IQ22.] P. C. Mahalanobis : Analysis oj Stature. 93 



observers. 1 But it must be remembered that the distribution is 

 only approximately normal and is almost never exactly so. We are 

 thus obliged to study other types of frequency distribution. 



It is often found that the maximum frequency does not occur 

 at the Mean value of the character concerned. In such cases, the 

 most frequent size, that is, the position of the maximum ordinate, 

 is called the Mode. In anthropometric measurements it is very 

 usual to find the Mode different from the Mean. When this hap- 

 pens, the distribution is no longer symmetrical about the Mean. 

 Such asymmetrical distributions are called skew distributions 2 



The distance between the Mode and the Mean is one obvious 

 measure of skewness, or better still (for purposes of comparison), 

 this distance divided b} T the Standard Deviation. The mathemati- 

 cal measure of skewness depends on the third moment /% obtained 

 by cubing the deviations from the Mean and taking the average. 

 The positive and negative deviations (from the Mean) must, by the 

 very definition of the Mean, balance exactly ; so that the sum of 

 all deviations is zero. For a symmetrical curve this is also true 

 of the cubes of deviations. But in the case of an asymmetrical 

 curve, the sum of all the cubes of deviations is not zero. Hence 

 the third moment, which is merely the average-sum of the cubes 

 of all deviations, is not equal to zero. Thus p s or more con- 

 veniently /?, =/j^jfx 9 s is a precise measure of the degree of asymmetry. 

 If /?i is significantly different from zero, then the curve must be 

 considered skew. 



Frequency distributions may differ from the normal curve in 

 another particular. The change of slope of the normal curve is 

 a characteristic feature of the curve. Now a frequency curve may 

 differ from the normal as regards the manner in which its slope 

 changes. For example, if a curve rises more abruptly than the 

 normal curve, it is then called a lepto-kurtic curve. While if it is 

 more flat-topped than the normal, it is called a platy-kurtic curve. 

 Curves with the same degree of abruptness as the normal are 

 known as meso-kurtic curves. The kurtosis is measured by fi% — 3. 

 For meso-kurtic curves & is equal to 3, and the kurtosis is zero. 

 For lepto-kurtic £ 2 is greater than 3, and for platy-kurtic it is less 

 than 3. A frequency curve may also differ from the normal in 

 having a definitely limited range. The curve may be limited in 

 one or in both directions. With these curves there is a definite 

 theoretical limit to the size of deviations. 



The Coefficient of Variation. — Pearson 3 says, " In dealing with 

 the comparative variation of men and women. . . . , we have con- 

 stantly to bear in mind that relative size influences not only the 

 means but the deviations from the means. When dealing 

 with absolute measurements, it is, of course idle to compare the 



'■ For references see pp. 42-44. 



2 For literature on the subject see references quoted on p. 16. Also J. C. 

 Kapteyn : " Skew FYequency Curves in Biology and Siatistics." 



' 6 Karl Pearson: "Regression, Heredity and Panmixia," Phil. Tran*. 

 Roy. Soc, Vol. 187 A. 1896, pp. 276-277. 



