APPENDIX 351 



No. 0/ trumps in his own and partner's hands — 0, i, 



2, 3. 4. 5> 6, 7, 8, 9, 10, II, 12, 13. 

 No. of times this hand was held — o, o, o, i, 9, 29, 



53. 52, 35. 14. 6, I, o, 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 Vv^e 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 



