Search

The method of least squares offers a precise method ofc fitting a curve describing the relation between two or more related, measurable variables, but certain criteria must be fulfilled to justify its application. First, the type of equation selected for fitting must be the true mathematical expression of the law governing the relationship of the variables. Secondly, all errore of measuremeiit, made in obtainiing the observed values of the variables when the data were collected, must be distributed according to the wellknown laws of probability.' This paper is concerned with the latter of these two criteria. The fundameintal theory upon which the method of least squares is based can be found in any text-book on the subject and need not be elaborated upon here. However, it may be well to point out a very pertinent, if somewhat elenientary, aspect of the theory which facilitates the ready visualization of the fundamental concepts involved.

THE DESIG N OF A SAMPLE usually involves some method of taking account of heterogeneity in the universe to reduce the size of the random sampling error. The degree of success with which this objective is attained depends upon the extent and precision with which various contributions to the variability of the individual units in the universe are isolated and measured before the sample is designed. The amount of pertinent information about the universe that is available before a sample is selected is sometimes meager and sometimes quite far-reaching; it may have become available from previous sampling investigations or from previous contacts with every individual in the universe. But regardless of how much information is available, or how it was obtained, an experienced sampler takes advantage of it. Mathematically, the heterogeneity of the individuals in a universe with respect to a variable X can be expressed as a series of sums of squares which add up to the total sum of squares S(X m)2, where m is the average of all values of X in the universe. The number of component sums of squares into w-hich such a total can be resolved is limited only by the amount of available information concerning the universe and the number of individuals in the universe. This simple algebraic identity serves as a basis for constructing the formula for the sampling variance of a mean, x, computed from a sample with a specified design. The synthesis of such a formula can be accomplished in more than