Suppose that t treatments are given whose properties (yields, responses, effects, etc.) we wish to compare when they interact with a given set of rs experimental units, the latter being classified into r blocks, each containing s = pt units. Suppose, further, that an experiment is carried out in which the treatments are applied at random to the experimental units, with the restriction that each treatment appears with p units in each of the r blocks. We refer to this design as the generalized randomized block design and note that it includes as special cases the completely randomized design (r = 1, p > 1) and the randomized block design (r> 1, p = 1). The object of this paper is to study the basis for statistical inference which is provided by the randomization procedure.