274 On the Probable Errors of Frequencij Comtantx 



Problem I. Let there be n frequency groups, containing in Ute total population of N individualt, 

 i/t> 1/21 2/3 — i/n i'idit'idiKils reapeclively. Toßnd the Standard deviation 0/ y„when random mmples 

 are inadc of m individuals front the whole popuiition. 



The cliancc of au individual being drawn from the s"' grouii ou one trial ='^r=P saj', »»d of 

 its not being drawn = 1 - ■%=5', say. Thcn in »1 trials the distribution of frequency of this group 

 will be given by (p + (/)"', with the known Standard deviation o'„,= ^'"'/'y = */ «i . — ( 1 - — j , where 



y/=^y, and is accordingly the proportion of y, which we should cxpect in the typical group of 



m out of the ^V individuals. But in actual practico we havc only the sample and do not know 

 N or y,. If yi' be the observed freipiency of the «"> group y," will almost certivinly lie within 

 y,±3(r, . Hence in the above formula for (r„ we may replace, if o-j, be siuall coiupai-ed with 

 y,, y,' by y,", the observed frequency of the sample. Or, we read 



'^\-'J'{'-t} «' 



where y, is now takcn as the frequency of the «•"' group in the sample. 



Problem II. To find the correhition beticeen deviation» in y, and y,; or hetween deviationt in 

 the frequencies of the s"' and s'"" groups. 



Let 8y, = deviation from y, the niost probable value in the «"' gi'oup, then since 



Now if our sample has givcn hy, too many in y,, it is projK^r to supiK).se that this error 

 will be distributed aniong the other groups in the proportion of their relative frequencies*. In 

 other words we should have 



Thus; 



Summing for all samplings 





Hence 



<, y>' V'V'', rs 



o'u ,<^v.>'u V .= , ;— = by (1). 



".- "s ".".■ ;„ l-y,jm m •' ^ ' 



y^ye 



(ii). 



TO 



Problem III. To find the Standard deviation o-j of the nuan h of a System of observations. 



Measured from a fixed ix>int of reference, we have : 



S{xy)_Sixy) 

 S{y) m ' 



where y is the frequency of individuals of size x. 



* This of conrse assumes that the error is merely diic to randoni .«ampling, and not to defeetive 

 measurement or classificatory judgment. In such casea the error in Sy, might, for example, be cspecially 

 drawn from the "adjacent" classes y,.i and y^, . 



