494 Notices respecting New Books, 



theoretical frequency distribution meant to describe the above, giving 



theoretical values f x = (j>(a\), f 2 = <p(# 2 ) .... Then if x 2 = 2 l-^j- ) 



■be calculated, P corresponding to this ^ 2 is found from Table XII, 

 and gives the probability that in a repeated set of observations the 

 actual frequencies will differ from the theoretical ones (this differ- 

 ence being measured by ^ 2 ) as much or more. If this probability 

 is high, the lit is good. 



The second group is intended to simplify the calculation of the 

 constants in the group of frequency curves introduced by Prof. 

 Pearson as generalizations of Gauss's normal curve. Gauss 

 assumed (i.) the equal probability of errors in excess of the 

 mean and errors in defect, (ii.) the continuity of magnitude in 

 the errors, and (hi.) the independence of all the small contributions 



!.%_ 



2-2 



to the total error. This leads to -• ~ = — — , or y=ce 2^' 



y ax a 2 



Prof. Pearson drops these assumptions and finds 



1 cly _ x-\-d 

 y dx a + bx + cx 2 '' 



leading to curves of seven types (five more have since been dis- 

 covered) of which Gauss's is one, and 



(x\ ma / x\ m ^ 

 1 J (14-t) is another. 



Tables XXVI-XXXI. and XLIX. include a table of Gamma 

 functions, logarithms of factorials, the powers and sums of powers 

 of natural numbers. The latter are especially useful for solving 

 numerical algebraical equations of high degree. 



The third group deals with correlations and Avith probable errors 

 of the constants employed. Briefly, if 



s-Acxd O-O 2 fa-™*? 

 s-Aexp 2 ^ 2 - 2ff , 



gives the frequency distribution of deviations of a; and y from 

 their means, then these deviations are independent ; but if 



(ay-mj 2 _ (y— m 2 f _ 2(x—m l )(y-m 2 ) 

 z — Aexp 2 9 2 o 5 



the deviations are no longer independent : r is the coefficient of 

 correlation ranging in value from —1 to +1 and measures the 

 dependence, and most of the tables in this section are intended 

 to give the value of r and the degree of its accuracy with a 

 minimum of labour from a minimum of data. Particular attention 

 is devoted to the determination of r from a fourfold table, as this 

 case frequently occurs in the current work of medical statistics 



