TABLE XXII.

THE RELATIONSHIP BETWEEN AGE AND HEIGHT AND BETWEEN AGE AND

when we compare age and height among boys or among girls, or age and weight among boys or among girls.

Similarly in Table XXIII are given the correlation coefficients indicating the relationship between height and weight. The maximum difference between two

NOTE. The numbers on which those correlation coefficients are based range from 168 to 545. The probable error is thus in every case of the order 01.

* In this school Standards I and II are not available.

schools is eight times its probable error, and thus is certainly significant, but the regression lines are even more divergent, so that there seem to be significant differences between the schools in this respect; this is probably due to the influence of the selection of age already indicated.

(9) THE RELATIONSHIPS BETWEEN MENTAL CAPACITY AND HEIGHT AND WEIGHT. In considering these relationships, we must of course allow for the influence of age on height and weight. This is done by means of partial correlation coefficients. If TMH, TMA, THA, be the direct correlation coefficients between mental capacity and height, mental capacity and age, and height and age, then the correlation between mental capacity and height for a constant age is given by

and a similar formula holds for the correlation between weight and mental capacity for a constant age. The values of TMA, THA, TWA have already been given in Tables XII and XXII, and in Table XXIV and XXV I have given Tн, wн, RMн, ARMw for the TWH, MH individual schools. It will at once be noticed that to allow for the influence of age makes no significant difference on the average of the correlation coefficients, .but that the effect is really that of a reducing factor on the separate correlations. We have, for instance, twenty-five correlation coefficients between mental capacity and height, with a range of from 24 to +29, and a standard deviation of 17, while the partial correlation coefficients have a range of from -03 to +18, and their standard deviation is only .05, less than a third of the former value. Exactly the same result is obtained from the correlations between mental capacity and weight. The range of the direct correlation coefficients is from 21 to +28, and the standard deviation is 16, while the range of the partial correlation coefficients is from -04 to +13, and the standard deviation is 05. The effect of allowing for the influence of age is thus to leave the average correlation practically unaltered while materially reducing the range and variability of the correlation coefficients. The mean values are in all cases positive, but are very small, and no marked stress can be laid upon these relationships. In other words, if the height and weight of the children be primarily a result of their home environment and not largely due to racial differences, this environmental influence does not substantially affect their mental capacity.

This result is confirmed by examination of the individual returns, which are very irregular. Thus in the School B. 5 and G. 5, the boys show one of the relatively high relationships between height and mental capacity, while the girls show no relationship; a similar difference between the sexes is found in School No. 9 when dealing with the relationship between weight and mental capacity. It may be presumed that the home environment is the same for both boys and girls, and yet if mental capacity be affected by the state of nutrition,

and this by home conditions, the effect is quite different for boys and girls. If it be suggested that the bigger girls reach puberty sooner and that their mental energies may be thus lessened, some solution must be offered to the difficulty that these relationships rise to their highest positive values in G. 1 and G. 3 for weight, and in G. 3 and G. 7 for height. Further, not more than some eight of the fifty partial correlation coefficients can be considered significant, having regard to their probable errors. In other words, unless we are to consider the mental capacity scale as wholly valueless relatively as well as absolutely, i. e. unless we are to assume that no reliance is to be placed on the teachers' classification into intelligence groups in the individual schools, there is really no basis for the belief that mental capacity is substantially associated with the height and weight of the children.

(10) THE RELATIONSHIP BETWEEN MENTAL CAPACITY AND QUALITATIVE CHARACTERS.

Having discussed the relationship between mental capacity and those characters which can be expressed on a quantitative scale, we must now deal with purely qualitative characters, such as the condition of the teeth and of the clothing, the state of nutrition, cleanliness, &c.

Two methods only are available for determining the relationship between mental capacity and any one of those characters, by forming a contingency table or a fourfold table. The contingency table was used in all cases where at least a twelvefold division, i.e. a three x fourfold division of the material could be made, and where this could not be done the fourfold table was used.

But before discussing in detail those relationships it will perhaps be profitable to look at the question in quite another way. Let us assume again that the distribution of mental capacity obeys the Gaussian law, and let us find for each grade of intelligence the centroid vertical and plot up on those verticals the percentages of children who possess in a given degree a certain character. To take an example, we find that in B. 7, of those children who are classed as brilliant 90 per cent. obtain marks IV or V for the condition of the clothing, i. e. in those cases the condition of the clothing is at least above the average. Of the children who are above "the average " in intelligence 91 per cent. reach the standard "above the average for the condition of the clothing; while in the "average", "under the average and " very dull and backward” groups of intelligence, the percentages are 84, 72, and 44 respectively. It is clear then that, so far as this school is concerned, there is some relationship between mental capacity and the condition of the clothing, and, further, that this relationship is positive, i. e. the higher the intelligence the better the condition of the clothing. If these numbers are expressed graphically we obtain a diagram which Professor Pearson has termed an analograph.* * See Biometrika, vol. v, p. 129.

If the percentages increase or decrease uniformly with mental capacity, a more or less close relationship between mental capacity and the character under consideration is indicated. In such cases the analograph is said to be "homoclinal ", while if the percentage does not reach its maximum with the maximum or minimum of intelligence it is said to be "heteroclinal ".

The analograph is also useful in another way; as has already been pointed out, the contingency coefficient has essentially no sign, but if we are to consider the correlation between two characters to be determined by a contingency coefficient we must prefix a positive or negative sign to the arithmetical value found from the contingency table, and this sign can usually be determined from the analograph. The procedure to be adopted in doubtful cases has already been discussed.

Analographs have been constructed to show the relationship between mental capacity and seven other characters for the boys' and girls' departments of School No. 7, and these are given in Fig. 12.

It should be noted that the deductions drawn from these analographs apply only to those two schools, and in general cannot be extended to the other schools where the analographs may show actually reversed conditions.

In the boys' department, five groups of mental capacity are used, but in the girls' department there are only six girls who are classed as "brilliant", so that here the two groups "above the average" and "brilliant" have been combined, another illustration of the difficulties of the investigation.

To take individual cases there is clearly, as has been stated, some relationship between intelligence and the condition of the clothing, and this is seen to be true both for the boys and the girls of this School, No. 7.

Similarly, I have plotted on the centroid verticals of the mental capacity groups, the percentage of children who obtain the marks IV or V for their state of cleanliness. Here, again, the relationship is homoclinal and the percentage decreases as we pass from the "brilliant " to the “dull”.

The percentage of boys who have normal hearing decreases slightly with decreasing intelligence; among the girls there is no sensible difference between the groups.

The relationship between the condition of the teeth and mental capacity is more marked among the boys than among the girls; similarly the percentages of boys whose glands are enlarged increase from 45 to 72 as we pass from the "brilliant " group to the "dull", i. e. with increased intelligence the condition of the glands improves; among the girls, on the other hand, the differences are hardly significant and the relationship is heteroclinal.

The condition of the tonsils and adenoids, again, is slightly correlated with mental capacity in the case of the girls, and not at all in the case of the boys.

G