Past Experience on Future Expectation. 369 



by the means, from which they may differ, as just shown, by 

 as mnch as 5 to 50 p. c. o£ their values. 



I now pass to the standard deA-iation, o - , of the frequency, 

 measuring it as usual about the mean. The value of 

 //, 2 = o- 2 is given in the paper referred to (p. 239, eqn. (13)) 

 for a hypergeometrical series. We find, substituting the 

 present values of the symbols, that : 



It is clear that this can take values differing comparatively 

 widely from the customary o- 2 = mpq. This latter carries on 

 the face of it its unfitness to express the desired facts. The 

 variability about the expected average must depend on the 

 extent of the previous experience, but no sign of this is 

 visible in <r= s/mpq. 



Suppose, for example, that the characteristic does not 

 occur in a disproportionately large or small section of the 

 population, then if the samples are considerable as they 

 mostly will be in safe practice (# — p)/(n+2) may be 

 neglected and (viii.) gives : 



a~ = m 



pq 





™ P q(l+^}. 



if m and n are considerable. Hence a first sample n having 

 given a percentage value of the characteristic p, a second 

 sample of m individuals may be expected to give : 



100p±6rU9\/frq(- + -\ per cent.* . . (ix.) 



This is the correct value, and not lOOf + 67'M9a/— as is 



frequently adopted. To justify the latter the first sample 

 must be indefinitely larger than the second, and in many 

 cases this is not true. The first result given shows at once 

 why a small first sample is quite useh-ss in determining the 

 expectancy in the case of a second large sample. The 



absence of the term - in the second formula proves its 



unfitness. 



* This value has been long used in iny laboratory and by iny 

 statistical students. 



Phil. Mag. S. 6. Vol. 13. No. 75. March 1907. 2 D 



