Classical Texts in Psychology
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"GENERAL INTELLIGENCE," OBJECTIVELY DETERMINED AND MEASURED
C. SPEARMAN (1904)
First published in American Journal of Psychology 15, 201-293
THE PRESENT RESULTS.
1. Method and Meaning of the Demonstration.
As the reader will have noticed, the formulae given at the end of the previous chapter are equations whereby from several observed correlations we are able to deduce a single true one. This latter alone is of real scientific significance, and under the ordinary unsystematic conditions -- such as governed the great majority of work reviewed in the second chapter -- the actually observed correlations will rarely be of much interest in their primitive raw state; for after passing through the proper corrections, they would come forth transfigured in every conceivable manner; some would increase in size, some diminish, some entirely disappear, and some even become inverted. Nevertheless, our true correlation in no way deserves the reproach of being a theoretical abstraction, for it only represents the limit to which the observed correlation itself will continually [p. 257] approach as improvement is made in the experimental procedure; and not even the most perfect methodics can afford to dispense with the formulae, seeing that these are the sole means by which the perfection may be adequately ascertained.
Our method of demonstration implies four distinct steps, all of which are believed to be absolutely indispensable for work intending to be more than merely suggestive. First, we must exactly determine the quantity of correlation actually observable, and we must compare it with the probable error; then, if the former be no more than about twice as large as the latter, the whole experiment may indeed have produced a substantial negative result, but cannot possibly warrant any positive conclusion other than to suggest the desirability of extending the investigation until it acquires more evidential value; but if, on the other hand, the observed correlation be four or five times greater than the probable error, we may then consider a prima facie case of correspondence to have been established and we may legitimately go on to the corrective processes so as to bring our raw figure to its most probable real amount. Accordingly, the second step will be to form an estimate of the errors in observing the two series compared; for this purpose we must have obtained two or more independent sets of measurements for each series, or at least must be acquainted with the relations found between other such sets under sufficiently similar circumstances; the influence of these errors can then be eliminated by the formulae given on page 253, and at the same time an opinion can be formed as to the presence or not of the grave fallacies discussed on pages 253 ff. The third proceeding is to look for any factors irrelevantly admitted (or, more rarely, excluded); any suspicions must be carefully verified in succession, and, if necessary, employment must be made of the eliminating equations given on page 256. Finally, we have to critically review the whole argument, paying particular attention to such disturbing factors as have not been disposed of very satisfactorily; in this way we come to a final estimate, not only as to the most probable amount of real correspondence, but also as to the degree of confidence to which our evidence is entitled; for these two things are by no means always parallel, a high apparent correlation often having but small evidential value and vice versa.
A few words may now be said concerning the eventual meaning attachable to the result which we hope to obtain. To put it briefly, the usual direction of inquiry is in the present work reversed. The customary procedure consists in determining some matter of research subjectively, say, "Perception," "Attention," "Imagination," "Fatigue," etc., and then ascertaining its relation to other similarly pre-determined psy- [p. 258] choses or neuroses. Here, on the contrary, although every effort has been made to render the mental phenomena as unequivocal and significant as possible, yet in the beginning not the least note is taken of any psychological import beyond such as is barely necessary to define the subject of discussion in the most positive and objective manner; while the structure of language necessitates the continued use of such terms as Discrimination, Faculty, Intelligence, etc., these words must be understood as implying nothing more than a bare unequivocal indication of the factual conditions of experiment. For the moment we are only inquiring how closely the values gained in the several different series coincide with one another, and all our corrections are intended to introduce greater accuracy, not fuller connotation; the subjective problems are wholly reserved for later investigation. It is no new thing thus elaborately to deal with and precisely measure things whose real nature is concealed from view; of this nature, for instance, is obviously the study of electricity, of biology, and indeed of all physical science whatever.
Let us, then, consider the extent of connection between two series of things implied by this sole fact of their presenting a numerical correlation with one another; such a correspondence, when beyond the range of mere chance coincidence, may be forthwith assumed to indicate and measure something common to both series in question. Such a community may often consist of a definite so-called "substance;" A's changes of wealth will show some correlation with those of B, if both possess some shares in the same stock. Or, on the other hand, the community may derive from a more complicated interaction of forces; thus, the weather is supposed to correlate with the state of the spots on the sun. But this distinction is superficial even in physical matters; thingness may well be an indispensable crutch for popular thought, and indeed in metaphysics becomes a serious enough topic, but it has no place in strict natural science and still less in psychology, where fast limit has never been securely traceable between things, qualities, and conditions.
But the same simple mathematical formulae which have brought us so far will take us yet farther. As from several sets of inaccurate measurements it has been found possible to arrive at the accurate correlation of the two real series, so now in a similar manner from any number of real series we can proceed on to dealing exclusively and precisely with any element that may be found common to these series; from ascertaining the inter-correlations of, say, auditory discrimination, visual discrimination, the capacity for learning Greek, and that for playing the piano, we can arrive at estimating the [p. 259] correspondence of whatever may be common to the first pair of faculties with whatever may be common to the second pair. By combining such correlations of higher order, it is feasible to execute any required amount of elimination and selection, so that eventually a dissociation and exactness may be introduced into psychology such as can only be compared with quantitative chemical analysis; even in the present work, it is hoped to obtain results of sufficient fineness to be independent of local conditions of experiment, and therefore to be precisely verifiable by any other workers. All the time, the relations discovered by us will wholly retain their impartial objective character; however accurately we may learn the distribution of community, it will remain as a later and very different task to detect and analyze its psychical nature. But we shall find that the successive positive ascertainment of objective relations continually reduces and simplifies the thinkable explanatory hypotheses, so that practically our method of investigation is bringing us towards the introspective psychological solution also -- and perhaps in the end by the shortest route.
2. Correspondence between the Discrimination and the Intelligences.
(a) Experimental Series I. We will begin by dealing with the subject broadly and considering the general average correlation between the various forms of Discrimination and those of Intelligence. To establish our prima facie case, we note that Discrimination has been tested in three senses and that Intelligence has been graded by three different persons; thus we have nine correlations which, if no correspondence exist, should all be small (about half of them under 0.09) and approximately as many should be inverse as direct. Far from this being the case, we find that every single one is direct, that the smallest amounts to 0.25, and that the average comes to 0.38 with a probable error of about only 0.02. Now, a correlation thus more than nineteen times the size of its probable error would not occur by mere accident in millions of trials, so that chance, as a possible cause of the apparent correlations, may at once be put completely out of court. Our result has thus made good its right to further elaboration.
But when we consequently proceed to discount the errors of measurement, it unfortunately becomes clear that our data are far from being adequate for the purpose; we have, indeed, a duplicate set of observations for Common Sense, but none for School cleverness nor for any of the Sensory Discriminations. The excuse for the deficiency lies to some degree in practical difficulties, but still more in the fact that at the time of the experiments I was only just beginning to realize the necessity of [p. 260] such duplication. In default, then, of better information, the other errors will provisionally be taken as being about the same size as those for Common Sense (later on we shall have opportunity of partially checking this assumption); on this basis, an elimination of the observational errors by our first or theoretical formula brings the required correlation to 0.60. This result can now be compared with that given by the second formula; for this purpose we utilize the fact that Common Sense and School Cleverness prove to be not very different criteria, so that all three lists may be used as measurements of practically the same intellectual faculty; if we accordingly amalgamate these three lists into one, the latter shows an average correlation with the Discriminations amounting to 0.44 and our required correlation comes to 0.54, or somewhat smaller than by the other way. Such a decrease by the empirical as compared with the theoretical formula could be produced by two causes: either the estimation of School Cleverness might have been more accurate than those for Common Sense, or else there could have occurred the fallacy a priori feared by us, namely that the three critics had been warped by the same prejudices and therefore not able to judge with sufficient independence of one another. As, however, the total divergence only amounts to 0.06, we can conclude that neither of the above disturbances can have existed to any appreciable degree, and we might well ascribe even the small apparent difference to mere chance variation; but to be on the safe side we will adopt the lower value, 0.54.
We next pass to the inquiry into irrelevant factors and will commence with the conspicuous one of Sex. This to all appearance manifests a connection with both Discrimination and Intelligence, and therefore might conceivably by the sole cause of the two latter being congruous with one another. But closer inspection alters the aspect of affairs; for while Sex and Discrimination only show a correlation with one another to the [p. 261] extent of 0.26 (after correction for errors), they respectively correlate with Intelligence to the amount of 0.59 and 0.54; so that the true correlation between Sex and Discrimination comes to - 0.07, that is to say it entirely disappears. Thus it would seem that the correspondences of Discrimination and Intelligence with Sex are in no degree the causes but merely effects of their correspondence with one another and that the fluctuating differences of Sensory Discrimination observable in connection with Sex at the various stages of growth are chiefly and perhaps altogether a mere consequence of similarly fluctuating differences in their Intelligence. This hypothesis tallies well with other indications; in the experiments of Gilbert, for instance, as to the most characteristic discrepancies between boys and girls at the various ages, the two senses tested (visual and muscular) present an almost identical progress, as if both were depending on some common influence. The same conclusion can be more directly derived from the fact that either the boys or the girls, taken separately, present correlational values very similar to the above; but such a subdivision so reduces the number of conjoined cases that the probable error becomes too formidable for the attainment of sufficiently regular results. Hence it has been thought both permissible and advantageous to throw the boys and girls together into the one collective experiment.
The next obvious irrelevant factor is that of Age, which apparently exhibits correlations with Discrimination and Intelligence of 0.37 and 0.42 respectively. But as regards the former, first, the true value obtained in the same way as before descends down to only 0.18, thereby not indeed this time disappearing but still coming almost within reach of some bias in the operation of grading; possibly, then, the connection of Age with Discrimination is at least in the main, like that of Sex, no more than an effect of their common correspondence with Intelligence; this would accord with the strange phenomenon noted above, that Intelligence appears temporarily to diminish about the eleventh year, for precisely the same occurs to their powers of Sensory Discrimination (a fact first pointed out by Gilbert and noticeable in the present experiments also). To turn to the other above indicated correspondence, that between Age and Intelligence, all those who furnished me with their personal estimation of the children's comparative intellect had been particularly requested to do so entirely regardless of Age, and they had anticipated no difficulty on this head declaring [p. 262] that their opinions were naturally formed quite independently of any such consideration. But we have seen good reason for being very strict in this respect, and when we actually examine the figures, the observed deviations are often far greater than can be attributed to mere chance; from seven to ten years it is the little ones who are favored and to the large extent of 0.65, which is, however, in this case only three times the abnormally big probable error; but from ten to fourteen the above correlation of 0.42 is found in the opposite direction and upon being corrected only descends to 0.38, which is more than seven times its probable error and therefore would not occur by mere coincidence in many thousand times. Thus we are impelled to believe, either that judgment of Intelligence is to a great though unconscious extent biased by consideration of Age, or else that there is a stage of development somewhere near the eleventh year where Intelligence temporarily declines; in connection herewith we have the curious fact that from about eleven to twelve years there appears to ensue a suspension in the growth of children's heads. Returning finally to our main topics with these two values, 0.18 and 0.38, the corrected correlation between Discrimination and Intelligence now comes to the slightly reduced amount of 0.52.
Passing to the next irrelevant factor, Practice, this influence has in the third chapter shown itself to be only moderate as regards distinguishing Light, and still smaller as concerns Weight; moreover, there is no reason to suppose that the children differed appreciably from one another in their amount of previous practice with these two kinds of activity. But in the matter of Sound all this is reversed, for even the most homogeneous school presents a wide diversity of musical education, and we have already seen that such circumstance is of enormous influence. Taking the quantitative estimates given on page 231, in conjunction with the average and its mean deviation in Table I, the correspondence of Discrimination and Practice can be reckoned out to probably amount in such a school to something like 0.70. This very large factor must therefore be eliminated be- [p. 263] fore we can hope to obtain an approximately true result. As it only affects Auditory Discrimination, the total average will thereby be finally increased to about 0.58.
So far, we have broadly taken the general average correlation between all the kinds of Discrimination and those of Intelligence. Let us next briefly consider the individual relations between the several specific sensory and intellective faculties. Common Sense and School Cleverness present practically the same amounts, their raw correlations being 0.39 and 0.36 respectively, so that there appears no object in treating these separately. The various sensory departments show apparently much larger discrepancies: for Weight, which has a raw correlation of 0.34, after correction we eventually get 0.43; for Light, with a raw correlation of 0.44, we come to 0.58; and for Sound, with a raw correlation of 0.37, we arrive at no less than 0.71. To this disparity, however, no great evidential value can be attached, until more positive estimates have been obtained as to the errors of measurement; the above rank might really only mean that the accuracy of experimentation had been least in the thresholds for Weight and greatest in those for Pitch.
Lastly comes the process of reviewing the whole argument. Clearly enough, it has in many respects been of a rough character. The arbitrarily assumed observational errors for School Cleverness and for Discrimination were but inadequately checked in the former case and not at all in the latter; while the irrelevant influence of previous practice on Auditory Discrimination was based upon solid but too general data. Also, there is always the danger that other unsuspected irrelevant factors may exist in harmful magnitude; on this head, however, the precautions taken appear fairly adequate; chapter III only shows the results of the preliminary investigation as regards those factors which finally proved most formidable, but a great number of others have been thoroughly examined and their influence has been found to be inappreciable for our present purpose. Among all these sources of inaccuracy, it will be manifest how insignificant is here the rôle of the probable error of the raw main correlation (0.02); thus, though the reagents were only twenty-four in all, it would have been worse than useless to augment their number at the expense of correctness in other respects; for the present, increased precision must be chiefly sought by other means. To sum up, the most likely value for the average correlation between the Discriminations and the Intelligences comes to about 0.58, but this final conclusion must be considered as having a large total probable error, say, 0.10.
(b) Experimental Series II. On first inspection, the results here would seem diametrically opposed to our last ones, the [p. 264] correlation turing out to be on the minus side, so that Discrimination would appear correlated to the extent of 0.39 not with Intelligence but with Stupidity.
But we again notice that both Intelligence and Discrimination are irrelevantly correlated with Age, and to the large amounts of 0.69 and 0.81 respectively. On applying the corrective formula, the true correlation turns round to + 0.41, whereby the paradox is readily resolved into a result under the circumstances perfectly normal; we should expect the amount to be smaller than in the former experimental series, because the observational errors must have been greater and have required larger compensation than before, whereas we have given the same.
Thus it is once more evident that the influence of irrelevant factors, though sometimes of moderate magnitude (as in the preceding series) may at other times assume such dimensions as to wholly reverse the conclusion. Nor would such an effect be in the least diminished by increasing the number of reagents experimented upon.
Except for this lesson concerning the danger of irrelevant factors, there is little information to be gleaned from the present series; for while the observed correlation is only - 0.25, the probable error comes to no less than 0.18; in other words, such a correlation would turn up about every third time, either when no correspondence existed at all, or when there really was one twice as large. Hence we see that though a short series of cases may be managed in such a way that the rôle of the probable error becomes insignificant, yet this is not necessarily so; when the data are gained in the customary unscientific manner, the results of these brief experiments are worthless for persons versed in correlational methodics and delusive for those who are not so.
(c) Experimental Series III. Here the probable error has been reduced to more reasonable dimensions (0.06 for the average of two correlations); also the disturbances from Sex and Age have been eliminated. But on the other hand (as we have seen on page 248), the sensory tests were so unsatisfactory that the attention by errors must be estimated of enormous magnitude. Quite accordingly, the observed correlations of Intelligence with Visual and Tactual Discrimination are no more than 0.13 and 0.12 respectively. Upon this diminutive basis we cannot attempt to pile the very large and vague corrections that would be necessary.
Thus we see that an inadequate way of testing the reagents, [p. 265] whether due to circumstances, want of skill, or undue hurry, will so increase the attenuation by errors that the raw observable correlation is brought down to a vanishing minuteness. This effect, so far from being remedied by increasing the range of the experiments and the number of cases, will almost inevitably be augmented thereby.
The principal information to be gleaned from this series is that the correspondence between Discrimination and Intelligence cannot well be appreciably due to Zeal, for the latter faculty had upon this occasion become the paramount differentiating influence.
(d) Experimental Series IV. On this occasion circumstances were as favorable to accuracy of experimentation (see page 249) as in the last two cases they had been the reverse. Accordingly, we find that Discrimination correlates with Talent in the four branches of study, Classics, French, English, and Mathematics, by an average of 0.51 with a probable error of only 0.03. This proportion of over 17 to 1 is amply sufficient to warrant us in proceeding to determine the real correspondence with considerable precision.
We will again take first the errors of measurement. As regards Intelligence, there was no difficulty in obtaining the required data; for the gradings were based upon the regular school examinations, so that if several of these be taken simultaneously into consideration, each will constitute a sufficiently independent set of measurements. The errors prove much smaller than those in the first experimental series; one examination paper correlates with another in the same branch of study by an average of 0.86; and one total examination correlates with another total examination by an average of 0.95; so that whereas before we had to make a compensation of 24%, we now only require those of 7% and 2%. From this it would appear that examination papers form a test that is far more constant and free from accidental errors of judgment than are teachers' subjective impression as to the "brightness" of their wards; but still this only refers to the reliability of the testing process and does not prejudice the question to be subsequently discussed as to whether the kinds of intelligence tested are different and of unequal value. As concers the grading of Discrimination, unfortunately the same impediment again frustrated the attainment of several independent sets of observations, so that we once more have no precise measure of compensation; we shall therefore have to make the same free estimate as before. Correcting, then, for both Intelligence and Discrimination, we get a true value of 0.69.
[p. 266] Proceeding next to irrelevant factors, these appear to have been reduced in the present series to such a minimum that there is no more any necessity for theoretical corrections. The matter of Sex does not again come into the question, since this school is only of boys. Nor is Age this time a disturbant, for its correlation with Discrimination works out to be the insensible amount of - 0.07. The latter circumstance completely disposes of another possible objection, for it might be urged against Experimental Series I that Age is by no means identical with stage of Growth, some children being more precocious than others, so that our correction to eliminate the former factor does not necessarily suffice to nullify variety in the latter one; hence, it might be argued, the whole correspondence between Intelligence and Discrimination could conceivably be due merely to the brighter children being also more forward with their senses. But though it must certainly be admitted that Age and Growth do not always keep level with one another, yet they at any rate correspond to the extent that when the former has ceased to exercise any influence at all the latter also must have become entirely inoperative; and as throughout the school the older boys do not in the least surpass the younger ones in Discrimination of Pitch, we may safely say that their faculty in this respect no longer depends to the smallest degree either on their Age or even on the stage of their Growth.
Still it may be interesting to know what would have occurred had the school lists not been artificially modified, but allowed to retain the factor of Age. In such case, the correlation comes to only 0.45. Thus by eliminating Age we had increased the correlation from 0.45 to 0.69; but to obtain such a rise it can easily be calculated that we must remove an irrelevant factor amounting to about 0.76; now, in the unmodified lists the actual correspondence of Age with Class Order turns out to be precisely this amount. Thus it would appear that the influence of Age was wholly irrelevant; not actual Proficiency but Native Capacity is the factor directly correlated with Discrimination; our apparently somewhat theoretical modification of the original school order was no empty abstraction but had after all a solid enough basis in present fact; if with regard to the educational curriculum it merely represented future possibilities, yet in other directions it showed itself to correspond with already efficient powers.
It is now also evident that the whole process of modifying [p. 267] the school lists could have been avoided; we could have left them in their raw state and simply have eliminated the irrelevant factor of Age by our usual formula; in this way, indeed, the theoretical precision would have been far greater, for our artificial treatment of the lists is only even approximately correct when Age is a paramount influence in deciding school place, and if applied, say, to university students, would produce an improper decrease (never increase) in the observed correlations; but practically the advantage is generally the other way, as in the present case, for the above theoretical incorrectness is more than compensated by a reduction gained in the probable error, owing to our being able to actually observe more of the correlation and thus leaving less to obtain by calculation.
As regards possible irrelevant correspondence with Practice, the present experimental series is again favorably situated; for though this factor has its usual large influence upon Discrimination of Pitch, there are in this case some positive data wherewith to measure it. Out of the thirty-three children it was ascertained that twenty-two were taking lessons in music, and these not unnaturally showed a much finer Sensitivity (a median of 2.3 v. d. as against one of 5 v.d.). We can therefore reject the eleven not learning music, thus confining the experiment to reagents on nearly the same level as concerns Practice; upon doing so, the correlation makes a further rise to 0.78. Or else we can work out the correlation of Discrimination with Music Lessons, and then remove this irrelevant factor by means of our formula; thereby we arrive at a similar value.
Let us now sum up this fourth series. With respect to the correlations of Discrimination with the School Studies separately we have arrived at an average of 0.78, of which figure 0.57 has been actually observed; to this aggregate 0.89 is contribted [sic] by Classics, 0.88 by French, 0.73 by English, and 0.61 by Mathematics; here the order may be considered as well enough evidenced, the ambiguity present in the first series having disappeared, since the observational errors have been calculated separately for each study. The correlation of Discrimination with the Total School Ability can be calculated with equal ease; the average raw correlation is 0.68, while the successive Total School Orders correlate with one another by 0.95, so that the corrected required correlation becomes 0.87.
In this series we have had the good fortune, not only to ob- [p. 268] tain reliable estimates of almost all the perturbing influences, but even to eliminate them in practice so that an average of no less than 0.68 can be actually observed. In the further calculation the only value that cannot be approximately relied upon is that of the errors in measuring Discrimination, and even this cannot well be very much smaller than the amount here adopted and, as is subsequently proved, cannot possibly be much larger. The existence of other sources of fallacy yet lurking, either to reduce or to still further augment the total cannot of course ever be categorically disproved; but at any rate a careful search has been made and has so far failed to reveal them. The whole system of results exhibits such regular unconstrained compliance with the definite laws governing correlations (and also present such other remarkable uniformities to be discussed later on) that they appear to offer every guarantee of being perfectly normal.
The correlational value is, however, considerably larger than that found for the same sense in Experimental Series I. The readiest explanation of this discrepancy would appear to lie in a fact which I have often had occasion to notice, namely that when the reagents are very unpractised in any form of sensory Discrimination, the latter correlates with their Intelligence by a much smaller amount.
(e) Conclusions. On the whole, then, the results of all four experimental series appear sufficiently concordant with one another. Whenever we have succeeded in obtaining a fairly pure correction between Sensory Discrimination and Life Intelligence, we have found it amount to a very considerable value. In the case of Pitch, it came to as high as 0.87. Very possibly other discriminative functions would show similar results, while some would prove much more specific (and usually dependent on factors peripheral to the nervous system).
3. Correspondence between General Discrimination and General Intelligence.
Up to now, we have only discussed the correspondence of the various Intelligences with the various sensory activities, Hearing, Sight, Touch, etc. Such isolated facts are interesting enough, but quite otherwise important is the relation of any common and essential element in the Intelligences to any common and essential element in the Sensory Functions. For brevity, we will term these common elements "General Intelligence" and "General Discrimination," but always with the reservations made in the first section of this chapter.
Curiously, this more general correspondence can in the present case be settled with much greater precision than was possible for the specific relations. This is due to our now [p. 269] having adequate data wherewith to measure the errors of observation, seeing that all the experimentally obtained gradings of specific Discrimination constitute so many one-sided independent attempts to grade the General Discrimination; the amount of observational error will be quantitatively revealed in the correlations between one grading and another.
(a) The Village School. Here our calculation is as follows. The average of the nine correlations between the Intelligences and the Discriminations comes, as we have seen, to 0.38; the two kinds of intellective gradings correlate with one another by an average of 0.55; and the three gradings in Discrimination do so by 0.25. Therefore by the theoretical formula the true correlation between General Intelligence and General Discrimination comes to
Checking this by the second or empirical method, we find that on taking an amalgamation of the three intellective gradings with an amalgamation of the three gradings in Discrimination, the correlation rises to 0.66. Therefore the true correlation between General Intelligence and General Discrimination comes in this way to
This again may be further checked by taking our amalgamation two instead of three lists at a time; in this way we get nine different correlations which present an average of 0.55, so that our required result now becomes 0.96. Therefore an average again gives us as nearly as possible 1.00.
Thus we arrive at the remarkable result that the common and essential element in the Intelligences wholly coincides with the common and essential element in the Sensory Functions.
(b) The High Class School. Here, also, the children were tested in the three senses, but unfortunately, as we have seen, the results for Light and Weight are not seriously usable, so that we no longer have sufficient material for constructing a "General" Discrimination. [p. 270]
This default, however, has been made good by what appears to be a very happy substitute. Our main correlations have dealt with reagents all undergoing musical instruction, and I have kindly been furnished with a complete order of their relative abilities in this department. Musical talent has always been recognized as being not so much an intellective as a sensory function; whole nations appear almost devoid of it, without therefore showing themselves any less intelligent; lunatic asylums, on the contrary, often contain a surprising share of the faculty. We will, then, take this as our second sensory function, will note whether it presents any community with Discrimination of Pitch, and if so will compare this common element with that obtaining between the intellective functions. As regards the first point, it may be noted that hitherto very conflicting opinions have been stoutly maintained; the great majority of writers have held Musical Talent and Pitch Discrimination to be very intimately connected and even go so far as to directly term the discriminative power "musical sensitiveness;" while a few, but including perhaps the ablest judges, flatly deny any such correspondence whatever. The actual facts would at first sight seem to lie wholly on the side of the former tenet, seeing that the correlation works out to the substantial amount of 0.40 (or about 0.63, when corrected for errors). Next, these two auditory functions correlate with the Intelligences by 0.57 and 0.55 respectively, and the latter correlate with one another to the amount of 0.71. Thus the relation between the element common to the two former and that common to the four latter will be given by
We can now check the result by the empirical formula; for we find that the amalgamated order derived from the two sensory faculties correlates with the amalgamated order derived from the four Intelligences by 0.72; so that the required correlation comes to
Taking as usual the mean, we again reach a final correlation of precisely 1.00, and therefore once more must conclude that the element common to the sensory activities also wholly coincides with that common to the intelligences. [p. 271]
Before passing, it may be remarked that thus after all those were virtually in the right who maintained Musicality and Pitch Discrimination to have no correspondence with one another; for though a correspondence really does exist, yet it is not to the smallest degree of the specific character contemplated by those who talk of "musical sensitivity." It must here also be noted that this surprising intellectuality of musical talent by no means annihilates the many well-evidenced phenomena seeming to indicate the contrary; one fact cannot destroy another, and any apparent conflict merely proves our imperfect acquaintance with their true nature.
(c) Practical Verification of the Argument. The conclusion above arrived at is so important and the method or argument is so new, that I have endeavored to reproduce analogous circumstances artificially, so that any one may easily test any portion of the reasoning.
The main argument was repeated as follows. A target was constructed of a great many horizontal bands, numbered from top to bottom. Then a man shot successively at a particular series of numbers in a particular order; clearly, the better the shot, the less numerical difference between any number hit and that aimed at; now, just as the measurement of any object is quite appropriately termed a "shot" at its real value, so, conversely, we may perfectly well consider the series of numbers actually hit in the light of a series of measurements of the numbers aimed at. When the same man again fired at the same series, he thereby obtained a new and independent series of measurements of the same set of objects. Next, a woman had the same number of shots at some set numbers in a similar manner. If, then, our above reasoning and formulae are correct, it should be possible, by observing the numbers hit and working out their correlations, to ascertain the exact resemblance between the series aimed at by the man and woman respectively. In actual fact, the sets of numbers hit by the man turned out to correlate with those hit by the woman to the extent of 0.52; but it was noted that the man's sets correlated with one another to 0.74, and the woman's sets with one another to 0.36; hence the true correspondence between the set aimed at by the man and that aimed at by the woman was not the raw 0.52, but
that is to say the two persons had fired at exactly the same series of bands, which was really the case. I repeated this ex- [p. 272] periment, testing three times by the first or theoretical formula and four times by the empirical one; by both methods the average came to just upon 1.00, with a mean variation above and below of precisely similar dimensions to those in our instances of Discrimination and Intelligence. Thus the experimental justification of our method of argumentation was as complete as could well be desired.
(d) Conclusion. On the whole, then, we reach the profoundly important conclusion that there really exists a something that we may provisionally term "General Sensory Discrimination" and similarly a "General Intelligence," and further that the functional correspondence between these two is not appreciably less than absolute.
Besides its intrinsic value, such a general theorem has the enormous advantage over the specific results of the last section of being independent of any particular conditions; it has nothing to do with the procedure selected for testing Discrimination and Intelligence, nor even with the accuracy of its execution, nor indeed even with the homogeneousness of the experimental subjects; if correct, the proof should be reproducible in all times, places, and manners -- on the sole condition of adequate methodics.
4. Universal Unity of the Intellective Function.
In view of this community being discovered between such diverse functions as in-school Cleverness, out-of-school Common Sense, Sensory Discrimination, and Musical Talent, we need scarcely be astonished to continually come upon it no less paramount in other forms of intellectual activity. Always in the present experiments, approximately, 
I have actually tested this relation in twelve pairs of such groups taken at random, and have found the average value to be precisely 1.00 for the first two decimal places with a mean deviation of only 0.05. All examination, therefore, in the different sensory, school, or other specific intellectual faculties, may be regarded as so many independently obtained estimates of the one great common Intellective Function.
Though the range of this central Function appears so universal, and that of the specific functions so vanishingly minute, the latter must not be supposed to be altogether non-existent. [p. 273] We can always come upon them eventually, if we sufficiently narrow our field of view and consider branches of activity closely enough resembling one another. When, for instance, in this same preparatory school we take on the one side Latin translation with Latin grammar and on the other side French prose with French dictation, then our formula gives us a new result; for the two Latin studies correlate with the French ones by an average of 0.59, while the former correlate together by 0.66 and the latter by 0.71; so that the element common to the Latin correlates with the element common to the French by
That is to say, the two common elements by no means coincide completely this time, but only to the extent of 0.86 or 74%; so that in the remaining 26%, each pair must possess a community purely specific and unshared by the other pair.
We therefore bring our general theorem to the following form. Whenever branches of intellectual activity are at all dissimilar, then their correlations with one another appear wholly due to their being all variously saturated with some common fundamental Function (or group of Functions). This law of the Universal Unity of the Intellective Function is both theoretically and practically so momentous, that it must acquire a much vaster corroborative basis before we can accept it even as a general principle and apart from its inevitable eventual corrections and limitations. Discussion of the subjective nature of this great central Function has been excluded from the scope of the present work. But clearly, if it be mental at all, it must inevitably become one of the foundation pillars of any psychological system claiming to accord with actual fact -- and the majority of prevalent theories may have a difficulty in reckoning with it.
Of its objective relations, the principal is its unique universality, seeing that it reappears always the same in all the divers forms of intellectual activity tested; whereas the specific factor seems in very instance new and wholly different from that in all the others. As regards amount, next, there seems to be an immense diversity; already in the present examples, the central factor varies from less than 1/5 to over fifteen times the size of the accompanying specific one. But all cases appear equally susceptible of positive and accurate measurement; thus we are becoming able to give a precise arithmetical limitation [p. 274] to the famous assertion that "at bottom, the Great Man is ever the same kind of thing."
Finally, there is the exceedingly significant fact that this central Function, whatever it may be, is hardly anywhere more prominent than in the simple act of discriminating two nearly identical tones; here we find a correlation exceeding 0.90 indicating the central Function to be more than four times larger than all the other influences upon individual differentiation. Not only the psychical content but also the external relations of Sensory Discrimination offer a most valuable simplicity; for it is a single monotonous act, almost independent of age, previous general education, memory, industry, and many other factors that inextricably complicate the other functions. Moreover, the specific element can to a great extent be readily eliminated by varying and combining the kind of test. For these reasons, Discrimination has unrivalled advantages for investigating and diagnosing the central Function.
5. The Hierarchy of the Intelligences.
The Theorem of Intellective Unity leads us to consider a corollary proceeding from it logically, testing it critically, and at once indicating some of its important practical uses. This corollary may be termed that of the Hierarchy of the Specific Intelligences.
For if we consider the correspondences between the four branches of school study, a very remarkable uniformity may be observed. English and French, for instance, agree with one another in having a higher correlation with Classics than with Mathematics. Quite similarly, French and Mathematics agree in both having a higher correlation with Classics than with English. And the same will be found to be the case when any other pair is compared with the remainder. The whole thus forms a perfectly constant Hierarchy in the following order: Classics, French, English, and Mathematics. This unbroken regularity becomes especially astonishing when we regard the minuteness of the variations involved, for the four branches have average correlations of 0.77, 0.72, 0.70, and 0.67 respectively.
When in the same experimental series we turn to the Discrimination of Pitch, we find its correlations to be of slightly less magnitude (raw) but in precisely the same relative rank, being: 0.66 with Classics, 0.65 with French, 0.54 with English, and 0.45 with Mathematics. Even in the crude correlations furnished by the whole school without excluding the non-musicians, exactly the same order is repeated, though with [p. 275] the general diminution caused by the impurity: Classics 0.60, French 0.56, English 0.45, and Mathematics 0.39.
Just the same principle governs even Musical Talent, a faculty that is usually set up on a pedestal entirely apart. For it is not only correlated with all the other functions, but once again in precisely the same order: with Classics 0.63, with French 0.57, with English 0.51, with Mathematics 0.51, and with Discrimination 0.40. Ability for music corresponds substantially with Discrimination of tones, but nevertheless not so much as it does with algebra, irregular verbs, etc.
The actual degree of uniformity in this Hierarchy can be most conveniently and summarily judged from the following table of correlation; the values given are those actually observed (theoretical correction would modify the relative order, but in no degree affect the amount of Hierarchy or otherwise). Each number shows the correlation between the faculty vertically above and that horizontally to the left; except in the oblique line italicized, the value always becomes smaller as the eye travels either to the right or downwards.
Altogether, we have a uniformity that is very nearly perfect and far surpasses the conceivable limits of chance coincidence. When we consider that the probable error varies between about 0.01 for the ordinary studies to about 0.03 for music, it is only surprising that the deviations are not greater. The general Hierarchy becomes even more striking when compared with [p. 276] the oblique line, which is no measure of the central Function and where consequently the gradation abruptly and entirely vanishes.
The above correlations are raw, and therefore do not tell us either the true rank of the respective activities or the full absolute saturation of each with General Intelligence. For the former purpose we must eliminate the observational errors, and for the latter our result must further be squared. Thus we get:
   [Classics Note: footnotes 3, 4 and 5 in table correspond to footnotes 129, 130 and 131 respectively]
It is clear how much the amount of any observable raw correlation depends upon the two very different influences: first, there is the above intellective saturation, or extent to which the considered faculty is functionally identical with general Intelligence; and secondly, there is the accuracy with which we have estimated the faculty. As regards the ordinary school [p. 277] studies, this accuracy is indicated by the oblique italicized line, and therefore appears about equal in all cases (not in the least following the direction of the Hierarchy); but in other cases there is a large divergence on this head, which leads to important practical consequences. Mathematics, for example, has a saturation of 74 and Common Sense has one of about 96; but in actual use the worth of these indications becomes reversed, so that a subjective impression as to a child's "brightness" is a less reliable sign than the latter's rank in the arithmetic class; almost as good as either appears a few minutes' test with a monochord.
In the above Hierarchy one of the most noticeable features is the high position of languages; to myself, at any rate, it was no small surprise to find Classics and even French placed unequivocally above English (note that this term does not refer to any study of the native tongue, but merely to the aggregate of all the lessons conducted therein, such as History, Geography, Dictation, Scripture, and Repetition).
However it may be with these or any other special facts, here would seem to lie the long wanted general rational basis for public examinations. Instead of continuing ineffectively to protest that high marks in Greek syntax are no test as to the capacity of men to command troops or to administer provinces, we shall at last actually determine the precise accuracy of the various means of measuring General Intelligence, and then we shall in an equally positive objective manner ascertain the exact relative importance of this General Intelligence as compared with the other characteristics desirable for the particular post which the candidate is to assume (such as any required Specific Intelligences, also Instruction, Force of Will, Physical Constitution, Honesty, Zeal, etc.; though some of these factors cannot easily be estimated separately, there is no insuperable obstacle to weighing their total influence as compared with General Intelligence). Thus, it is to be hoped, we shall eventually reach our pedagogical conclusions, not by easy subjective theories, nor by the insignificant range of personal experiences, nor yet by some catchpenny exceptional cases, but rather by an adequately representative array of established facts.
6. Outer Factors Determining the Amount of Correlation.
The values given in the preceding section show the correlations found for various specific activities. These amounts, however, are not wholly constant; the apparent or raw correlations, as we have abundantly seen, deviate in every direction, depending almost entirely upon the number and kind of impurities suffered to enter and vitiate them; but even the true corrected correlation appears to admit of no inconsiderable variation, [p. 278] according to the conditions of experiment. Generally speaking, the amount seems to be always greater in proportion as the reagents are enabled to manifest their finest powers; nor is such factor readily resolvable into the greater regularity thereby obtained. It is especially conspicuous in the following phenomena:
The correlation is augmented when Discrimination is calculated according to the reagents' acutest perception under the most favorable circumstances, rather than according to their freedom from accidental slips. The correlation is larger when all the reagents have the function in question well developed, either by general habit or by careful fore-exercise, than when all are in a comparatively backward stage. So far, this increase has only manifested itself for the lower grades of practice; it might conceivably be reversed, on approaching the higher grades of special training.
The correlation increases, when the conditions of examination are such as least to distract or puzzle the reagent.
The other factor appearing at all likely to determine the amount of correlation is Age. Within the narrow range of most of our experiments, however, namely from nine to fourteen years, this influence is not apparent. To measure exactly any such change, correlations have been calculated of a secondary order, that is, between the above correspondence on the one hand and Age on the other; even among the village children, who are still in the process of developing their sensory acuteness, this correlation of secondary order only amounts to - 0.15, thus indicating that the correspondence between Discrimination and Intellect is almost as great among the older children as among the younger ones. For the boys of the preparatory school, who have already reached their full sensory powers, this secondary correlation comes to - 0.07, testifying that the above correspondence is as nearly as possible uniform throughout the different ages. Finally we have seen evidence that this correspondence is the cause of the correspondence between one kind of Discrimination and another, and the latter has proved of exactly the same value for adults as for children (0.25 raw).
It must, however, be mentioned that the opinion has been arrived at and stoutly defended by Wissler, that any corre- [p. 279] spondence between Intelligence and Mental Tests can only exist among young children and must disappear with advancing adolescence. He bases his view firstly upon the fact that his Columbia experiments with University students show no appreciable correlations of this nature; this argument will be dealt with in the next section, criticising these same experiments. And secondly, he truly enough remarks that even the correlations found by Gilbert in children from six years upwards are no longer evident in their seventeenth and eighteenth years. But Wissler does not seem to have been able to measure these correlations of Gilbert quantitatively, the latter not having furnished the data required for the standard formula; they may, however, easily be reckoned by means of the method of "class averages;" they will then be seen to be all very irregular and throughout in dangerous proximity to the amount of the probable error; though the seventeenth and eighteenth years do indeed show little correspondence, yet the sixteenth year exhibits the very highest of all while the twelfth year has the lowest; as regards the general tendency of these correlations, it is really if anything to increase with Age (such tendency only amounts to 0.15 ± 0.08 and therefore is probably a mere chance).
7. Previous Researches Conflicting with the Present Results.
The great bulk of past experiments do not admit of direct comparison with the present ones; for while the latter have been expressly confined to the most elementary forms of laboratory procedure, the former have continually striven to cover vast territory so as to summarily exhaust the whole problem. These more ambitious researches, therefore, can only be criticised in the general manner attempted in Chapter II.
But three investigations -- luckily such as to represent the best work accomplished in this department -- have also included our present topic, simple Sensory Discrimination. These three will now be discussed in more detail.
Gilbert. The first is that of J. Gilbert, whose valuable and already frequently mentioned experiments upon over 1,000 school children in 1893 included an inquiry into "muscle sense" and "color differences." The apparatus for the former, as for my own, consisted in a number of small boxes which looked and felt exactly similar but were really a set of finely graduated weights. Also his "color differences" were very analogous to my sight tests, for both utilized a series of objects each slightly darker than the preceding one and both were executed by day- [p. 280] light. His grading of "mental ability" lay in the usual classification of the children by their respective masters into "bright," "average," and "dull."
As regards his results, Gilbert is perhaps the most conspicuous among those investigators who do find an appreciable correspondence between "mental tests" and "general ability." Concerning the above two Sensory Discriminations, he unfortunately confines himself to the following indirect information: "The curves for reaction-time gave the most positive results showing that the brighter the child the more quickly he is able to act. In discrimination the same relation is noticeable but to a less degree." Thus we are referred on to the correlation for reaction-time, which is everywhere emphasized. "The difference between the reaction-time of those who were bright, of average mental ability and dull respectively" becomes "very noticeable." Again, "the bright children react much more quickly than the dull," and "it is shown here that we judge of a child's mental ability by the quickness or rapidity with which they were able to act." Though Gilbert contents himself with these utterances of rather vague character, he here carefully furnishes most of the essential data for more precise conclusions; he tells us the average reaction-time of "bright," "average," and "dull" for each of the twelve years tested; also he gives us all the mean variations.
With this knowledge, we are easily enabled to work out the exact correlation for ourselves, and are surprised to find that it after all averages no more than 0.19 ( ± 0.04). The correlation of general ability with sensory discrimination, being even less noticeable, must indeed be minute.
To explain this low figure we must note that from the described mode of procedure the conditions would seem to have here been at least as unfavorable as they were in the present Series III. In confirmation of this view, it may be seen that [p. 281] the average threshold found by Gilbert for discrimination of weight is as coarse as 1/11, whereas even that in Series III comes to 1/15 and the more carefully executed Series I shows 1/20. The natural consequence would necessarily have been to produce a similar attenuation by inadequate representativeness and therefore a similarly small correlation as in that series. Moreover, such an effect would have been materially enhanced by the heterogeneity of Gilbert's reagents, which must inevitably have introduced serious irrelevant correlation. If these considerations be justified, his observed correlations must be taken as being very much smaller than the true ones would have been.
Seashore. The next results to be criticised are those of Seashore, which include an important investigation into the same Pitch Discrimination with which our own experiments have been so much occupied. Seashore, as we have seen, comes to the resolutely negative conclusion "that there is no functional relation between any one of these (mental tests) and general mental ability."
To support this verdict in the case of Pitch, he draws up the following table, remarking, "the distribution of the results practically coincides with the most probable distribution according to chance, which is indicated in the parentheses."
But this ingenious mode of calculating correlation is of a somewhat disseminated nature, and one may be pardoned for thinking it hardly adapted for giving very accurate results. If, instead of trying to consider fifty amounts all at the same time, we sum them up a little and compare averages we shall find after all a fairly definite tendency for the higher ability to be also accompanied by higher place in Discrimination. If we desire a still more unified and really usable result, we can easily obtain it by any of Pearson's or my auxiliary method; it works out by Pearson's method to 0.24 ( ± 0.07). Thus here, where functional relation has been so categorically denied, it is in reality greater than in Gilbert's reaction-time where it was held up as being so complete. [p. 282]
But this value, though plain and positive enough, must nevertheless be admitted to be somewhat small as compared with that in my own experiments. Nor can this be altogether explained in precisely the same way as in the last example, for here the method of procedure seems to have been much more deliberate; the experimentation was skillfully designed, and the seven minutes allowed to each reagent was perhaps sufficient for its proper execution, though about a quarter of an hour appears to me better. But when we consider the composition of the sample of persons experimented upon, we come upon an irrelevant correlation of great magnitude; for we find that though Seashore remarks a great discrepancy between the children under and over ten years of age, yet he throws them all together into the same correlation and thus introduces an irrelevant connection with Age which comes to no less than 0.73 (reckoned again by Pearson's auxiliary method). This irrelevancy has evidently just the same effect, whether it be really due to difference of age; or, as I have suggested, partly to disparity of culture; or even, as Seashore himself supposes, to imperfection of experimentation. In all cases alike, the real correlation comes to about 0.35, which is almost exactly the same result as obtained in my own Series I before allowing for further errors.
Columbia. We now pass to the third and last series of experiments bearing on our particular question. This is the very extensive and in many respects important one that has been continuously conducted at Columbia University for the past ten years. For our present purpose, it includes "Perception of Weight" and "Perception of Pitch," while an intellectual grading is obtained from the students' average standing in the various university courses.
Again the general conclusion is an unqualified negative, reading as follows:
"The markings of the students in college classes correlate with themselves to a considerable degree, but not with those made in the laboratory."
Here our critical review is rendered much simpler by the correlations having been properly calculated and plainly stated. Wissler's statement is fully borne out by the values quoted, [p. 283] the average correlation of the tests with Intelligence being only 0.06 and that of the tests among themselves being 0.09, thus in neither case much exceeding the size of the probable error, though the great majority of instances are at any rate positive. The amount for Perception of Weight is unfortunately not given, but that for Perception of Pitch comes to 0.16.
Above we have seen that Wissler would account for this minuteness of correspondence by the greater age of the students as compared with the children usually experimented upon. But so far our available evidence is not at all that correspondence diminishes with age, but rather that it is completely independent thereof. Moreover there does not appear any urgent need of introducing such a factor, seeing that perhaps sufficient explanation is forthcoming otherwise. To begin with, university students are not exactly the average from the schools but more or less a selection of the most able; hence they intellectively form a more homogeneous class, so that all their intellective correlations will be somewhat reduced in amount. More serious, probably, is the fact that the test of Discrimination has here been so impurified by alien elements, that even Wissler himself prefers to call it a test of Memory.
But perhaps the chief source of the lowness of the correlations will be found in the following circumstances, namely: that the subjects were examined three at a time, each being managed by some "student or officer of the department;" that no less than twenty-two different tests were carried out, many of a most difficult character, besides measuring the length and breadth of each reagent's head; that during the leisure moments afforded him in the course of these tests the observing 'student or officer of the department" had to note in writing the contour of the reagent's forehead, the character of his hair, the nature of his complexion, the color of his eyes, the shape of his nose, the description of his ears, of his lips, of his hands, of his fingers, of his face, and of his head -- and that this whole procedure is considered to be satisfactorily completed in forty-five minutes.
On the whole, then, the apparent conflict of previous researches with the present ones does not appear fundamental or such as in any way to invalidate the evidence now produced.
8. Summary of Conclusions.
To conclude, the following is a brief summary of the principal conclusions indicated by the foregoing experiments:
I. The results hitherto obtained in respect of psychic correlation would, if true, be almost fatal to experimental psychology as a profitable branch of science. But none of these results, as at present standing, can be considered to possess any [p. 284] value other than suggestive only; this fact is not so much due to individual shortcomings of the investigators, as to the general non-existence of any adequate system of investigation.
II. On making good this methodological deficiency, there is found to actually occur a correspondence -- continually varying in size according to the experimental conditions -- between all the forms of Sensory Discrimination and the more complicated Intellectual Activities of practical life.
III. By this same new system of methodics, there is also shown to exist a correspondence between what may provisionally be called "General Discrimination" and "General Intelligence" which works out with great approximation to one or absoluteness. Unlike the result quoted in the preceding paragraph, this phenomenon appears independent of the particular experimental circumstances; it has nothing to do with the procedure selected for testing either Discrimination or Intelligence, nor with the true representativeness of the values obtained by these tests, nor even with the homogeneousness of the experimental reagents; if the thesis be correct, its proof should be reproducible in all times, places, and manners -- on the sole condition of adequate methodics.
IV. The above and other analogous observed facts indicate that all branches of intellectual activity have in common one fundamental function (or group of functions), whereas the remaining or specific elements of the activity seem in every case to be wholly different from that in all the others. The relative influence of the general to the specific function varies in the ten departments here investigated from 15:1 to 1:4.
V. As an important practical consequence of this universal Unity of the Intellectual Function, the various actual forms of mental activity constitute a stably interconnected Hierarchy according to their different degrees of intellective saturation. Hence, the value of any method of examination as to intellectual fitness for any given post is capable of being precisely ascertained, since it depends upon:
(a) the accuracy with which it can be conducted;
(b) the hierarchical intellective rank of the test:
(c) the hierarchical intellective rank of the duties involved in the post.
Methods have been given whereby all these three points can be sufficiently ascertained.
VI. Discussion as to the physical nature of this fundamental Function has been reserved until a more complete acquaintance has been gained concerning its objective relations. Among the latter, the principal and determining one is its [p. 285] unique position as indicated in paragraph IV. The chief further evidence is to the following effect:
The function appears to become fully developed in children by about their ninth year, and possibly even much earlier. From this moment, there normally occurs no further change even into extreme old age.
In adult life, there would seem no appreciable difference between the two sexes.
The Function almost entirely controls the relative position of children at school (after making due allowance for difference of age), and is nine parts out of ten responsible for success in such a simple act as Discrimination of Pitch.
Its relation to the intellectual activity does not appear to be of any loosely connected or auxiliary character (such as willingness to make an effort, readiness in adaptation to unfamiliar tests, or dexterity in the fashion of executing them), but rather to be intimately bound up in the very essence of the process.
Even if we neglect the slight discrepancy between the two sorts of intelligence, the result will not be very different, for
Note that as we have here an amalgamation for only one of the two compared series, we must take the square instead of the fourth root of the number of amalgamated lists.
 This result depends upon only one short set of observations; also no detailed rank had been furnished, but merely that favorite but particularly bad classification into "bright," "average," and "dull."
This method of correcting three inter-correlated terms in succession beginning with the smallest of them, though far from being theoretically exact, nevertheless appears sufficiently applicable to the large majority of actual cases including the present one.
 If this small difference of value between the theoretical and empirical results be minutely investigated, it can be clearly proved to be solely attributable to mere chance, as indeed might well be expected from its small dimensions.
 Where rpq= the mean of the correlations between the members of the one group p with the members of the other group q,
rpp= the mean of the inter-correlations of the members of the group p among themselves,
and rqq= the same as regards group q.
 Of course, notable instances will easily be found where musical ability is apparently divorced from General Intelligence; in this very school, for example, the best musician is far from standing high intellectually. But not even the most extreme cases necessarily contravene the above rule. A correlation does not state any absolute coincidence between two faculties, but only a limited and precisely measured tendency in this direction; so far from excluding deviations, it proclaims them and even estimates their exact probability. If we may assume the normal law of frequency to approximately hold good and may abstract from further influences, then the proportion of persons with any given amount of musical talent who will attain to any given degree of stupidity (or vice versa)
where h is a measure of the correlation between Musicality and Intelligence, and a = the given inferiority in the latter faculty.
 The only other data of this kind with which I am acquainted are some comparisons made between the different branches of study at the Columbia University in the course of the research quoted on page 218. The correlations there obtained, which were throughout somewhat smaller than the above, manifest only a limited concordance with our above principle of Hierarchy. But a university is clearly not the place in which to look for natural correspondence between functions; at that time of life, strong ties of a wholly artificial sort have intervened; each student singles out for himself that particular group of studies tending to his main purpose and devotes to them the most judicious amounts of relative energy. To determine natural correlations, we must go to where the pupils meet each other in every department on relatively equal terms.
 The opposite and more usual view, namely, that mathematics form an entirely independent faculty, will be found expounded in 331 pages "Ueber die Anlage zur Mathematik" by the well-known psychiatrist, Möbius. Similar evidence is brought by him to the effect that this talent is proportional to the development of the upper outer orbit of the eye, especially the left.
 This seems to indicate an opposition between the sensory acuteness due to Intelligence and that arising from Practice. This, again, would evidently conflict with Fechner's principle of measuring the sensory threshold by means of Gauss' formula.
 It must be mentioned that Binet has arrived at the contrary conclusion, namely that correlations with Intelligence are only observable on first trial and almost disappear when the reagents are again tested. See L'année psychologique, Vol. VI.
 Here we have an illustrative instance of operating with large numbers of cases. This correlation of Gilbert was based upon an examination of no less than some 1,100 children; therewith we may compare the chief correlation in the present Series IV based on examination of only 22. To the layman, the latter result would seem the immensely more exposed of the two to the danger of being a mere chance coincidence. But when the matter is worked out precisely according to the true and established laws of chance, a correlation like that found by Gilbert, being less than four times its probable error, would occur by mere accident about once in 200 times; whereas the correlation in Series IV, being over twenty times its probable error, would not so occur in millions upon millions.
 Once more Dr. Seashore has an easy opportunity of practically testing these theoretically gained conclusions concerning this work; if he will exclude the disparate cases under ten years of age, and still more if he will then confine his consideration to such as have learnt music, he will be able definitely to ascertain whether or not the correspondence does not thereupon become very apparent.