The situation ia considered where the multivariate distribution of certainvariablesXI,X,, . . ., X, is changing with time in a population because natural selection related to the X's is taking place.
It is aseumed that random samples taken from the population a t times t,, b, . . ., t, are avaihbh and i t is desirable to estimate the fitness function ro,(z,, z2,. . ., zp) which shows how the number of individuals with &=zi, i = l . 2, . . ., p at time t is related to the number of individuals with the same X values a t time zero.
Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating woI.
The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case them seems to have been very little selection on the population dispersion but considerable selection on means.