It is hoped that the discussion of the statistical problems connected with the quantitative study of mental development has given more meaning to the different attempts to devise scales for measuring mental ability. It should be noted that the same relative development at different ages, expressed relative to the distance from lowest to highest ability measured in equal objective units, does not correspond to the same relative development measured in percentages of the groups, as soon as the forms of the distributions change. The theoretical considerations show that we have available at once a perfectly definite and clear method of stating relative development in terms of corresponding percentages of corresponding groups. If the groups distribute normally these units are translatable into units of the standard deviation of the group. If the distributions change in symmetry the only equivalent units of deficiency available are in terms of corresponding percentages reading from either end of the group. On the other hand percentile units are not equivalent in _amount_ of change for the same distribution, so they are of most importance for comparing different age distributions of uncertain forms. Until we have a scale of equal objective units for mental ability, it is not possible to obtain a measure of relative development which shall take into account the _amount_ of relative change. We must be content to measure the change in percentile rank (changes in serial position) of an individual relative to those of his own age. Having clarified our conceptions of mental development and brought them into harmony with certain suppositions regarding the distribution of ability and its change from year to year, we are in a better position to evaluate in the following chapter the different objective methods of defining the borderline of feeble-mindedness. CHAPTER XIV. QUANTITATIVE DEFINITIONS OF THE BORDERLINE On the basis of the detailed conception of the developmental curves and distributions of ability at different ages, which we have been considering, we can now compare the percentage method with other quantitative methods of describing the borderline on developmental test scales. A. DIFFERENT FORMS OF QUANTITATIVE DEFINITIONS