370 Mr. F. Y. Edgeworth on 



deviation from the Mean in the case of Departmental returns 

 of the proportion between male and female births is signifi- 

 cant and indicative of a difference in kind, provided that we 

 select at random a single French Department; but that the 

 same deviation may be accidental if it is the maximum of the 

 respective returns for several Departments. There is some- 

 thing plausible in De Morgan's * implied assertion that the 

 deficiency of seven in the first 608 digits of the constant it is 

 theoretically not accidental ; because the deviation from the 

 Mean 61 amounts to twice f the Modulus of that probability- 

 curve which represents the frequency of deviation for any 

 assigned digit. I submit, however, that Cournot is right, and 

 that De Morgan, if he is serious in the passage referred to, has 

 committed a slight inadvertence. When we select out of the ten 

 digits the one whose deviation from the Mean is greatest, we 

 ought to estimate the improbability of this deviation occurring 

 by accident, not with De Morgan as 1 — 0(1*63), corresponding 

 to odds of about 45 to 1 against the observed event having 

 occurred by accident ; but as 1 — # 10 (1*63), corresponding to 

 odds of about 5 to 1 against an accidental origination. 



II. (2) Prof. Chauvenet's criterion differs from Prof. 

 Stone's in that he makes the a priori probability of a mistake 

 - — instead of being small and undetermined — definite and con- 

 siderable. In effect he assumes that a mistake is as likely as not 

 to occur in the course of m observations, where m is the number 

 of the set which is under treatment. It is not within the scope 

 of this paper to consider whether this assumption is justified 

 in the case of astronomical or of any other observations. It 

 suffices here to remark that this assumption coupled with 

 hypothesis (a) commits us to the supposition that huge mis- 

 takes occur on an average once in the course of 2m observa- 

 tions. Upon this supposition no doubt Method II. (2), is a 

 good one. Hypothesis (J3) expressly J excludes this suppo- 

 sition ; the mistakes which, according to II. (2), are as likely 

 as not, must, according to this second hypothesis, be of 

 moderate extent. Thus, in the case above put of sums of ten 

 digits, suppose that the number of such sums under observa- 

 tion is ten. According to Prof. Chauvenet's criterion we 

 must reject any sum which lies outside 45+&, where 



/^\ 2n-l = 19_ 

 °\1S/ In 20 - ^ 



' Budget of Paradoxes,' p. 291. 

 t If we take many batches of random digits, each batch numbering 



608, the number of sevens per batch ought to oscillate about the Mean 61, 



cording to a prob 

 % Above, p. 366. 



according to a probability-curve whose Modulus is,* / — — 608 = 10*4. 



