure and applied can be recognized in almost any science. A good example is Newtonian dynamics, according to which the motions of all bodies in the universe were presumed to obey certain axioms and postulates, namely Newton's laws of force and motion and the law of gravitation. Later experiments, more numerous, more delicate, more comprehensive, suggested that this formulation, though describing almost all observed dynamical phenomena with a precision unprecedented in history, did not sufficiently account for certain exceptional facts, such as the precession of the perihelion of Mercury. The discrepancies between prediction and actuality were extraordinarily small, but they were persistent. There thus arose a theory, or rather a succession of supplementary theories, of relativity, formulated on a new axiomatic basis by which the discrepancies of the earlier one might be reconciled, or removed. This reformulation of hypotheses still proceeds, is still incomplete, and undergoes modification from time to time.
What is the axiomatic basis of the science of statistics, and what are the facts upon which the inductive synthesis is based? The facts are certain regularities which have been observed in the proportionate frequency with which certain simple events happen or do not happen, when the circumstances under which they may occur are reconstructed again and again in repeated trials; and the axioms, and the structure of theorems founded upon them, constitute the subject called mathematical probability. As for the facts, anyone who is interested can collect a few for himself. Spin an ordinary coin a large number of times, and one can hardly fail to notice that the proportions of heads and of tails are very nearly equal; or shake a well-made die repeatedly from a dice-box and one will find that after many trials each face of the die has turned up in about one-sixth of the total number of trials."