August 6, 1920] 



SCIENCE 



129 



JMcGinnis has been for nine years professor of 

 physics and electrical engineering in the Uni- 

 versity of New Brunswick, Fredericton, N. B. 



Dr. Haery B. Tocoji, of the department of 

 biology of the College of the City of New York, 

 has been appointed assistant professor of zool- 

 ogy in the University of Oregon. 



Dr. F. Francis, professor of chemistry, has 

 been appointed pro-vice-chancellor of Bristol 

 University, in succession to Professor C. 

 Lloyd Morgan, who is about to resign office. 

 Dr. Lloyd Morgan has been appointed emeritus 

 professor of psycholog-y and ethics. 



DISCUSSION AND CORRESPONDENCE 



A PRIORI USE OF THE GAUSSIAN LAW 



To THE Editor op Science: Mr. Michael^ 

 in interpreting Dr. Johnstone's results- for 

 twenty counts of bacteria in polluted shell- 

 fish deplores certain naive errors to which the 

 lay statistician is prone, but is not, so it 

 seems to me, free from statistical illusion him- 

 self. I had hoped, at least, that the identi- 

 fication of the Gaussian law with the ideal 

 "chance" distribution was a custom of the 

 past, and that the prevalence of this practise 

 in the literature was simply due to the inertia 

 of thinking. May I submit the following 

 relevant observations ? 



1. The sole condition of " change " is 

 ignorance.' In science the thing to do with 

 ignorance is to admit it, not to posit the form 

 of distribution that a variable assumes un- 

 der it. 



2. Biological and mental phenomena, of 

 whose conditions of variability we are thus 



1 E. L. Michael, ' ' Goneernmg Application of the 

 Probable Error in Cases of Extremely Asynnnetri- 

 cal frequency Curves," Science, N. S., 51, 89-91. 



- J. Johnstone, ' ' The Probable Error of a Bac- 

 teriological Analysis," cited as Eept. Lane. Bea- 

 rish. Lab., No. 27, 1919, 64-85. 



3C/. J. Venn, "Logic of Chance," 1888, espec. 

 119 fe.; B. Bosanquet, "Logic," 1911, I., 322 fC. 

 If the scientist prefers not to go to the logician, 

 let him see if he can formulate for himself, with 

 scientific rigor, the conditions of "chance." 



ignorant, do not necessarily give sjrmmetrical 

 distributions when observed. Pearl showed 

 that the amount and direction of skewness 

 and the dependence of skewness on known 

 conditions might be the significant biological 

 fact.* The Gaussian law does hold for coin- 

 tossing, but the relationship has been scien- 

 tifically observed,^ not posited a priori. 



3. Moreover, there can be no reason to ex- 

 pect a Gaussian distribution a priori when 

 we are ignorant. A form of distribution is 

 always function of the unit of measurement; 

 and, since the choice of a biological unit is 

 ordinarily arbitrai-y, the chances of getting 

 the normal distribution are very small.® 

 Galton pointed out, furthermore, that chance 

 distributions of two related variables, when 

 the relationship is not linear, can not both be 

 Gaussian.^ 



4. When we observe a skew distribution and 

 are in ignorance of the conditions that cause 

 the variation, it is useless labor to factor the 

 skew distribution into a Gaussian " chance " 

 distribution and a skewing factor, as Mr. 

 Michael does. The two factors that we so 

 obtain are meaningless. The Gaussian fimc- 

 tion is biologically meaningless because there 

 is neither a priori nor observational ground 

 for taking it as the curve of chance (igno- 

 rance). Mr. Michael's logarithmic function 

 is biologically meaningless because it is 

 merely a measure of the manner in which the 

 observed data depart from the meaningless 

 Gaussian law. Pearson saw this point in 

 1900 and noted the fallacy.^ He also made 

 fun of the Gaussian " fetish," although the 

 position of the Biometric School has since 

 become less definite. 



5. Probability in science means frequency 

 and nothing more. Fundamentally in science 



* E. Pearl, ' ' Variation and Bifferentiation in 

 Ceratophyllum," 1907, espec. 90 f. 



5 E. g., see H. Westergaard, ' ' Grundziige der 

 Theorie der Statistik," 1890, 21-38. 



6 J. Bertrand, ' ' Oalcul des probabilites, ' ' 1889, 

 180 f. 



7 P. Galton, Froc. Soy. Soc, 29, 1879, 365-367. 



s K. Pearson, Philos. Mag., 5th ser., 50, 1900, 

 173. 



