The Binomial Expansion: Simplifying the Evaluation Process

Abstract

Individual, small group, and program effectiveness can be evaluated quickly and painlessly using the binomial expansion. This powerful, but seldom used, nonparametric method is described and specific examples are used to illustrate its simplicity and elegance. An abbreviated table of probabilities is included. E valuation and accountability are frequently viewed as dark clouds looming over program development and individual staff appraisals. Every service provider faces challenges to prove that he or she has been effective in his or her job. But designing, conducting, and analyzing studies to evaluate services can be time consuming and complex, typically requiring changes in routines, identification and use of sophisticated measurement devices, and esoteric statistics long since forgotten, repressed, or joyously discarded. This is an unnecessary state of affairs because there are a number of simple procedures available. Perhaps because of their simplicity, these procedures have not received the attention they deserve. The subject of this article, the binomial expansion (BE), is one of these simple techniques. It requires only the identification of two numbers and a table of probabilities (see Table 1).In the behavioral scientist's world of statistics, there are three concerns: (a) magnitude of effect, (b) relationships among effects, and (c) consistency of effect. Magnitude of effect is determined by computing ratios of individual differences, such as the t test, analysis of variance (ANOVA), and regression. Relationships among effects are assessed through correlation coefficients and their extensions. Consistency of effect, however, is often overlooked when examining and analyzing data. The BE, described by the great mathematician, Blaise Pascal (1623-1662), is probably one of the most underused tools of analysis. It addresses the issue of consistency. Jenkins (1969) suggested that "the binomial expansion may well be the greatest all-around analytical instrument available for the treatment of behavioral data." Siegel (1956) claimed that if the data are basically dichotomous, there may be no other alternative as powerful as the binomial expansion.THE BINOMIAL A binomial is a mathematical expression, consisting of two numbers, which describes a series of observations or events. For example, if a coin is tossed 10 times (this is the series of events), the binomial is the number of times heads appears and the number of times tails appears. If heads turns up four times and tails six times, the binomial would be expressed as (4 + 6). In situations in which data can be described as consisting of only two classes or categories (i.e., heads-tails, male-female, member-nonmember, success-failure), the binomial can be used to analyze these data (Siegel, 1956). By mathematically manipulating the binomial (called expansion; see any basic text on probability), the researcher can determine the probability of obtaining the observed results by chance. This technique determines the probability of obtaining X objects or outcomes i...

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