168 Prof. Karl Pearson on Deviations from the 



Hence x 2 = 43'87241 and X = 6*623,625. 



As there are 13 groups we have to find P from the formula : 



l+| + 



7^ + 



x 8 , x 10 



X X 



271 + 271 . 6 ' 2,4.6.8 ' 2. 4. 6. 8. 10 



+ 



-P = e~ix 2 ( 



which leads us to 



P = -0000 16, 



or the odds are 62,499 to 1 against such a system of devia- 

 tions on a random selection. With such odds it would be 

 reasonable to conclude that dice exhibit bias towards the 

 higher points. 



Illustration II. 

 If we take the total number of fives and sixes thrown 

 in the 26,306 casts of 12 dice, we find them to be 106,602 



instead of the theoretical 105,224. Thus 1o 106 'f.°^ >g = "3377 

 nearly, instead of $. U X 2b ' 6()b 



Professor Weldon has suggested to me that we ought 

 to take 26,306(-3377 + -6623) 12 instead of the binomial 

 26,306(^ + f) 12 to represent the theoretical distribution, the 

 difference between "3377 and ^ representing the bias of the 

 dice. If this be done we find : 



> 



Group. 

 



m'. 



m. 



e. 



e'^fm. 



185 



1149 



3265 



5475 



6114 



5194 



30(57 



1331 



403 



105 



14 



4 







187 



1146 



3215 



5465 



6269 



5115 



3043 



1330 



424 



96 



15 



1 







- 2 

 + 3 

 + 50 

 + 10 

 -155 

 + 79 

 4- 24 

 + 1 



- 21 

 + 9 



- 1 

 + 3 







•021,3904 

 ■007,8534 

 •777,6050 

 •018,2983 



3991,8645 



1-220,1342 

 •189,2869 

 •000,7519 



1-040,0948 

 ■841,8094 

 •666,6667 



9 







1 



2 



3 



4 



5 



6 



7 



8 



9 



10 



11 



12 





Hence: % 2 = 17-775,7555. 



This gives us by the first formula in (ii.) of art. 4 : 

 P=-1227 ; 



or the odds are now only 8 to 1 against a system of deviations 

 as improbable as or more improbable than this one. It may be 

 said accordingly that the dice experiments of Professor Weldon 

 are consistent with the chance of five or six points being 

 thrown by a single die being "3377, but they are excessively 



