of army recruiting a series of tables and frequency curves has been 

 prepared for European conscripts and American recruits. A few typical 

 frequency curves of stature, for the purpose of convenient illustration, 

 have been worked out for me by Mr. Arne Fisher, the well-known 

 author of a standard treatise on "Probabilities." The point of view that 

 the mean or average value of a large number of measurements may be 

 relied upon unconditionally as a measure of comparison is a serious 

 fallacy common in general statistical practice. The mean at best repre- 

 sents a norm around which the other values of the variate are grouped. 

 The mean frequency gives not the slightest clue as to the possible 

 tendency of the statistical material to cluster around a particular value 

 which for practical purposes may be of governing importance. As 

 pointed out to me by Mr. Fisher in his observations on the frequency 

 curves of recruits that if, for illustration, the mean stature of American 

 recruits and of Norwegian conscripts are 67.52 inches and 67.49 inches, 

 respectively, we are by no means justified in assuming that the statures 

 of the two populations are precisely the same, although the difference be- 

 tween the means is less than .03 part of an inch. It is quite probable 

 that in the one we would find 70 per cent, having a stature between 

 65 and 70 inches, while in the other the percentage distribution 

 would amount to only 55 per cent. The mean, under such conditions, 

 is therefore often a fictitious mathematical measure which without 

 qualification is practically certain to prove misleading. A more satis- 

 factory method is to determine the possible presence of a clustering 

 tendency, or a constant known as the dispersion, or the standard devia- 

 tion around the mean value. If, for illustration, it is found that the 

 mean stature of Norwegian conscripts is 67.5 inches and the disper- 

 sion is 2.33 inches, this means that about 68 per cent, of the Norwegian 

 conscripts measure between 65.17 and 69.83 inches. * More precise 

 statistical analysis readily disproves the common error that all statistical 

 frequency curves are true symmetrical curves. This point of view was 

 first advanced by the German mathematician Gauss, and unfortunately 

 widely accepted among statisticians of modern times. As a matter of 

 fact, however, the symmetrical distribution is the exception rather than 

 the rule, and the correct ascertainment of frequency distribution 

 requires in addition to the mean and the dispersion the computation of 

 at least two additional parameters, the skewness and the excess. As 

 explained by Mr. Arne Fisher, "These two statistical characteristics are 

 purely abstract numbers." A positive skewness indicates a tendency to 

 a heavier clustering of values greater than the mean ; negative skewness 

 indicates a heavier clustering of values less than the mean. A positive 



* The importance of the dispersion (also called the standard deviation) as the best measure 

 of a clustering tendency about the mean value of a variate is emphasized in the formulas of 

 the mathematical theory of probability. Through a simple application of the Bernoullian 

 theorem, or the so-called "law of large numbers," it can be shown that a range of six times 

 the dispersion will include more than 99 per cent, of the bulk of the observations, while two 

 ordinates drawn both to the right and to the left of the mean at a distance from the mean 

 equal to the dispersion will include about 68 per cent, of the area of the frequency curve. 



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