182 
Skew Variation, a Rejoinder 
heterogeneous they are merely arguing in a circle. The distributions are as 
continuous and smooth as those which occur in the case of the Gaussian curve, and 
they occur for characters in the same group of individuals wldcli present for other 
characters tlie normal distribution. 
Thus the length of meropodite of right claw in Gelasimus pugilator* is quite 
sensibly normal, but the length of the carpodite of the right claw is almost as 
certainly platykurtic and skew. If two characters are normal, a third character 
Avhich is their difference, whether they be correlated or not, should have a normal 
distribution, yet in the case of Gelasimus ■pugilator for the whole series of 
measurements the difference distributions are essentially platykurtic -f*. 
The size of the disc in Oplncoma nigral has a modal difference in the distribution 
of 1000 cases of "271 rum. and tlie probable error on the basis of a normal curve 
is 056. The deviation is thus five times its probable error and the asymmetry 
undoubtedly significant. 
The outer diameter oi Arcelhi vulgaris^ in 504 cases gives a modal difference 
of 3'226 mikrons, and the probable error of this difference is only •211. The 
asymmetry is therefore undoubtedly significant. 
The distance between the mean and mode in the case of the length of shell of 
Nassa. obsoletaW from Lloyd Point, Long Island, U.S. was "68 mm. for 368 individuals, 
the probable error of this modal difference was 08. The asymmetry is therefore 
significant. Other characters of N. obsoleta were as definitely asymmetrical, while 
some from exactly the same individuals were sensibly normal. 
The transverse arc in cT Naqada skulls has for 115 individuals a modal 
difference of 2'34 mm. and the probable error of this difference is "78 mm., or it is 
probably significant. Yet the breadth of the male Naqada skulls is significantly 
symmetrical. 
The height/length index of 117 English $ skulls has a modal difference of "SS 
and a probable error of only "22, the skewness is therefore significant. 
The same is true of the distribution of many internal organs in man. For 
example, if we exclude recognised diseased hearts, we obtain a markedly skew 
distribution such as is given in the broken line of Fig. 1. This is for 1382 heart- 
weights. If this be supposed to be due to the great variety of ages, we have only 
to look at the continuous curve for hearts of 358 young adults, 25 to 35, to see the 
same asymmetry. This is drawn for four times the scale. 
G. Duncker : Biomctrika, Vol. ii. p. 313. 
t There is another point to which I will only refer briefly here. If characters were always 
distributed according to the Gaussian law the regression curves must be straight lines. The generalised 
Mendelian theory of determinants I have developed makes them, however, hyperbolas, and I have given 
instances in a recent memoir of a variety of curved regression lines 
+ Mcintosh : Biomctriha, Vol. ii. p. 470. 
§ K. Pearl and F. J. Dunbar : Bionirtrika, Vol. ii. p. 327. 
li A. C. Dimon : Blometrika, Vol. ii. p. 29. 
