On the Probable Errors of Frequency Constants. 

 (EDITORIAL.) 



Introditdori/. 



In all rea.soning on Statistical data we havo to deteriuine whether diÖ'erciiees between Statistical 

 constants are signiticant or not. Half the blunders made in superficial Statistical investigatious 

 arise froui neglecting the values of the j)robable errors of the results obtaiued. But the 

 calculation of probable errors becoraes soniewhat coniplex, when we have to deal not merely 

 with the probable errors of means, biit with those of constants dcpending in a mnch more 

 intriciite maiiner on the monients of tlic niaterial. 



Tlie fundamental menioirs on the subjcct are W. F. Shcppard ; " On the Application of the 

 Theory of Error to cases of Normal Distribution and Normal (Jorrelation," Phil. Trans. A., 

 Vol. 192, pp. 101—167. L. N. G. Filon and K. Pearson : "On the Probable Errors of 

 Frequency Constants and on the Influence of Random Selection on Variation and Correlation," 

 Phil. Tram. A., Vol. 191, pp. 229—311. 



The present discussion of the probable errors of frequency constants presents, perhaps, little 

 that is uovel, but it endeavours to give simple proofs of the main propositions of the subject. It 

 is published in Biometrika, because several readera of that Journal have written to the Editors 

 making enquiries on these points, recognising that the iion-pulilicatiim of iirobablc errors in 

 Statistical memoirs is a serious disadvantage. 



The simple idea involved iu the probalile error of a Statistical constant is of the following 

 kind : If the whole of a [)opulation were taken we should have certain values for its Statistical 

 constants, but in aetual practice we aro oiily able to take a sample, which should if possible be a 

 " random sample." If a numbor of random samples be taken an}- Statistical constant will vary 

 from sample to sample, and its Variation is distributed according to some law round the aetual 

 value of the constant for the total population. This Variation would be very properly measured 

 by the sUmdard deviation of the constant for an indefiuitely great series of random samples. 

 Unfortunately custom has not taken this Standard deviation as the mea.sure of the goodness of 

 the sample, but tlie whole theory having ultimately developed from the normal curve, the 

 probable error instead of the Standard deviation has been chosen, i.e. "67449 x Standard deviation. 

 The adoption of the "probable error" of a constant as a measure of its exactness must not, 

 however, be taken as equivalent to asserting the validity of the normal law of errors or deviations, 

 but merely as a purely conventional reduction of the Standard deviation. It would be equally 

 valid providcd it were customary to omit this reduction or indeed to multiply the Standard 

 deviation by any other conventional factor. 



Biometrika ii " 35 



