. The distribution and relations of educational abilities . nied by some measureof deviation. In order that the general amoimt of overlap may be visible at a glance,the various distributions have boon amalgamated and smoothed by a processof averaging. The average overlap in Mechanical Arithmetic, for any twoadjacent age-groups thus obtained, is represented in Figure 10. The method of averaging employed is shown in Table XXVIT. Fromthe average j)ercentagos in the last line of the table the average differencesbetween the averages for each age can bo computed in terms of the standarddeviation of
. The distribution and relations of educational abilities . nied by some measureof deviation. In order that the general amoimt of overlap may be visible at a glance,the various distributions have boon amalgamated and smoothed by a processof averaging. The average overlap in Mechanical Arithmetic, for any twoadjacent age-groups thus obtained, is represented in Figure 10. The method of averaging employed is shown in Table XXVIT. Fromthe average j)ercentagos in the last line of the table the average differencesbetween the averages for each age can bo computed in terms of the standarddeviation of a nonnal group. These are shown by the fine vertical lines inFigure 10. The curve (continuous line) for the given age is drawn aboutthe median, not about the average. The median for the second curve in thefigure (interrupted lines) is then found by averaging the differences betweenthe average of the given age and both the average of the age above and that ofthe age below. Figure (FUNDAMENTAL PROCESSES)DISTRIBUTION OF ABILITY WITHIN EACH AGE 13 AVtSASE FOR tACH AGE 7 8 10 li E 13 YEARS SCALE OfMARKS 0 100 200 {To face p. 68.) 69 To nioasiiro the general degree of overlapping we need sonie simple equationor formula. The area shaded in the figure would natmally bo taken as in-dicatuag the amount of overlap. If wo assmne that both ciuves are normal,ami have tho same stan<lard deviation, this area can readily bo ha\-e merely to find the difference between the two medians, halve it, findfrom a table of values for tho normal probability integral tho poicentago oftho whole area falliiig beyond this midpomt, and doviblo it. Accordingly,twice the percentage of either group, which falls beyond the pomt midwaybetween the medians of the two groups, may be taken as the measure of overlap.^ If the overlap is complete, the figure thus calculated Mill clearly be 100per cent. If there is no overlap, then with age-groups or classes of about50 children, the figi
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