XXXV— XLVI] Introduction lxv 



Hence we select the grade of working probability we require, roughly 1 in 23, 

 1 in 44, 1 in 114 or 1 in 555, and this determines q. Divide iVthe total frequency 

 by q and look up in Table V, Xi f° r ^ll> multiply this by the 1177 ViV2 2 of 

 Table XLV, p. 84, and we obtain the semi-major axis of the required ellipse- 

 Multiply the same Xi by 1177 ViV2i of Table XLIV and we have the semi-minor 

 axis. We can then construct round the point ft, ft this ellipse and ascertain if 

 it cuts critical boundaries on Diagram XXXV, p. 66, the orientation being given by 

 Table XLVI, p. 86. Less accurately, but for practical purposes effectively, we may 

 work on Diagram XL VII, p. 88. We proceed just as before, to select our q and so 

 determine our \2 2 and \2i. Then we take the ratio of 2i/2 2 . We now pick out 

 of the ellipses on p. 88 the set having the nearest 2,/2 2 value and out of this set 

 the ellipse with the nearest \2 2 value of its semi-major axis. This ellipse or if 

 necessary an interpolated one is transferred to tracing paper and placed with its 

 centre at the given point (ft, ft), and its major axis touching the dotted curve. If 

 this ellipse does not cut a critical line, we can be certain that to the given degree of 

 probability the curve is of the type into the area of which its (3 U ft point falls. 



It would be impossible in an Introduction to these tables to give the whole 

 theory of frequency curves*. But one or two formulae may be usefully placed 

 here for reference. 



Distance d from mode to mean = , .. _ * ttt (lxxiii), 



2 (oft, - 6ft -9) 



Skewness sk = g '_^ & + _ (lxxiv), 



JVV = ft (4ft - 24ft + 36 + 9ftft - 12ft + 35ft) (lxxv), 



NZtf = ft - 4ft ft + 4ft 3 - ft/ + 16ft ft - 8ft + 16ft) (lxxv Us), 



SftSfl.Eftp, = 2ft - 3ftft - 4ftft + 6ft 2 ft + 3ftft- 6ft + 12ft 2 + 24ft (lxxvi). 



It is from the above formulae that the Tables now under discussion have been 

 calculated. 



Illustration. The following percentages of black measured with a colour top 

 are stated to occur with the recorded frequencies in the skin colour of white 

 and negro crosses f. 



Discuss the type of frequency curve suited to the data and determine the chief 

 physical constants of the distribution and their probable errors. 



* The general theory is given in " Skew Variation in Homogeneous Material," Phil. Trans. Vol. 186 

 (1895), A, pp. 343—414: Supplement, Vol. 197 (1901), A, pp. 443—459; "On the Mathematical Theory 

 of Errors of Judgment," Phil. Trans. Vol. 198 (1902), pp. 274—279 ; "Das Fehlergesetz und seine 

 Verallgemeinerungen durch Fechner und Pearson," A Rejoinder, Biometrika, Vol. iv. pp. 169—212. 

 " Skew Frequency Curves," A Rejoinder to Professor Kapteyn, Ibid. Vol. v. pp. 168—171, and " On the 

 curves which are most suitable for describing the frequency of Random Samples of a Population," 

 Ibid. Vol. v. pp. 172—175. 



t Extracted from C. B. Davenport, Heredity of Skin Color in Negro-White Crosses, Carnegie 

 Institution of Washington, 1913. 



B. * 



