130 



SCIENCE 



[N. S. Vol. LII. No. 1336 



it means observed frequency. The value of 

 the statistical constants is simply that they 

 provide a conventional method of summar- 

 izing frequencies of observed data. To shift 

 the meaning of probability from observed 

 frequency to predicted frequency is preca- 

 rious, although we are always attempting it 

 in scientific generalization. However, it takes 

 more than a process of division by the square 

 root of the ntunber of cases — the obtaining 

 of the probable error of the mean — to bridge 

 the gulf between observation and prediction. 

 The lay conviction that the probable error of 

 the mean is actually a prophecy is hard to 

 overcome. That it is not prophetic will be- 

 come clear to any one who will take the trouble 

 to fractionate a large body of data, compute 

 the probable errors of the means of each 

 fraction and note how they vary, and then 

 compare all these discordant predictions with 

 the actual probable error of the means com- 

 puted from the array of means. The prob- 

 able error of the mean is a usefid constant 

 since it summarizes the variability of data 

 in relation to their amount; but it is not a 

 key to the future. 



All this is negative. Actually what was 

 Dr. Johnstone to do? First, observe and 

 report, I should say; and let him predict who 

 will. Certainly there is no need for much 

 statistics to summarize his twenty cases. He 

 wishes to know the most probable nrnnber of 

 bacteria per cc. in this emulsion. Scientific- 

 ally by the most probable number is meant 

 the most frequent number; and his data show 

 that 6-10 counts were more frequent than 

 any other. Why obscure the simple fact by 

 a statistical superstructure? If now he 

 wishes to risk prediction on the basis of 20 

 cases, he may say that 6-10 counts will occur 

 more often in his 250 cc. than any other 

 grroup, 16-20 counts next most often, 11-15 

 and 21-25 counts less often, and so on. This 

 course has the simple merit of telling the ob- 

 served truth and doing very little more. 



In predicting the total number of bacteria 

 within the 250 cc. one must multiply the 

 arithmetic mean of the counts by 250. We 

 have given the distribution of 20 counts and 



we have no alternative to assmning that it 

 is the most probable distribution of 20 counts. 

 Hence we must take the observed distribution 

 as many times over (12|times) as 20 will go 

 into 250 and simi all the frequencies. Dr. 

 Johnstone found 366 bacteria in 20 cc. The 

 most probable ntunber in 250 cc. must be 

 250/20 X 366 = 4,575. Mr. Michael gets 4,005 

 by the erroneous assumption that the most 

 probable (most frequent) logarithm is the 

 logarithm of the most probable (most fre- 

 quent) count, which is plainly impossible since 

 the logarithmic relation is not linear. The 

 illusion arises because we take it for granted 

 that any most probable natural number must 

 be inseparably connected with the most prob- 

 able logarithm. When we substitute the word 

 " frequent "' for " probable " we may see our 

 mistake, for the logarithms of the small nmn- 

 bers are more frequent than the logarithms of 

 the large numbers.^ 



Concerning the general problem of obtain- 

 ing "the probable error of extremely asym- 

 metrical frequency curves," I would urge that 

 in simple cases it is unnecessary to depart far 

 from the observed facts. Usually one is most 

 interested in the value of the most frequent 

 (most probable) case and in the amount of 

 deviation on either side. The values of the 

 mode and of the upper and lower quartiles 

 give this information, as well as the range 

 within which half the cases have fallen and 

 an indication of the skewness. Except the 

 gift of prophecy, what more could one want?^" 



Edwin G. Boring 



ClABK TJNIVERSITT, 



WoKCBSTER, Mass. 



ALBINO VERTEBRATES 



In July 1919, on the Beaver Eiver near the 

 mouth of the Dore River in Saskatchewan, I 

 shot a pure albino grackle (Quiscalus quiscula 

 wneus). It was a young male, 10.5 inches 

 long, and was associated with a flock of 

 grackles. It seemed much less shy than the 



» S. Newcomb, Amer. J. Math., 4, 1881, 39 f . 



10 See in general, ' ' The Logic of the Normal 

 Law of Error in Mental MeaBurement, " Amer. 

 J. Psychol, 31, 1920, 1. 



