APPENDIX 351 



No. of trumps in his own and partner's hands — o, i, 



2, 3> 4> 5> 6, 7, 8, 9, io, n, 12, 13. 

 No. 0/ times this hand was held — o, o, o, 1, 9, 29, 



53, 52, 35, I4> 6, 1, 0, o. 



He should note also the number of times that 

 trumps were spades, clubs, diamonds, and hearts : he 

 will get some such results as the following : spades, 46 ; 

 clubs, 53 ; diamonds, 51 ; hearts, 50. 



The numbers in the lower line of the first series 

 form a " frequency distribution," for they tell us the 

 frequency of occurrence of the hands indicated in the 

 numbers above them. " No. of trumps " is the in- 

 dependent variable, and "no. of times these nos. of 

 trumps were held " is the dependent variable. 



A frequency distribution represents the way in 

 which the results of a series of experiments differ from 

 the mean result. A particular result is expected from 

 the operation of one, or a few, main causes. But a 

 number of other relatively unimportant causes lead to 

 the deviation of a number of results from this mean or 

 characteristic one. Yet since one, or a few, main 

 causes are predominant, the majority of the results of 

 the experiment will approximate closely to the mean ; 

 and a relatively small proportion will deviate to vari- 

 able distances on either side of the mean. If a pack of 

 cards were shuffled so that all the suits were thoroughly 

 mixed among each other, then we should expect the 

 trumps to be as equally divided as possible between 

 the four players. But a number of causes lead to 

 irregularities in this desired uniform distribution, and 

 so the results of a large number of deals deviate from 

 the mean result. It is possible, by an application of 

 the theory of probability, to calculate ideal, or theoreti- 

 cal frequency distributions, basing our reasoning on the 

 considerations suggested above. We then find that the 



