Mathematical Contributions to the Theory of Evolution, lib 



are calculated. The formulas obtained have been cited and used in a 

 memoir by Karl Pearson and' Miss Alice Lee, " On the Distribution 

 of Frequency (Variation and Correlation) of the Barometric Height 

 at divers Stations."* We may note the following results : — 



Probable error of mean = \<rj */ n. 



Probable error of standard deviation a 



= 4= - 



v/^V 2 (p+i) 3 s/' 



B B B 



where S= — H — \ — &c, and B 1} B 3 , B 5 .. ..are Bernoulli's 



numbers. 



Probable error of skewness S& 



Correlation of errors in mean and standard deviation 



2. 

 p + l 



i+iv- 



Correlation of errors in mean and skewness = 0. 

 Correlation of errors in skew r ness and standard deviation 



{l + 2(p- r l) 2 S} i 



Thus a random selection of size differing in its mean value from the 

 population mean gives in all probability an alteration in variability, 

 but none in skewness. Increased size means decreased variabiliiy. 

 A random selection altering variability alters also both size and 

 skewness of distribution. 



Probable error of modal frequency y 



= X 7l7 l ( 1+ iiJ' approxiniately ' 



which is shown to be always less than the probable error of the 

 mean. 



The correlation of errors in the moments and their probable errors, 

 up to the fourth are also worked out. 



4. As a limiting case the probable errors of the skewness, the 

 moments up to the fourth, and of the distance between mean and 



* About to be published in ' Phil. Trans*' 



