178 



New Tables uf tlie l'rubabilitn /uta/ral 



Examples of Application. 



5. Expression of data in terms of x. Whcu a distribution is nearly normal, 

 we may statc tlic data by cxpressing A' (soc § 1) in terms of x. For an exaniple, 

 take tho head-brcadths of 3000 criminals, givun on p. 214 of BiometHka, Vul. I. 

 The interval iu X is '1 of a ceutimetre : but, for brevity, we shall take intervals of 

 •3 of a centimetre. 



We sliould first note tbc probable eiTore. If, of the N values, i\'', lie below A' 

 and iVj above it, thc probable error in A'i or iV„ is + Q '^^■N'i^'JN, where Q = '67449 ; 

 i.e. it is an eveu cliauce that N tiines the truc pruportion of values below A' lies 

 between N,- Q'</N^N,/N and N, + Q'^N'7N^. Thus, for A'=14ö.J cm., the 

 i>. E. is ± Q \/479 X 2.521 -f- 3000 = 13*.5. Calculating thc probable crrors, the data 

 may be expressed thus: — 



Now calculatc the values of x and of z correspouding to A', = ^A'^(l + ot). An 

 error of 6 in x is equivalent to an error of zd iu ^ (1 + a), so that the above values 

 of the P. E. have to bc diviilcd by 30005 to give the P. E. iu x. We thus get the 

 data in the form : — 



This, it shoidd bc obscrvcd, is mcrely a statcment of facts, and docs not involvc 

 any assumption as to the distribution being really normal. 



6. Interpolation*. By means of these values of x, we can intcrpolate for 

 values of A lying towards tho extremities of thc ränge, where thc differences 



* For a fuUer discusgion of the methods employed in this and thc ncxt two sections, Bee Journal 

 0/ the Royal Statigtical Society, Vol. LXiii. pp. 433 — 451. 



