Scientific Method

Like other scientific methods statistical method aims at the discovery of connections between natural phenomena. And it does so by a close study of their concurrences or sequences. Unlike the so-called method of simple enumeration, it notes and records carefully not only actual concurrences and sequences, but also exceptions; it makes observations over as large and varied a field as possible; and cautiously draws conclusions that will fit all the observed facts. The observation of only a few cases of concurrence, or sequence, or concomitant variation, among certain phenomena, especially when the conditions are not under control and the full circumstances are not known, makes it im possible to distinguish a causal connection between the phenomena from a casual coincidence between them. But the observation of a large number of cases over a wide and varied range of circum stances, an exact record of positive and negative cases, and of variations between series of instances, may justify a highly prob able conclusion about a causal connection between the phenomena concerned. The assumption on which the statistical method pro ceeds is this : If two phenomena, say A and B are not really con nected, then their concurrence (or sequence, or concomitant variation) is a mere coincidence. In that case, the concurrence, etc., of B with other things than A should, as a matter of prob ability, be about as frequent as with A. But if the concurrence, etc., of B with A is appreciably greater than with things other than A, then the two are probably connected. Such greater con currence of B with A than with non-A is called their correlation or association (according as the reference is to "variables," that is, things that can be present in various measurable magnitudes, or "attributes," that is, what can only be present or absent, what can be counted but not measured). And the main line of enquiry by means of statistical method, as an independent method of science, is into correlations and associations as clues to causal connections. The degree of such correlation or association may vary considerably, and is expressed by certain "co-efficients." When it is complete we get a general law of the ordinary type (If A, then B) ; if partial we get what is more especially called a statistical law stating that B occurs in such and such a per centage of cases of A, or A=c (B), where c stands for some ascertained constant.

In view of the tendency to exaggerate the importance

of statistical method as an independent method of science, it may, perhaps, be well to point out that it is only one of the methods of science, and really only a substitute for more cogent methods when these are inapplicable, for reasons already indicated. This seems obvious from the fact that as soon as the law of certain phenomena is discovered (by the other methods mainly) there is no further use for the statistical method in that field of enquiry. There was a time, for instance, when statistical records were kept of solar and lunar eclipses, just as they are still kept of meteoro logical phenomena. These records were useful and valuable, for they enabled the ancients already to note certain empirical cycles on which they could base fairly accurate anticipations of eclipses, although they did not understand them. But since the laws of the occurrence of eclipses have been discovered there is no further need of statistical records of them—they can be foretold with accuracy and confidence. This, of course, does not affect the great value of statistical method as one of the methods of science, and as an auxiliary to the others. The important thing to bear in mind is that no amount of statistical technique can serve as an adequate substitute for a direct knowledge of, and familiarity with, the phenomena under investigation.

The Deductive-inductive Method.

Just as money makes money, so knowledge already acquired facilitates the acquisition of more knowledge. This fact has already been illustrated above in connection with the method of residues, etc. It is equally evident in the case of the method which will now engage our attention. The progress of science, and of knowledge generally, is frequently facilitated by supplementing the simpler inductive methods by deductive reasoning from knowledge already acquired. Such a combination of deduction with induction, J. S. Mill called the "Deductive Method," by which he really meant the "Deduc tive Method of Induction." To avoid the confusion of the "De ductive Method" with mere deduction, which is only one part of the whole method, it is better to describe it as the "Deductive Inductive Method" or the "Inductive-Deductive Method." Mill

distinguished two principal forms of this method as applied to the study of natural phenomena, namely, (I) that form of it in which deduction precedes induction, and (2) that in which induc tion precedes deduction. The first of these (I) he called the "Physical Method"; the second (2) he called the "Historical Method." These names are rather misleading, inasmuch as both forms of the method are frequently employed in physics, where some times, say in the study of light, mathematical (i.e., deductive) calculations precede and suggest physical experiments (i.e., induc tion), and sometimes the inductive results of observation or ex periment provide the occasion or stimulus for mathematical de ductions. In any case, the differences in order of sequence are of no great importance, and hardly deserve separate names. What is of importance is to note the principal kinds of occasion which call for the use of this combined method. They are mainly three in number: (I) When an hypothesis cannot be verified (i.e., tested) directly, but only indirectly; (2) when it is possible to systematize a number of already established inductions, or laws, under more comprehensive laws or theories; (3) when, owing to the difficulties of certain problems, or on account of the lack of sufficient and suitable instances of the phenomena under in vestigation, it is considered desirable either to confirm an induc tive result by independent deductive reasoning from the nature of the case in the light of previous knowledge, or to confirm a deductive conclusion by independent inductive investigation.

An example of each of these types may help to make them clear. (I) When Galileo was investigating the law of the velocity of falling bodies he eventually formed the hypothesis that a body starting from rest falls with a uniform acceleration, and that its velocity varies with the time of its fall. But he could not devise any method for the direct verification of this hypothesis. By mathematical deduction, however, he arrived at the conclusion that a body falling according to his hypothetical law would fall through a distance proportionate to the time of its fall. This consequence could be tested by comparing the distances and the time of falling bodies, which thus served as an indirect verifica tion of his hypothesis. (2) By inductions from numerous astro nomical observations made by Tycho Brahe and himself, Kepler discovered the three familiar laws called by his name, namely, (a) that the planets move in elliptic orbits which have the sun for one of their foci; (b) that the velocity of a planet is such that the radius vector (i.e., an imaginary line joining the moving planet to the sun) sweeps out equal areas in equal periods of time; and (c) that the squares of the periodic times of any two planets (that is, the times which they take to complete their revolutions round the sun) are proportional to the cubes of their mean distances from the sun. These three laws appeared to be quite independent of each other. But Newton systematized them all in the more comprehensive induction, or theory, of celestial gravitation. He showed that they could all be deduced from the one law that the planets tend to move towards each other with a force varying directly with the product of their masses, and in versely with the square of the distances between them. (3) H. Spencer, by comparing a number of predominantly industrial States and also, of predominantly military States, ancient and modern, inferred inductively that the former type of State is democratic and gives rise to free institutions, whereas the latter type is undemocratic and tends to oppression. As the sparse evidence hardly permitted of a rigorous application of any of the inductive methods, Spencer tried to confirm his conclusion by deductive reasoning from the nature of the case in the light of what is known about the human mind. He pointed out that in a type of society which is predominantly industrial the trading relations between individuals are the predominant relations, and these train them to humour and consider others. The result is a democratic attitude in all. In a State which is predominantly military, the relations which are most common among its members are those of authority, on the one part, and of subordination on the other. The result is the reverse of a democratic atmosphere.