An initial diagnosis of some educational and psychological capacities of students on arrival at university were studied. This enabled us to find out what factors had a greater influence on academic achievement at the end of the first year. Using the techniques of multiple regression we established the optimal achievement performances expected from each of the students. Secondary school marks, the academic achievement tests and the intermediate examinations at university were the best set of predictors of academic performance. Differential aptitudes of intelligence increase considerably the accuracy of the prediction. Values of R of between 0.71 to 0.88 were reached depending on the criteria used. The usefulness of the prediction equations as a tool for increasing personalized attention to students is pointed out and a case made for the establishment of objective mechanisms for admission to higher education.