The estimation of statistical coefficients from samples

Abstract

Statistics, it has been said, is the reduction of data. This definition fails, however, to bring out one of the most important and interesting aspects of modern statistical theory - the making of estimates. Let us consider a typical example. An investigator collects a body of data, consisting, say, of the heights, or other attribute, of some members of a community. He calculates a few quantities - such as a mean, a measure of dispersion, perhaps one of skewness - which enable him to apprehend the properties of the assemblage of figures. This is the reduction of data. But now curiosity or practical need poses a new question. What can be said regarding the heights of all the members of the community - not merely of those in respect of whom information is tabulated? Of course, nothing can be stated with certainty. The heights of the members who were measured may vary widely from those of the others. Nevertheless, intuition, or experience, suggests that at least a guess might be hazarded. One suspects, too, that in some instances a better guess is justified than in others. Can one, then, measure in some fashion the reliability of a guess? Questions such as these are the subject-matter of the Theory of Estimation.

This theory, as our example indicates, has nothing in common with the deductive problems frequently encountered.in Pure Mathematics, such as the deduction of a particular geometrical theorem, given a few general axioms. On the contrary, Estimation is concerned with the realm of induction; with passing from the particular to the general; with the making of inferences regarding a population from the data of a sample thereof. By their nature, these inferences are uncertain. Perhaps it is the insufficient appreciation of this which has led to the well-known criticism that statistics "can be made to prove anything." It would be fairer to say that statistics proves nothing - but that it may suggest a great deal.