Significance of the difference between two means when the population variances may be unequal

Abstract

THE purpose of this communication is to present a simple test applicable at the 5 per cent level to the general case of the significance of the difference between two means. If we have two samples of n1 and n2 variates from normal populations with the same, though unknown, scale factor, that is, σ1 = σ2, the significance of the difference between the means can be tested by the criterion: Chemical equation represents where Σ1 and Σ2 represent the sums of squares from the respective means and t is referred to Student's integral with n1 + n2 - 2 degrees of freedom. If the scale factors are unequal the use of this criterion involves a bias1 which is small if n1 = n2 but which can be large if n1 ≠ n2. When the relative scale factor of the two populations is known, appropriate weighting of the sums of squares gives an exact solution. In the case where the relative scale factor is unknown, Fisher2,3 advocates the use of a different criterion, Behrens' d4, which can be expressed as : Chemical equation represents and tables5,6 are available for the 5 per cent and 1 per cent points of this distribution for a lattice of values of n1 and n2 both greater than 6. © 1960 Nature Publishing Group.