. The distribution and relations of educational abilities . ivergence of a single year appears to signify far less at the higher ages thanat the lower. Hence, any comparable measure of backwardness or advancementmust not be couched in terms of actual years ; with increasing age the figureobtained in years must be progressively reduced ; and the amount of reductionwill depend inversely upon the average range of variation in each successive age-group. Accordingly, before our data are reclassified, we must find for each age,not only a more exact central value, but also a tmit of scale more nearly
. The distribution and relations of educational abilities . ivergence of a single year appears to signify far less at the higher ages thanat the lower. Hence, any comparable measure of backwardness or advancementmust not be couched in terms of actual years ; with increasing age the figureobtained in years must be progressively reduced ; and the amount of reductionwill depend inversely upon the average range of variation in each successive age-group. Accordingly, before our data are reclassified, we must find for each age,not only a more exact central value, but also a tmit of scale more nearly uniform. (M In a group that is slightly skewed, the choice of the mode as norm necessarilyenhances the apparent asymmetry, that of the mean reduces it, that of the median would, by definition, abolish it altogether. Figure 5. DIAGRAM TO SHOW INCREASE OF APPARENT VARIABILITY WITH INCREASE OF AGE RED LINE REPRESENTS THE EQUATION:—STANDARD DEVIATION = j^jj (AGE—^) - lb - 1-4 - 13 - 1-2 • II? lO?0-9?0 8?0 7•06•O 6 ?O 3 »o-z ?O I • oo. O I 2 3 4 5 6 7 8 9 lO II 12 13 K 15 ACE ace p. 31.) 31 The Standard Deviation, and its Correlation with Age. The extent to which children of a given age vary up and down thescholastic scale may be measured in several ways. The sunplest would beto take the individual deviations above or below the average for the group,ignore the direction of th9 deviation or sign, and calculate their arithmeticmean. For statistical purposes, it is better first to square the deviations,and then to take the squaxe root of their mean. Tliis more elaborate calculationautomatically eliminates the difference of sign, and gives greater weight toextreme deviations. The measure thus obtained for scatter, dispersion, varia-bility, or spread, within each group, is its Root-Mean-Square (or Standard )Deviation. The standard deviations for the several ages and grades are given inTable X. Except where the age-groups are small in nmuber or selected in kind,the increase
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