© 2015 Elsevier Inc.Abstract We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main question we are concerned with in this paper is "How much do we loose by considering standard information algorithms against those using general linear information?" For both types of the fields, the error of linear algorithms has been studied in great detail; however, the power of standard information was not addressed so far, which we do here. Our main result is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one. The results are obtained by randomization techniques.