1. THE object of this paper is to show how we can devise valid tests of significance which involve no assumptions about the forms of the populations sampled. It is also shown that precise fiducial limits can be determined for the difference of means of populations of the same form, no matter what the form of the populations may be. While only one test is discussed in this paper, the principle is applicable to all tests. The main idea is not new, it seems to be implicit in all Fisher's writings; * but perhaps the approach to the subject, frankly starting from the sample and working towards the population instead of the reverse, may be a bit of a novelty. Discordant, Concordant, and Neutral Separations.Suppose that we have m + n numbers (not necessarily all different), and that their mean is z. Numbers which are equal in value are supposed to be distinguishable from one another-we may think of the m + n numbers as painted on m + n different marbles.Consider a separation of the numbers into two different classes of m and n, (m ;;;;: n), U 1, U z . . . Um, with mean u,The number, N, of such separations is m+nCm, provided that when m = n the two classes, though equal in size, are regarded as different, so that two separations in which the classes are simply interchanged are regarded as different separations.We shall call Iii~1: ' 1 the spread of the separation. Since mu + nv= (m + n)z, an alternative expression for the spread isLet M be a fixed integer less than N. Consider any particular separation R. If the number of separations with a spread equal to or greater than that of R is less than or equal to M, we shall call R

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It is proved that if there exists a sufficient statistic for the estimation of an unknown parameter of a population, the frequency function of the population must be of a certain type.It is shown that some modification of previous theory of the intrinsic accuracy of statistics is necessary when the range of the population sampled is a function of the parameter to be estimated.Finally, the theory is extended to sufficient sets of statistics, i.e. sets of statistics which together contain all the information provided by a sample about an unknown parameter.