Probable in a Correlated System of Variables. 167 



sample may show that our theoretical frequency gives only an 

 approximate law for samples of a certain size. In practice 

 we must attempt to obtain a good fitting frequency for such 

 groupings as are customary or utile. To ascertain the 

 ultimate law of distribution of a population for any groupings, 

 however small, seems a counsel of perfection. 



(6) Frequency known or supposed known a priori. 

 Illustration I. 

 The following data are due to Professor W. F. It. Weldon, 

 F.R.S., and give the observed frequency of dice with 5 or 

 6 points when a cast of twelve dice was made 26V,306 times : — 



No. of Dice in 



Cast with 5 or 6 



Points. 



Observed 



Frequency, m'. 



Theoretical 

 Frequency, m. 



Deviation, e. 







185 



1149 



3265 



5475 



6114 



5194 



3067 



1331 



403 



105 



14 



4 







203 



1217 



3345 



5576 



6273 



5018 



2927 



1254 



392 



87 



13 



1 







26306 



- 18 



- 68 



- 80 

 -101 

 -159 

 + 176 

 + 140 

 + 77 

 + 11 

 + 18 

 + 1 

 + 3 







1 



2 



3 



4 



5 



6 



7 



8 



9 



10 



11 



12 







26306 



The results show a bias from the theoretical results, 5 and 6 

 points occurring more frequently than they should do. Are 

 the deviations such as to forbid us to suppose the results due 

 to random selection ? Is there in apparently true dice a real 

 bias towards those faces with the maximum number of 

 points appearing uppermost ? 



We have : — 



Group. 



e 2 . 



e 2 hn. 



Group. 



e 2 . 



e 2 /m. 







1 

 2 



3 ... 



4 



6 



324 



4624 

 6400 

 10201 

 25281 

 30976 

 19600 



1-59606 

 379951 

 1-91330 

 1-82945 

 4-03013 

 6-17298 

 6-69628 



7 



8 



9 

 10 

 11 

 12 



5929 



121 



324 



1 



9 







4-72807 

 0-30903 

 3-72414 

 0-07346 

 9-00000 

 ■00000 



Total... 





4387241 



