372 Prof. Karl Pearson on the Influence of 



with p and q not small, — in /3 X = or /3 2 = 3, i. e. in the sym- 

 metry and mesokurtosis, which are for practical purposes 

 closely enough represented by the Gaussian curve. But if 

 m and n be commensurable, and either f or q moderately 

 small, this result by no means follows. Thus suppose we 

 take : — 



Illustration I. A sample of 100 of a population shows 

 10 p. c. affected with a certain disease. What percentage 

 may be reasonably expected in a second sample of 100 ? 



In this case m = ?z = 100, ^=10, </ = 90, and we easily 

 find: 



Mode=10. Mean = 10-78. o-=4'344. 



Probable error of mean = 2*930. 



Further: /3, = *2749, /3 2 = 3'3255. 



There is thus sensible skewness and sensible leptokurtosis. 



If we suppose the probable error to still give a sufficiently 

 definite meaning, we should put the answer in the form 

 10*784 + 2*930, or assert that the percentage of the character 

 in the second sample was as likely to fall outside as 

 inside the limits 7*85 to 13*71. If we used the value 

 mp± *67449 Vmpq, often adopted, we find the limits 7*98 

 to 12*02. The first result gives a very much more con- 

 siderable chance of percentages arising in the second sample 

 considerably beyond that shown by the first sample. It 

 seems worth while investigating for this case, the actual 

 errors in expectancy introduced by using the normal curve 

 for our skew platykurtic frequency distribution. I have 



accordingly calculated out series (iv.). I find C= 9 k<w 



nearly, and obtain the distribution of frequency given in 

 Table I. In this table the second column gives in a 1000 

 second samples of 100 each, the numbers which would show 

 the percentage in the first column ; the third column gives 

 the numbers, which would show this percentage or any less 

 percentage. This table proves at once the marked skewness 

 of the distribution which with a modal value at 10 stretches 

 from on the defect to upwards of 30 on the excess side. 

 If we interpolate to find the quartiles we have lower quartile 

 at 7*65 and the upper quartile at 13*48. The median is at 

 10*40. Thus the lower quartile has 2*75 and the upper 3*08 for 

 range ; yalues in excess of the percentage observed are thus 

 to be expected. Since -J (2*75 + 3*08) =2*92, we see that the 

 " probable error " thus found is practically identical with 

 the value found from the standard deviation. The range, 

 however, as obtained from placing this probable error either 



