338 Life and Letters of Francift Galtou 



by Galton to have the median value of the character (m). The two men with 

 75 '/^ and 25 7, of the population alxive them are said to have the lower and 

 upper (luartile values (</, and </,). If the distribution Ije synunetrical about 

 the meaian then »i — 7, and 7, — m will he enual ; if the distribution obeys the 

 so-called normal curve of deviations, then all the constants i>f the distribiition 

 can be found by measuring the intensity of the chai-acter in the median and 

 in the quartile individuals. Thus Galton would place a hundred and one 

 savages in a row, the curves formed by the apices of their heads would be 

 his "ogive'" for their stature, and by measuring only the 25th, the Slst and 

 7Gth men he would obtain a reasonable distribution for the stature of adult 

 men in that tribe. 



Theoreticaliy there are difficulties about Galton's "ogive," if we suppose 

 it to correspond to a normal curve of deviations, in particular at the terminals. 

 Galton endeavoured to get over these difficulties by replacing the normal 

 curve by a symmetrical binomial, which has a finite range. He treats of this 

 matter in a paper on "Stjitistics by Intercomparison with Remarks oti the 

 Law of Fretiuency of Error'." In this paper after mentioning that Quetelet 

 had shown that a binomial to the it99th [xtwer was practiadly a normal curve 

 of deviations, Galton goes on to indicate that the same holds very closely 

 for symmetrical binomials of quite low powers. Thus he plots (p. 39) the 

 Binomial Ogive of 17 elements agjiinst the Binomial Ogive of 999 equal 

 elements, which is practically identical with the Exponential Ogive, and argues 

 therefrom to the binomial of the 17th power being very close, indeed (which 

 is a fact), to the normal curve'. Galton then pa.sses to some suggestive 

 remarks on the origin of the distribution of deviations according to the 

 normal law. He rejects any idea of its source in a very large number of 

 small and independent contributory causes. He supposes the exponential 

 curve to arise because it nearly resembles the curve based upon a binomial 

 of moderate power, i.e. he supposes that in nature the contributory cause- 

 groups are relatively few; but he has to suppo.se in this case that nature 

 works all her processes by equal additions or subtractions, i.e. prefers the 

 mathematics of coin-tossing to those of the dice-box. 



"1 shall show," he wriU-.t, "by quite a different line of argument that the exponential 

 view contain.s inherent contradictions when nature is appc<iled to, that the binomial of a 

 moderate power is the truer one and tliat we have means of ascertaining a limit which the 

 number of elements [independent cause-groups as the individual coins of n coinhincd toss] 

 cannot exceed." (p. 40.) 



Galton takes the mean (m) and divides it by the quartile deviation 

 (7, — m or m — 7,) and computes the ratio ni/{q, — m). In the case of the 

 binomial (a + b)" this will be 



nb/-67i49j^b=ViS257 J~ , 



' Philosophical Magaziiu, January, 1875, Vol. XLix, pp. .33-46. 

 . ' If A" Ik- the abscissa, i.e. the rank, and V the ordinate or value of the variate, m the mean, 

 <r the standanl deviation of the population - 1-48^57 (y, — m), and N the total po})iiln.tiim lln-n 



the equation to Gallon's "ogive" will heX=\+ j -f^ ,-i(l'-"«)V£/r. 



* I have not been able to agree with the values given in the Table on p. 42. 



