November 12, 1920] 



SCIENCE 



465 



the peak is at 5 ft. 8 in. The inversion occurs 

 Just above the peak: there are fewer men re- 

 oorded at 5 ft. 9 in. than at 5 ft. 10 in. That 

 the inversion is not due to the inclusion of men 

 of diiierent ages in the same curve is shown 

 by the fact that it occurs for ten of thirteen 

 •ge-groups (five-year range) taken separately. 

 The question arises as to the cause of this 

 •triking feature of the curve. 



35000 



Heights 



3000li — OF 



221819 



Men 



29000 — 



20001 — 



ism— 



lOOOC — 



5000 — 



We are at liberty to conclude that the ob- 

 served curve is the "true" distribution of 

 American male heights, but most persons, 

 whether they believe in the a priori necessity 

 of the Gaussian form of distribution or not, 

 will be under a strong conviction that the curve 

 should at least be smooth and not bimodal. 



The alternative explanation is that the in- 

 version is an artefact. The sampling of the 

 records in the offices of the insurance com- 

 panies was carefully done. All risks accepted 



during January of the odd years and July of 

 the even years throughout a sixteen-year period 

 were included. It would seem probable, there- 

 fore, that the selection must have occurred 

 when the measurements were made. Many 

 persons who were nearest 5 ft. 9 in. must have 

 been recorded as 5 ft. 8 in. or 5 ft. 10 in. 

 Here we would have a case of artificial selec- 

 tion that depends on the factors involved in 

 obtaining measurements, a case analogous to 

 the predilection that occurs when estimates are 

 made in " round numbers " or, as happened in 

 this investigation, to the predilection to give 

 weights as multiples of five pounds.' We may 

 hazard that the error occurred not so much in 

 reading the measuring stick as in the accept- 

 ance by the examining physician of the per- 

 son's own statement of his height. There may 

 be a tendency for a person to prefer an even 

 8 or 10 in. height to an odd 9. There is, how- 

 ever, no similar inversion obvious at 4 ft. 8-10 

 in., although this may be obscured by the ef^ 

 fort of men of this height to have themselves 

 recorded as 5 ft.; there is a suspicious bump 

 in the curve just above 5 ft. And the cases at 

 6 ft. 8-10 in. are too few to show. 



The further question arises whether the pre- 

 dilection is simply against the 9 in. or whether 

 all even heights are favored. It is not pos- 

 sible to determine this accurately, since an 

 inversion can not so readily appear in the 

 steeper parts of the curve. If we take the 

 Gaussian distribution (dotted line in the fig- 

 ure) as ideal we see that it is not true that 

 even inches fall above this ideal and odd be- 

 low. But then it is doubtful whether the 

 Gaussian distribution should be ideal. Tou 

 can not, at least, prove it from these data, 

 since the probability that an ideal Gaussian 

 distribution would turn out as this observed 

 distribution has is only (by Pearson's chi'- 

 criterion) about one chance in 10". 



We appear to have, then, a special predilec- 



2 For a complete discussion of the influence of 

 tiese oonununal mental habits upon scientific meas- 

 urements and other quantitative judgments, see 

 J. E. Coover, "Experiments in PBychical Bo- 

 search," 1917, 229-290. 



