A Reduced Formula for Point-Biserial Coefficient of Correlation

Abstract

TWO OF THE most common types of correlation coefficients for the purpose of item analysis are biserial r and point-b i s e r i a 1 r. The first of these, biserial r, assumes that the item is represented by a dichotomization of a continuous and normally distributed variable and that the criterion is a continuous but not neces sarily normal variable. Whether or not the s e assumptions can be justified is often question able. Richardson and Stalnaker (5:463-64) in dicate that the biserial r is not a suitable meas ure for item analysis where the responses are dichotomized because it assumes:(l) that the distribution of the dichotomized variable is normal, and that (2) the dichotomized variable is continuous. They indicate that unless these conditions can be established and be shown to fit the case being studied, that use of the biser ial r is not warranted. In such cases the point biserial r is more suitable. The laborious computations which are re quired for most methods of item analysis hav e been a deterrent to their use. In order to re duce the labor of computing the biserial coeffi cient, Flanagan devised a procedure using on ly the upper and lower portions of the group (3: 674-80). Dunlap (2:51-58), Chapanis(l:297-304) and others have proposed methods for shorte n ing the amount of computation in determining the biserial index. Such shortened methods are pre ferable if they yield equally satisfactory results, and in some cases may be preferred because of the saving in time and effort even though a cer tain degree of accuracy may be sacrificed. A re duced formula for the point biserial coefficient is developed and shown below. The point-biserial coefficient of correlation is given by the following formula (4:353):