Effective Size and Number of Components

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Abstract
Two families of indices are defined which may be used to characterize the effective number of components in systems with unequal component sizes. A general formula is staled relating size, effective size, and effective number of components. Special cases of this formula are considered which yield formulae identical or closely related to a variety of other expressions including entropy, the Greehberg-Lieberson index of diversity (also known as the Rae-Taylor fractionalization index), the Herfindahl-Hirschman concentration index, the coefficient of variation, the ordinary mean, the weighted mean and the harmonic mean. Applications of these formulae are considered for a variety of problems, including measurement of population and GNP concentrations, distinguishing between density and crowding, and reconciling differing student and faculty perceptions of average class size.

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