Stati»ttit',al fnventigatious 405 



in this case, to an erroneous conclusion. The study of popular judgments 

 and their value is an iniportant matter and Galton rightly chose this 

 material to illustrate it. The result, he concludes, is more creditable to 

 the trustworthiness of a democratic judgment than might be expected, 

 and this is n»ore than confirmed, if the material be dealt with by the "average" 

 method, not the "middlemost" judgment, the result then bemg only 1 lb. in 

 11!)8 out. 



Among other mattei*s which much interestetl (Walton wa« the verification 

 of theoretical laws of frequency by experimejit. He considered that dice 

 were peculiarly suitable for such investigations', as easily shaken up and 

 cast. As an instrument ibr selecting at random there was he held nothing 

 superior to dice'. Each die presents 24 ef|ual possibilities, for each face has 

 four edges, and a ditterential mark can be placed jvgainst each edge. If a 

 number of dice, say four, are cast, the.se can without examination be put, by 

 sense of touch alone, four in a row, and then the marks on the edges facing 

 the experimenter are the i*andom selection. Galton uses another die, if 

 desirable, to determine a plus or minus sign for each of the inscribed values. 

 On the 24 edges of this die he places the possible combinations of plus and 

 minus signs four at a time (1(5), and of plus and minus signs three at a 

 time (8). Then, when he has copitnl out in columns his data from the facing 

 edges of the first type of dice, he puts against their values the plus or minus 

 sign according to the facing edge of the sign-die, which gives either three or 

 four lines at a cast. The paper is somewhat difficult reading, and there are a 

 good many pitfalls in the way of those who wish experimentally to test 

 theories of frequency, especially those of .small sampling. The importjince of 

 distinguishing between hypergeometrical and binomial distributions, between 

 sampling from limited and from unlimited or very large populations, and the 

 question of the returning or not of each individual l)efore drawing the next, 

 are matters which much complicate experimental work with dice. 



Galton, however, was not unconscious of the many pitfalls which beset 

 the unwary student of the theory of chance. There is an interesting short 

 paper by him on "A plau.sible Paradox in Chances," written in 1894*. 

 The paradox is as follows: Three coins are tossed. What is the chance that 

 the results ai'e all alike, i.e. all heads or all tails ? 



"At leiist two of the coins must turn up alike, and as it is an oven <-liancc wluahi-r a tliini 

 coin is hea<ls or tiiils, therefore the chance of l>oing all alike is 1 to 2 and not 1 to 4." 



If the reader can distinctly specify oflf-hand, without putting pen to 

 paper, wherein the fallacy lies, he has had some practice in probability or 

 lias a clear head for visualising permutations. We leave the solution to him 



' Ordinary dice do not follow the rules usually laid down for them in treatises on prol>ahility, 

 because the pips are cut out ou the faces, and the lives and sixes are thus more frequent tlian 

 ivces or deuces. This point was demonstratetl by W. F. K. Weldon in 25,000 throws of 12 ordi- 

 nary dice. Galton had true culxjs of hard ebony made as accurate dice, and the.se still exist 

 in the Oaltoniawt. 



» "Dice for SUtistical Experiments." NvUxire, Vol. xlu, pp. 13-14, 1890. 



' Nature, Vol. xLix, pp. 365-6, Feb. 15, 1894. 



