Contribution to Applied Statistics: Dedicated to Professor by J. O. Irwin (auth.), Walter Joh. Ziegler (eds.)

By J. O. Irwin (auth.), Walter Joh. Ziegler (eds.)

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S. C. Author's address: Tore Dalenius, University of Stockholm. 45 w. Edwards Deming On Variances of Estimators of a Total Population Under Several Procedures of Sampling w. 1. Edwards Deming Introduction The aim here is to compare several simple plans of sampling that often appear to be equal, but which may give widely different degrees of precision when put into use. For example, it is well known that if we draw with equal probabilities and without replacement a sample of pre-determined size n from a frame of N sampling units (Plan I ahead), and if N be known and used in the estimator X = N X, in the notation set forth in the next section, then X is an unbiased estimator of the total population A of the frame, and the Var X is given by Equation (5) ahead.

Therefore we have two kinds of distribution-i. e. the number-distribution (which is the ordinary one in statistics) and the weightdistribution. Are these the only two distributions? The answer is no; people are occasionally measuring other kinds and quite unconsciously they think that what they are handling is one of these two. 64 I. Higuti The aim of this short article is to make clear the basic concepts of various size-distributions in small particle statistics and then the methods of their measurement, as well as the relations between them, and to suggest the possibility of building up new concepts of distribution and their measurement.

We now list the 4 possible outcomes of the sampling procedure for P = Q = 1(2. Their expected proportions are equal. We observe that: 1. E Ii = 1 (0 + 1 + 1 + 2) = 1. NP = 2 . 1/2 = 1, in agreement. 2. Av X = 3 = A, which illustrates the unbiased character of the sampling procedure. Likewise, Av Y = 8 = B. 3. Var X =i{ (0 - 3)2 + (4 - 3)2 + (2 - 3)2 + (6 - 3)2} = 20/4 =5. 52) + a2 ) = 20/4 = 5. 5 2 • Table 5 Table of all possible samples selected from Plan II from the frame shown above, with P = Q = 1/2.

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