A TABLE OF THE DOUBLE INTEGRAL OF THE GAUSSIAN PROBABILITY FUNCTION1

In the course of studies2 on the eruption of the permanent teeth in children, it was found that a mathematical description of the age distribution of eruption of the separate morphological types of teeth could be obtained through the use of the normal probability (Gaussian) frequency function. In these investigations it was shown that the percentages of children at successive chronological ages who had a particular permanent tooth erupted into the mouth followed an S-shaped curve which could be fitted satisfactorily by the integral of the normal probability function. This finding leads to the observation that the area under the S-shaped curve (the double integral of the probability function) to any chronological age represents the total number of years of exposure of the tooth in the mouth per 100 children. The values of this area, which provide a measure of accumulated post-eruptive tooth age, have been found of considerable utility in studies on dental caries. The determination of post-eruptive tooth age values requires estimates of the double X X -X2 integral of the normal probability function ( f e XdX). So far as the authors are aware, no tabled values of this double integral are available. Since it seems likely that these values may be useful in connection with other studies, it has seemed worth while to publish in full Table 1, which was prepared in connection with the studies on dental caries.