The proof and measurement of association between two things

All knowledge—beyond that of bare isolated occurrence—deals with uniformities. Of the latter, some few have a claim to be considered absolute, such as mathematical implications and mechanical laws. But the vast majority are only partial; medicine does not teach that smallpox is inevitably escaped by vaccination, but that it is so generally; biology has not shown that all animals require organic food, but that nearly all do so; in daily life, a dark sky is no proof that it will rain, but merely a warning; even in morality, the sole categorical imperative alleged by Kant was the sinfulness of telling a lie, and few thinkers since have admitted so much as this to be valid universally. In psychology, more perhaps than in any other science, it is hard to find absolutely inflexible coincidences; occasionally, indeed, there appear uniformities sufficiently regular to be practically treated as laws, but infinitely the greater part of the observations hitherto recorded concern only more or less pronounced tendencies of one event or attribute to accompany another. Under these circumstances, one might well have expected that the evidential evaluation and precise mensuration of tendencies had long been the subject of exhaustive investigation and now formed one of the earliest sections in a beginner’s psychological course. Instead, we find only a general naïve ignorance that there is anything about it requiring to be learnt. One after another, laborious series of experiments are executed and published with the purpose of demonstrating some connection between two events, wherein the otherwise learned psychologist reveals that his art of proving and measuring correspondence has not advanced beyond that of lay persons. The consequence has been that the significance of the experiments is not at all rightly understood, nor have any definite facts been elicited that may be either confirmed or refuted. The present article is a commencement at attempting to remedy this deficiency of scientific correlation. With this view, it will be strictly confined to the needs of practical workers, and all theoretical mathematical demonstrations will be omitted; it may, however, be said that the relations stated have already received a large amount of empirical verification. Great thanks are due from me to Professor Haussdorff and to Dr. G. Lipps, each of whom have supplied a useful theorem in polynomial probability; the former has also very kindly given valuable advice concerning the proof of the important formulæ for elimination of “systematic deviations.” At the same time, and for the same reason, the meaning and working of the various formulæ have been explained sufficiently, it is hoped, to render them readily usable even by those whose knowledge of mathematics is elementary. The fundamental procedure is accompanied by simple imaginary examples, while the more advanced parts are illustrated by cases that have actually occurred in my personal experience. For more abundant and positive exemplification, the reader is requested to refer to the under cited research,1 which is entirely built upon the principles and mathematical relations here laid down. In conclusion, the general value of the methodics recommended is emphasized by a brief criticism of the best correlational work hitherto made public, and also the important question is discussed as to the number of “cases” required for an experimental series. The most fundamental requisite is to be able to measure our observed correspondence by a plain numerical symbol. There is no reason whatever to be satisfied either with vague generalities such as “large,” “medium,” “small,” or, on the other hand, with complicated tables and compilations. The first person to see the possibility of this immense advance seems to have been Galton, who, in 1886, writes: “the length of the arm is said to be correlated with that of the leg, because a person with a long arm has usually a long leg and conversely.”2 He then proceeds to devise the required symbol in such a way that it conveniently ranges from 1, for perfect correspondence, to 0 for entire independence, and on again to −1 for perfect correspondence inversely. By this means, correlations became comparable with other ones found either in different objects or by different observers; they at of to and in a they now a scientific it is that other of are from to One for has been and by to is and most from all the by the important that any number of for be which are all any then the will the a or of the of the that of the and is the measure of the correlation. But more may to one the of the this that a measure might be of the of the for that and from and each of his from some to of will have a to and this correspondence will in any of the of be some but in only one of them will it actually be in such a and of from the the between these two will also as and to be found correlated with that of by then there is a that of with of to either The observed the of the amount of for and the In the this to the same of by by But this is and different from the and not the same of from the might his while some different in which it will be found that the is the between the two be to all his the then only in to the same as since the between and is to in as in most other and are not to be as in the of our first but as in the of our in to be correlated with by be considered to have only of of his to this and to have for which will an to of In not measure of but the the of the in any observed correspondence as with the of in other this of of for the measure of the be the of of any one of latter, have various and of which the is of to the between two series of is an mathematical value in our able to a of cases a we to our as a one at all to investigation be will in such a as to the of some more reveals that the observed was due to and has no whatever to this one of be to the that such are different in are the and in to the of The consequence is that they more or less one another, and present an however, since no amount of is absolutely but beyond it so that for practical we to a person at that such a series not by in and will in the to some to the evidential value of any other observed we only require to nearly the have to some such as to as any be more or less it and is to a not the but the or may then find the of from the the of this upon the number of cases but also upon the mathematical of correlation. a in the be made by the of the such is not has other which will be discussed the best is that one which the For the of the this be more is required than a for while it is important to between a of perfect and one only it be a of to a to a of or to one of The most of are and to there under which they be at in to a which be for the there is of of under the most and with the But here a of not of For such are very and to will to some from one so that the may be the less the one most for his and upon his work an to which it is by no from the are only either absolutely or for or for may be as the of correlation. of is that by in and shown by in to be the best this that of is the of the two for any is the of such for all the is the of the of all the various of is for and is the required correlation. simple may this that it was to of with that of and that for this purpose as to the at which they and a and the to we so that and is no correspondence, or For of between whose from the as either or may be made to the may be that the of is or not the the of so long as the is In the by Galton, of correspondence between the length of arm and that of leg, it may be that the correspondence is a long arm has a to be accompanied by a leg not only but long to the same in such is by the of a is than one we in with the of are correlated to of that in the former there do not any a either or not even more are those cases does but practically will not of for it is that is to some a we well to the same amount of to the that we measure In all such cases we to the of find and are and has to the of a and and then to measure much to be in the same we have this simple we may of that is to the have a of to then with will have the same of to with while those with only a amount of will be correlated with also of only the may be expected to in only in as the is For of this a which may be that of has been by and The long and complicated to be in this will be found in the under cited with as by In it will have to be by one of the more to be of is not only and of but also it a nearly that of it be escaped in the of not by other way will be discussed at some as it has been by and is for some All may be from two either in of and by or by of we might that a in an while only or, we might that was of while was only the same of may be again in the the for the In the first it may be that the observed then only for of the same from one another. For to be correlated with then the of of and as is more to as the in of is than they are practically on the But the more the the less will be the of it be that the less perfect as the experiments made more however, be a experimental by on the actually the same as that in either the are by the amount of the is not upon the between the but upon the of this to the and of these with the number of the other hand, the will the be all from a more in a for for general will with other than be the in a for of the same but this as much to by The is that by upon an that all the from one by the same may from as much as from But such an only correspondence by considered to be to that by no such is the two are to be but is of the that they correlational in the may be a of the same of which is to the practical of the cases a number of The and only is that a less of correspondence than does and the even of this on that the two the same correlational The of these is the large of the In a the cases are much more than are those to the any the of these cases will have on the general of the and to this in the of the the will not one with the same as it is these cases there is most of and there very a different from that the of the the of this by from the which well be in any general there be no in it a amount of empirical the the of with to less than of that given by the same with and only about of that given by the of The is that any between the two as general of such a is not or in any way but merely an of the of the By of a series the be on even with series whose is entirely well be has also the useful of any two series to be and a the practical of view, it is so to the with a given number of that the of have the even we are with two series of comparable with one another. is at any to such a and as that between on the one and on the the in most psychological the only other be to by the of But this the of the and it that the be no less than as such an even be accompanied by infinitely the theoretical of The are with such as are by most For in these cases the of upon the is from the very and is of a In by in most cases a is for series them of evidential value to much ones treated by other it is in series that by is practically as has been are only for there is reason for not the number are correlational do not with those found by the but nearly so for most practical a the number of that is accompanied by the number of that is accompanied by the number of that is accompanied by the number of that is accompanied by to our that it was to the of by and by of each of have been shown to a of each had to by as as they The was but the of Of the while of the same number of ones only see that there is at any of some on the part of the the has been executed and any under sufficiently by other in a between and and between and we have a of the of this of the for the of the to so that of the various here to the some than the of to about one part for we demonstrating that may be by the of With this view, we have his to and after In the of them have the expected but on the have shown a than now to the of this at different of that after one than the had to we see that the of from to that from to of all the of in a has a very different scientific significance any that the had by so more our be and with any of the various other recommended more than all which present correlations and is the observed between the of the accompanied by and that of those accompanied by and is the that was as have had the correspondence been has been two about will be to the or from the in the series while has been after the such only the two ones be and then the in the series The of is valuable in the of other the and the are usually given but not the required for any of the other to that it seems to the of the with a that given by upon much each in the one from his in the other this will be the is and will as the is the number of and is the required correlation. The will then be even for of very and in has be only for and not for The given is only that correlations may be expected from there is no correspondence between the two does not for the of much the observed may be expected to by from any correspondence that does The various of are found to a of so that we in all the other here the of as so much correlation. be but there are also other in which this of the for our so that it is to as positive be the of one of the this value is not to the found by all the other but for the proof of this is only but it on a large number of cases however, only between and it be at be found from the In the first we have that any correlational only be as a of the immense and will present a amount of from the general we have that this is by the whose an requisite to all But now we are in of from For after cases and the there a very to from and at from to the of But or not such may be in it is at any in the of that we are at present we are not with but with a of work so that it may be by the of the is not the sole nor even the most there are other are by the most of in such of merely on the one or the in so in we may not only of or but also of or are of very the most as for one of our examples, that we have the to a number of and after and have found the between and it to that we executed our entirely or for it is to a series of some general either to them the or to such them in the of our may be this of is to be by our of for by it we definite which any may or it will be that a does not on the amount of in the two but on the between these on the one and the ones on the In our it not and each had one or a in the but only the amount was a or large of the be then the is that and the of the be sufficiently then any will infinitely a more to be a then the various they will be correlated to the various with which they But these be with the same of a then the will have but correspondence to the it will be that here the have a consequence but that discussed in the For they and the in a series to more and more one and in this is sole But here in they also have this which is in the of the correspondence and whose on the of the be in the by any of the series. The has general or that we to the correspondence between a series of and By practical we do not the and but only which we will and is less with than is with for the first only at all by the of the the correspondence between and has been the between one and of these series of for the same as and the required between the of and In some and very theoretical it may that either of the is with of or any other to the will be to that any in our will present large a greater practical is that of two series sufficiently of one another. For are to even two are to some by the same we are all on not only the but also the and the In such the is only does not at any on the will best the of this In a between two events, and observations of and of The for those of with those for was the of those for with one was the same for was the correspondence between the events, and by to so that the correspondence, of merely to be and the between the various for and those for and the of the series for with the series for In the the observations for series a and those for series this the two now a with one of no less than by this of the correspondence became more 1, so that this way also the correspondence advanced from to have that “the length of the arm is said to be correlated with that of the leg, because a person with a long arm has usually a long leg and also that this is by any which the of any definite of arm to the of the of the are very and we will the between the two But not the most investigation any for such even in the simple of and they have to be from of all the since this is the most of all of in In the they have to from of they from the to the also from the to the the even is are to a large number of the is confined to to to some In a we are to with a correlation. we to more these it that they are from For investigation will be found to be and by some as to the of the correspondence and of the will a of such as to the correspondence in the most But from these will at the same of In the first since the are not they are not in a the may be and For a series of experiments was executed by one of our best and his that some was correlated with and however, no had been to the due to of the only to be that as they and in the in to be the and The to be is For even the be in the of the the to be so that the inevitably and The is even with the this of the is an to by a knowledge of the we at all those to our all such in a as are to the or purpose the in it and will will in two the of one of the with a and admitted and the between and after for the of In this the more was of the by the to those and on a as the The then in this empirical way with the former again and have to the and to we now to a of that may well or is to the first consequence of discussed in the first part of this the latter, this does not in the to in the series of and the correlations of and with some and and the required between and after for the of but was under the of the about so that the one observed of have been entirely our of has been confined to from personal experience. But it might perhaps be that other such of by the of such does not to have been at all the not to the best of my has any between two psychological been in such a way as to present any evidential are with an to have with the work on since by the The consequence has been that they do not even to the first fundamental requisite of a precise indeed, in the and have of complicated numerical but we in in the of we find that they have the few facts which are so that we even work the for of more than merely in the have been to at all some have to an entire of correspondence, the has been have to the as a has been very and well the of these they have had no of and usually in such a way as to it that even the of correlational work in the first part of this article have been so we be to find that the more advanced of procedure discussed in the part have been so that the are and with of In this it is no to even the as a to be The now the very different of from criticism of by to an as to the and even the best correlational we will of at that it with our with and been a vast number of as to the amount of correspondence between in each upon to in to the of the with the for and we are to the definite conclusion, that the in are in the same as the and as much as our the of to and for our on the one with and on the other with the seems to have been to his as as and in one that is the to is no more than in that his to the that the correspondence between the be in measure due to under the same But such a be as For we do not anything precise about the of our observed correspondence is perhaps due to the been or even to such as well and the same amount of and less such a with that from has or no The of but be as and the of be as But even be the For the of has been admitted with to at it has been as For only those have been are at the same the of those also be are in a different have to The correlations are also this work of has been for it is from no to The and any other are of and this article has it is hoped, been of some that will no an immense amount of investigation and but in the his will all present is only to and to the of the methodics of correlational a of investigation which has so to and by of which is immense hitherto in the of The fundamental between his procedure and that here is that large of any series sufficiently to be treated here are but they are required to be and scientific are those of those here may be more to the important as to cases it is to for a series of In the has been in this have to sufficiently important correlations with less than experimental while have it to at a a series of experiments is a very whose to be as a of the immense the the the greater evidential value and the less of a for the of has been discussed in the first part and there shown to be strictly by the there that this be entirely by any so that it is to some less than as for all practical the of is considered to be a is observed as much as greater than the that not this in a the not only on the number of but also on the of the the more perfect the latter, the the cases that will be required to it and this is by the that the with does so to a with was shown in the same part that the of the also to the of to such an that cases treated in one of the as much as in more the be an even greater number is required to the same and the be not at but be in the to the general then no number of cases whatever appear to of the number of is not by any the sole in even the it has no whatever upon the more that it to an we are to be able to upon an number of cases and a we are in the same most a all we may that the number of cases be by the simple that the to be have to the same general of with is not the best nor is there any in upon a has actually been by an the to the in to the present of is one of of a few we require a large number of ones under and well considered At the same time, however, the be to at any for the of investigation to be For such a purpose a may at present be admitted much to about so by the of two to be for most The be by experiments under and then after a long of this we be in a to experiments and executed from the very on a vast