jAiNLAHY G, 1905.J 



SCIENCE. 



33 



or subgroups of the characters in a definite 

 order corresponding to the real (though in de- 

 tail undeterminable) quantitative scale in the 

 character or attribute itself. The order of 

 the classes appeared to be the important thing, 

 and consequently the method was assumed to 

 be limited to such attributes as could be ar- 

 ranged in a definite scale order. In recent 

 work, liowever, by varying the order of the 

 classes Pearson has found that so far as the 

 value of the correlation coefficient is con- 

 cerned this group order has practically no 

 influence. For the new conception of corre- 

 lation which arose from a consideration of 

 this fact Pearson proposes the term con- 

 tingency. 



As a measure of the contingency of any 

 classification of characters, it is proposed to 

 use some measure of the ' total deviation of 

 the classification from independent proba- 

 bility.' The practical method of making such 

 a measure Pearson develops in the following 

 way. 



" Let A be any attribute or character and 

 let it be classified into the groups A^, A.„ 

 A\, and let the total number of individuals 

 examined be N, and let the numbers which 

 fall into these groups be n^, n.,, n , re- 

 spectively. Then the probability of an indi- 

 vidual falling into one or the other of these 

 groups is given by nJN, nJN, • ■ ■, nJN, re- 

 spectively. Now suppose the same population 

 to be classified by another attribute into the 

 groups B„ B,, and the group 



frequencies of the N individuals to be m^, 

 m„, , respectively. The probability of 



an individual falling into these groups will be 

 respectively mJN, mJ'N, mJN, m /N. 

 Accordingly the number of combinations of 

 B^ with A " to be expected on the theory of 

 independent probability if N pairs of attri- 

 butes are examined is 



= V,,,, sav. 



N ^ N ~ N 

 " Let the number actually observed be n^^. 

 Then, allowing for the errors of random 

 sampling, 



is the deviation from independent probability 



in the occurrence of the groups A^, B^. 

 Clearly the total deviation of the whole 

 classification system from independent prob- 

 ability must be some function of the n — v" 

 quantities for the whole table." The value of 

 any function of these quantities will clearly 

 be independent of the order of classification. 

 The following- functions of the n — 



uo uv 



quantities were chosen for practical use. 



(a) 1 — P; the contingency grade, where P 

 is determined from by the use of Elder- 

 ton's tables.* The quantity is a measure 

 of the deviation of the observed results from 

 independent probability, depending on the 

 n — y quantities as shown by the equation 



where S indicates summation of like quanti- 

 ties over the whole table. A large value for 

 1 — P indicates that there is association be- 

 tween the attributes, while with a small value 

 of this function the chances are that the sys- 

 tem arose from independent probability. 

 (6) The function 



termed the mean square contingency. 

 (c) The function 



N 



where 2 denotes summation of all n — 



uv uv 



quantities having the same sign. This func- 

 tion xj) is called the mean contingency. 



In determining the functions of (j)' and xp 

 which shall be used practically, Pearson con- 

 siders the relation of those quantities in the 

 case of normal correlation. After some an- 

 alysis the result is reached that 



— 1 _o-2 » 



r — zcz -V — 



in the case of normal correlation. This re- 

 sult proves at once that ' the coefficient of 

 correlation is * * * entirely independent of 

 the arrangement of our classes on the basis 

 of any assumed order or scale.' 

 *' Biomctrika, Vol. I., p. 155. 



