The conventional outlier detection procedures, such as the methods of Baarda and Pope or the t-testing procedure, determine only one outlier reliably. The approach to robustifying these procedures is as follows: (I) To identify outliers by using an estimator that has a high breakdown point and a bounded influence Function; (2) to find "good observations" by separating outliers from whole observations; (3) to constitute the reduced samples obtained by systematically adding each single outlier in turn to the good observations; and (4) to apply the conventional outlier detection procedures to each single reduced sample separately. To test the approach, an M-estimator with Andrews weight function is chosen. Then it is studied using a coordinate transformation simulation. Only two outliers are able to be determined reliably.