Abstract: The residualt of observations,derived from least squares estimation can be defined in a great variety of ways and they have different probability distributions.This paper deals with their probability distributions,properties and applications.The table for the critical value of the redundancy numbers r_i=(Q_vP)_ii and the standar-dized residuals e_i are compiled.In addition,the independent residuals W=U^TV and S=D_w^-1/2W are derived by means of linear transformation.It is supposed that S can be used for the detection of gross errors,while the critieal value table carresponding to this test has been given.Finally,the adjustment for a trilateration network has been carried out,and the statis-fical tests have been performed by using a variety of the residuals obtained from the adustment.