L a ~ d ] fnti'in/in-tion ccxli 



TABLES 



Deviations to the 5/ and 0'5/ o point* in Type I Curves. (E. S. Pearson, / 



i, V..I. \vii. pp. 439 442.) 



Suppose y=f(x) is the frequency function of a variable x which lies between 



f* 

 the limits n and 6, so that I f(x)dx= I. Let 2, <r,, & and #, in the UHIIA! notation 



J a 



represent the moment-coefficients of this distribution; let 



em 



P = I f(x) dx and d = (x 3)/<r x . 



J a 



Values of P of '05, 95, '005 and '995 will be associated with values of d which 

 we may call d.^, d. w , d.QM and d.^; these are the deviations from the mean 

 measured in terms of the standard deviation as unit and up to the ordinates which 

 cut off tail areas from the curve of 5 / and 0'5 / respectively. Each value of fa 

 and fa (within certain limits) is associated with a particular member of the Pearson 

 system of frequency curves, and therefore with a set of four values of these d's. 

 For the normal curve we know that d.^ = d.& = T6449 and d.^ = d.^ = 2*5758, 

 but for platykurtic, leptokurtic and skew curves the values may be modified very 

 considerably. Tables L a ~ d give the deviations for a large range of Type I curves, 



The lower limit of each table corresponds very nearly to the position of the 

 Type III line; it is hoped to extend the tables so as to cover the Type VI and 

 Type IV areas, but in the meantime Tables L a ~ d will be found of use in many 

 problems. They are^ entered with the fa and fa of the curve; if the skewness be 

 positive (/* 3 and Vy9 t positive), then d.^ and d.^ are negative, and 6^.95 and ^.995 are 

 positive. For curves which are negatively skew the signs must be reversed. 



Illustration (i). The moment-coefficients of a frequency distribution will some- 

 times be known although the data themselves are not available. On the assumption 

 that the distribution can be represented approximately by a Pearson curve, it is 

 possible to obtain a good appreciation of the probable limits of frequency. 



Suppose that we were given 



x = 22-8361, <r x = 13-5078, & = -6783 (Vft is +), fa = 37342. 



Interpolation by aid of the tables, which may be done at sight, or by using the 

 first order forward difference relation only, shows that 



cU = - 1-35, d. K = + 1-87, c/,005 = - H59, d.^ = + 3'27. 



Hence, using the relations 



#05 = ^ + rf-05 X O-je, etc., 



we find that 90 / of the observations in the distribution should lie within the 

 limits 4-60 and 48'10, and 99 % between '00 and 67'01. 



The moment-coefficients are actually those of the distribution of 1086 observa- 

 tions of skin colour of white and negro crosses given as an example on p. Ixvi of 



B. II. /(// 



