- THEORY OF STATISTICS. 
perimental verification of these results. The following will serve 
as illustrations, but the student is strongly recommended to 
carry out a few series of such experiments personally, in order to 
acquire confidence in the use of the theory. It may be as well 
to remark that if ordinary commercial dice are to be used for the 
trials, care should be taken to see that they are fairly true cubes, 
and the marks not cut very deeply. Cheap dice are generally 
very much out of truth, and if the marks are deeply cut the 
balance of the die may be sensibly affected. A convenient mode 
of throwing a number of dice, suggested, we believe, by the late 
Professor Weldon, is to roll them down an inclined gutter of 
corrugated paper, so that they roll across the corrugations. 
(1) (W. F. R. Weldon, cited by Professor F. Y. Edgeworth, 
Encycl. Brit., 11th edn., vol. xxii. p. 394. Totals of the columns 
in the table there given.) 
Twelve dice were thrown 4096 times ; a throw of 4, 5, or 6 points 
reckoned a success, therefore p=¢=0'5. Theoretical mean //=6 ; 
Pe value of the standard-deviation oj, = #/05 x 0:5 x 12 = 
1-732. 
The following was the frequency-distribution observed :— 
Successes. Frequency. ! Successes. Frequency. 
0 Ro 7 847 
7 3 536 
- 60 9 257 
v 198 10 71 
430 11 11 
ae 731 12 —_ 
b a Total 4096 
Mean M = 6-139, standard-deviation ¢=1-712. The proportion of 
successes is 6:139/12=0512 instead of 0-5. 
(2) (W. F. R. Weldon, loc. cit., p. 400. Totals of columns of 
the table given.) 
Twelve dice were thrown 4096 times; only a throw of 6 was 
counted a success, so p=1/6, ¢=>5/6. Theoretical mean M=2, 
standard-deviation o = /1/6 x 5/6 x 12 =1-291. 
The following was the observed frequency-distribution :— 
Successes. Frequency. Successes. Frequency. 
0 447 5 115 
1 1145 6 24 
a 1181 ( 7 
J 796 8 1 
: 380 Total 4096 
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