374 Prof. Karl Pearson on the Influence of 



the calculation o£ the mode, the mean and the S.D. from 

 formula? (vi.), (vii.), and (viii.). For more accurate work 

 we must calculate the frequencies as I have done in Table I., 

 a laborious process, or we can note that the values of pi and 

 /3 2 correspond to a skew curve of Type I. * We have in fact 



ac 1 = 6 + 3A-^ 2 =-1737>0. This leads to r = 70*8325, 

 e=552-8582, m 1 =7'93 J m 2 = 60-90, 



and the equation to the curve is 



V 



(T \7'93/ v \ 60-90 



The origin is at 9*57 per cent, and the range from — 3*2 to 

 107*7 per cent., extending somewhat beyond the possible limits,, 

 with, however, no sensible frequency. Considering that the raw 

 moments have been taken to get the constants, the curve must 

 be considered as giving very good results, and exhibiting in 

 a manner amply sufficient for practical purposes the deviations 

 to be expected in second samples from the Gaussian symmetry 

 and mesokurtosis. The curves are shown in Diagram I. 



Illustration II. — I proceed now to a more extreme case. 

 Let us suppose that examination of 1000 members of a popu- 

 lation has shown 20 affected with a certain characteristic. 

 What is the expectancy if a second sample of 100 be 

 examined? In this case ^=20, # = 980, n = 1000, and 

 m = 100. We find: 



Mean = 2-096, Mode = 2, <r= 1-5015, 



and thus the probable error = 1*013. 



There is not much difference between the mean and the 

 mode, and the usual theory would give <r= 1*40, and state 

 the answer to be expected from the second sample as 2 + '94. 

 The second sample is so much smaller than the first, that 

 the standard deviation of second samples is fairly close to the 

 true value when calculated by the erroneous formula. But 

 the assumption of a Gaussian distribution falls wide of the 

 mark. In Table II. are the calculated frequencies given by 

 the series in (iv.) for a thousand samples f, the data being 



* Phil. Trans, vol. clxxxvi. A. p. 369. This is an interesting illus- 

 tration of the futility of Dr. K. Kanke's criticism of the worth of these 

 curves on the ground that they may give negative and unintelligible 

 values for the constants. In this particular case the n of Phil. Trans, 

 vol. clxxxvi. A. p. 361, has become the minus (n-\-2) of this paper, and 

 the pn and qn of that paper are here both negative. Yet we see that 

 the n, p, and q of our present notation are perfectly intelligible, and that 

 the skew curve follows at once from the series in (iv.). 



f C of (iv.) = -1328 nearly. 



