The quantitative definition of intellectual deficiency must be made with careful allowance for irregularities among different mental processes, among different individuals, and among different groups. Theoretically it is possible to place the borderline so low that a case with that degree of deficiency and without removable handicaps would be clearly feeble-minded. The chance that the diagnosis would be mistaken could be reduced to any minimum desired. Above this a wider region of doubtful deficiency could then be stated in similar form. This is the plan that we suggest in attempting the percentage definition. Practically, however, the plan assumes that a suitable allowance can actually be made for these variations and raises a number of problems as to variability which should be considered. Four of these sources of variation are discussed below: (1) the variation due to a limited sample of individuals measured, (2) the variation among different communities, (3) the variations arising from sex, race and social differences, (4) the variation of the same individual from one mental process to another. We do not have the problem of neglecting these variations, but of adequately allowing for them both in the percentage of presumably deficient and in the doubtful region. (1) _Variation among Samples of Individuals Measured._ The error introduced by the fact that measurements are made on a limited rather than an unlimited number of individuals, in establishing the standards with a system of tests, can be taken care of statistically fairly well by applying the theory of probability as to the error of a percentage in a single sample. The range of the error can then be indicated on the measurement scale. This supposes, however, that each sample to be measured is taken from a random group and not from a selected group. Allowance for this error of sampling is therefore complicated by the fact that the usual test data have been obtained from groups of _school children_, even when there has been no further selection within the school group. Data on school children are certainly reliable only within the years of compulsory school attendance. Ordinarily in this country, they are not reliable for children of 14 years of age or over. Moreover, the point of the scale which is reached by the lowest X percentage of school pupils will exclude a slightly larger percentage of all children of corresponding ages, since the idiots and some imbeciles are not sent to the ordinary schools. This slight discrepancy should be kept in mind. The problem of avoiding selected samples among adults is still more difficult; but we found that it was possible in one community at least to measure all the 15-year-olds in the lowest X percentage in certain districts, as we shall note later. By this age, mental processes are probably very much like those of adults, except for the amount of information and practise.