On the Elimination of Systematic Errors Due to Grouping
In the calculation of the moments of a frequency distribution it is often desirable, or even necessary, to consider not the distribution itself but another derived from it by certain groupings. As a first approximation to the moments of the original distribution we take the corresponding moment of the grouped distribution. But this first app)roximation is, not satisfactory, anld it is necessary to obtain some miiethod for the'elimination of part of the error committed in replacing the moments of the original distribution by the corresponding noments of the grouped distribution. This problem was first discussed by W. F. Sheppard in a paper: Ot the Calculation of the mnost Probable Valte of Frequency-Constants, for Data arranged according to Equidistant Divisions of a Scale.' If we denote the' n-th moment of the original distribution by a,u1 and the n-th moment of the grouped distribution by 7/ we will have Sheppard's corrections in the form: