Researches into the theory of probabiHty. 15 



The last part of the ealciiliis, (j) aitd (k) — by wliicli ß., and are obtained — 

 as ivell as (b) must be controlled through donhle calculation. 



A complete scheme for the calculation of a frequency curve according to the 

 above formulae is given on the preceding page. 



When a certain statistical material in respect to a » collective object* is to be 

 discussed, the first thing is to arrange this material into classes, all with the same 

 extension (range) as to the character in question. The class range is taken as unity 

 of the abscisste. By inspection a class in the neighbourhood of the mean is chosen 

 and considered as the origin from which the a;coordiuates are reckoned. The two 

 classes on both sides of that class, that is numerated with 0, get the number -\- 1 

 and — 1, and so on. The moments are calculated according to the equations 

 (a) — (h). It is not necessary to take into account the corrections given by Pearson 

 and SuEPPARD, if the class range is sufficiently small and coefficients of higher 

 order than ß^ are not taken into consideration. As a rule it may be advisable to talce 

 the class range snudler titan the standard deviation, the approximate value of which 

 is easily found from the frequency table (f of the material being included between 

 the limits h o and h — a). 



The corrected formulai for the moments given by Sheppard are: 



''j'' ~ (v., 



where (Vg), (v^) and (v,) design the corrected values of the moments (strictly the 

 moments divided with 



1" Example. For illustrating the above general theory I begin with a fre- 

 Ciuency curve discussed by Davenport, belonging to the type I of Pearson ^). 



Distribution of frequenci/ of glands in the right fore leg of ,9000 female sivine. 



Number of "lands 0 1 2 3 4 5 6 7 8 9 10 



Frequency 15 209 365 482 414 277 134 72 22 8 2 



We choose 4 glands as the ]irovisional origin of the ^'-coordinates. The cal- 

 culation scheme will then assume the following form. 



') ïlie frequency curve discussed in tills example belongs, strictly spoken, to the type JS, 

 tlie curve obviously being limited in one direction. It may, however, be used as an example of 

 such curves as, though belonging to the second type, may be conveniently represented througli 

 the formulae of type A. If notable differences occur at the limited end of the curve between the 

 observed and the calculated values, it will be necessary to use a curve of type B. I have treated 

 the sauie curve as a 7> curve beneath. 



) = ^2' — — tV = ^2 — T2- 



) = V3' — 3/>v; + 2//' = V, 



) = '^4' — 4?^V3' + G/rVg' — 3// — W — i 



