Thorndike - Educational Psychology - Distribution of Mental Traits

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    CHAPTER HLTHE DISTRIBUTION OF MBNTAX, TRAITS.

    Do the distributions of mental traits in groups of individualsfollow any regular law ? Are the differences between individualsin mental capacities and characteristics explainable by any simpleset of causes and amenable to any single type of description ? Ifsuch uniformity exists, the exact study of educational problemsis possible and even easy.

    It has been supposed, for more or less satisfying reasons, thatin any group of individuals representing a single species, in re-spect to any trait not then in-fluenced by natural selection,the distribution would be thatof a chance event, the surface -,of frequency being that of theprobability integral. The exactmeaning of this supposition andthe basis for it need not be dis-cussed here. Our interest is indiscovering whether any onetype of distribution does charac-terize all mental traits in hu-man beings. By using graphicrepresentations rather than alge-.braic formulae the answer and -the evidence for it can be made ' '^clear even to one who knows nothing whatever of the mathematicalproperties of the surface of frequency of a chance event or ofany other.

    Figure 4 gives the distribution or surface of frequency of thetype to which perhaps all the distributions of mental traits con-form. Figure 5 gives the same distribution as figure 4c, but witha coarser separation into grades. Figure 6 gives again the samedistribution, but this time with a very fine subdivision of grades.

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    14 EDUCATIONAL PSYCHOLOGYOur question is, Are mental traits commonly distributed

    after the type of figures 4r-6 ? We can answer it by comparingwith figures 4-6 figures 7-24:, each of which represents the actualdistribution found for some mental trait. To make the compari-son easy a light dotted line shows in each case the rough outlineof figure 6. The reader has then simply to note how closely theactual distributions follow the dotted lines.*

    In drawing these surfaces the median value is made to coin-cide with the median value of the normal surface of frequency.Whenever any distribution represents measurements by differentpeople or under different conditions or of different sexes or withdifferent tests, it is compounded of separate surfaces each drawnwith consideration of the mean and variability proper to the singlegroup and then so combined as to allow roughly equal weight tothe distributions from equal numbers of cases. No distributionsare thus combined unless they all individually represent the sametype as they do when combined. Barring the inclusion in thesame distribution of the different racial types found in schools,of children sometimes as much as 12 months apart in age in theage groups and children in the school groups as noted, there areno sources of the variability found save precisely those which weare trying to measure. The strongest proof of the approximationof the distribution of mental traits to the normal type is givenby a score or more of distributions from too few cases to appearhere which yet all follow the normal type.

    Since the aim of this section is simply to show the generalfact of distribution, not to' analyze it precisely, I have made noattempt to ascertain whether after all the combination of mentalspecies is not present That may well be the case, but for thepractical purposes of educational science it makes little differ-ence. Our use of the fact of approximation to the normal typewill be justifiable in either case.In all these cases there is a remarkable uniformity in the dis-tribution of mental traits -amongst individuals. In all cases the* For the sake of the leader versed in statistics, I may add that thescale of the base line for each distribution is so arranged that the deviationof mean square for each one is represented always by the same length andthe total number of cases by the same area. It is thus possible to compareany one with any other or with the normal frequency curve.

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    TEE DISTRIBUTION Of MENTAL TRAITS 15

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    16 EDUCATIONAL PSYCHOLOGYaverage ability is near the common ability and both are near thepoint above which. 50 per cent of the cases lie. The greaternumber of the cases lie near the average, mode or median point,and degrees of ability a certain amount above or below that pointare nearly equally common. The more remote a degree of abilityis from the average or median or mode, the fewer are the indi-viduals who possess it. The difference between the degrees ofability above and below the average, mode or median betweenwhich 50 per cent of the individuals are included is about twoninths of the difference between the lowest and highest degreesof ability found.-'' This type of distribution is called the normal distribution. Itapproximates the types found for most variable organs or func-tions in nature in the case of any single species when the organor function in question is not subject to selection.

    The so common fact of the approximately normal distributionof mental traits leads to many important theoretical considera-tions and gives many possibilities of studying human nature thatwould otherwise not exist But for our purposes most of its re-sults may be neglected. For us a knowledge of the existence andfrequency of normal distribution is of consequence first becauseit emphasizes the fact of human individual differences and givesus a precise idea of their amount; second, because it enables usto compare groups accurately. In the study of heredity, for in-stance, we shall compare the group *children of parents possessingsuch and such a degree of such and such a mental trait' withchildren of parents possessing a different degree thereof. In thestudy of sex differences we shall compare the group men with thegroup women, the group 10-yeaix>ld boys with the group 10-year-old girls. In the study of the influence of the environment weshall compare 'group with such and such training' with 'groupwithout it* In the study of growth and maturity we shall com-pare different age groups. In all these cases we can get muchmore illuminating and precise and extensive knowledge by com-paring the distribution curves for the two groups than by usingmere arithmetical averages. We shall also avoid a number ofmisunderstandings and fallacies by bearing in mind the fact of

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    THE DISTRIBUTION OF MENTAL TRAITS 17the variability of mental traits and the prevalence of variabilityof the normal type.The meaning of cases where the distribution of mental traitsdoes not follow the normal frequency curve will become dear ifwe examine, first, some cases of the distribution of a trait in agroup of individuals of two or more distinct species and, second,cases where some selective agency has been at work.

    FIGS. 25-30.

    Figure 25 gives the distribution of ability in the test in mark-ing A's of a group of children 8, 9, 14 and 15 years old. Fromfigure 26, which gives separately the distributions (1) for those8 and 9, and (2) for those 14 and 15, we see clearly that the pecul-iar flattening of figure 25 is due to the mixture of two specieseach of which approximates fairly to the normal type. The sameresult of mixture is shown even more emphatically by figure 27,which gives the distribution of a group composed of about 140third grade and about 180 seventh grade girls. Here the twomodes belonging to the two grades are easily distinguishable. Areal case of the same sort of distribution is pictured in figure 28,which gives the distribution of strength of arm in human adults.

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    18 EDUCATIONAL PSYCHOLOGYThe two species here are men and women.* When we find inthe surface of frequency of a mental trait a departure from thenormal toward a lowering and broadening of the surface or towardtwo or more modes we may com-

    monly expect to find a mixtureof species. Other illustrationsof such a condition are given infigures 29 and 30.

    Figure 31 gives the distribu-tion in a test of controlled associ-ation of the 12-year-old boys inthe 6A grade or higher. The lackof symmetry in the surface isobviously due to the fact that wdare dealing with a selected group ;that the duller and less matureboys have been eliminated. Theinfluence of the opposite sortof elimination is seen in figure32, which gives the distribu-

    tion in the same trait of 12-year-old boys in the grades lower thanthe 6A. By combining the two we should have a normal fre-quency surface. Figure 33 gives a real case of a distributiondistorted by selection comparable to those artificially produced inour examples. It is the distribution of mathematical ability inthe candidates for honors in mathematics at Cambridge Univer-sity,f Of course such candidacy implies that the poorer gradesof mathematical ability are eliminated. Any selective agencywhich works upon a species of individuals will alter the shape ofthe surface of frequency for any .mental trait unless its selectionsare random with respect to different amounts of that trait. Asthe selective action is commonly such as picks out the good or thebad, the result is commonly to produce a 'skewness' of the surfacetoward one extreme and a blunted condition at the other. Whena series of measurements in a group shows a deviation from the

    * Drawn roughly from the data given in Galton's ' Natural Inheritance/p. 200.

    tit is taken from Galton's 'Hereditary Genius/ 2d ed., p. 16.

    FIGS. 31-33.

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    THE DISTRIBUTION OF MENTAL TRAITS 19

    FIGS. 34-35.

    normal law of frequency toward conditions like those in figure34 and figure 35, it will be wise to look for some selective agencyat work upon the group. If theapproximately normal distribu-tions figured above are exam-ined carefully a slight elimina-tion of the least efficient will beapparent This is probably dueto the fact that children who arevery low in a scale of intelli-gence are eliminated from thepublic schools altogether and so 'are not represented in our tests.Figures 36, 37 and 38 present other samples of asymmetrical dis-

    tributions due to selection.It is likely that the statistics

    upon which were based the fre-quency surfaces on page 15axe slightly influenced by bothmixture of species and selec-tion, and that without these theywould approximate still doser tothe one simple law, and supportstill more emphatically the hy-pothesis that the distribution ofany mental trait in a homogene-ous species undisturbed by selec-

    FIGS. 36-38. **on is that given by the proba-bility integral.From this hypothesis, two important results follow. The

    measurement of a mental traitin a group does not, when distri- *-bution is normal, require the com- re-plete statement of the distribu- rftion, since from knowledge of the FIG. 39.average or median or mode andof the A.D. or some other measure of the variability of the groupabout the average we can reconstruct approximately the entire dis-

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    20 EDUCATIONAL PSYCHOLOGYtribution scheme. Thus given the facts that the average abilityof 12-year-old boys in a test of memory is 17.5 and that thestandard deviation is 3.0, and we know that the whole distribu-tion scheme for 12-yeaivold boys in the test is that of figure 39and table L The mathematical formulsB by which this is doneneed not concern us here. TABLE I.

    Percentage of cases Percentage of casesAbility. possessing It. Ability. possessing it.

    8 .1 18 13.19 .3 19 11.710 .6 20 9.411 1.3 21 6.712 2.5 22 4.313 4.3 23 2.514 6.7 24 1.315 9.4 25 .616 11.7 26 .317 13.1 27 .1

    The other result is that if we know that distribution is regularand have given the measurements in terms of relative position ofa large number of individuals chosen at random, we can turnthose measurements into terms of amount. Here again the mathe-matical formulae are best omitted. The reader may take it ontrust that such a transposition as the following is correct.

    Given the knowledge that 1,000 individuals rank in order ofexcellence in English composition as follows :Individuals 1 2 are indistinguishable in ability but better than anybelow. 3-5 are indistinguishable in ability but better than anybelow and worse than any above. 6- 10 are indistinguishable in ability but better than anybelow and worse than any above. 11- 20 are indistinguishable in ability but better than anybelow and worse than any above.

    21- 40 are indistinguishable in ability but better than anybelow and worse than any above.41- 70 are indistinguishable in ability but better than anybelow and worse than any above. 71- 120 are indistinguishable in ability but better than anybelow and worse than any above.121- 280 are indistinguishable in ability but better than anybelow and worse than any above.281- 720 are indistinguishable in ability but better than anybelow and worse than any above.

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    THE DISTRIBUTION OF MENTAL TRAITS 21Individuals 721- 880 are indistinguishable in ability but better than anybelow and worse than any above. 881- 920 are indistinguishable in ability but better than anybelow and worse than any above. 921- 960 are indistinguishable in ability but better than anybelow and worse than any above. 961- 984 are indistinguishable in ability but better than anybelow and worse than any above. 985- 993 are indistinguishable in ability but better than anybelow and worse than any above. 994- 997 are indistinguishable in ability but better than anybelow and worse than any above. 998-1000 are worse than any above.and given the knowledge that the distribution in this ability isregular. Take as the ability in composition above and belowwhich half of the individuals lie, and as 1 the ability excelled by159 of the group and as 1 the ability less than which 159 ofthe class have. Then the scheme of distribution is that of table H.

    TABLE H.2 rank between + 2.87 and probably + 3.00, perhaps more.+ 2.675

    -f 2.325+ 2.05+ 1.75+ 1.475+ 1.18+ 0.58+ 0.580.58

    1.181.4051.752.1452.532.74

    averaging close to +2.7 + 2.44 + 2.17+ 1.88etc.

    + 2.87+ 2.575+ 2.325+ 2.05+ 1.75+ 14.75+ 1.180.581.181.4051.752,1452.532.74

    probably 3.00, perhaps less.

    From this scheme we can get a measure in amount for anyindividual whose position compared with the rest is known. Forinstance, the ability of individual 7 is dose to + 2.44, that ofindividual 17 is close to + 2.17, while that of individual 33 is+ 1.88. No. 7 is nearly 30 per cent, farther above the averagethan No. 33. In this scheme of course we do not so far knowjust what 1.0 or 1.5 or any other measure represents. + 2.98may be the ability of Shakspere or of a fairly good high school

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    22 EDUCATIONAL PSYCHOLOGYcomposition according to the group of 1,000 we are studying. Butif we give a few samples of the compositions with the amountsassigned we make the hitherto arbitrary unit of amount a per*fectly definite thing, as definite as an inch or an ohm or kilogram.Our study of the distribution of mental traits thus provides uswith a means of accurately measuring such physical traits as colorof eyes or hair and such mental traits as courage, honesty, ambi-tion or eminence, provided we deal with homogeneous groups andhave reason to think that the distribution of tha ability in thegroup studied is normal.The reader will, I trust, have inferred already two corollariesof the law of distribution : the first that small differences betweenindividuals in the same species are far more common than largerones, the second that within any one species there is no cleardemarcation of ordinary from^x&eptional grades of ability. Itis a common error to distort the truth that in any school gradeor at any age there are great differences in ability between theextremes in any mental trait into the error that such great differ-ences are as much the rule as lesser ones. The inference is drawnthat teaching which is adequate for say one third the range ofabilities found would be adequate for only one third of the stu-dents. On the contrary, such teaching would be adequate for overtwo thirds of the students,* for over two thirds of the individuals inany homogeneous group are centered within the middle third orless of the total range of ability.

    Again it is a common error to imagine that nature has pro-vided distinct classes corresponding to our distinct words, e. jr.,normal and abnormal or ordinary and exceptional But withinany natural group grades of amount of any trait seem to be con-tinuous. Genius and idiot> precocious and retarded, musical andunmusical, bright and dull and all the host of descriptive wordsdo not mark off distinct varieties of beings, but artificial sectionsof a continuously varying group. The realization of this factwill prevent a multitude of errors in arguments about the processesand results of education.