This paper is concerned with estimation problems where there are several parameters all of the same type--for example, a set of means. It often happens that the knowledge of the parameters is exchangeable in the sense of de Finetti. This prior knowledge can be used to obtain estimates alternative to the usual ones. The particular problem studied here is the familiar analysis between and within groups. New estimates of the group means are obtained, and when the assumption of equality of variances within groups is omitted, new estimates of the individual variances are also derived.