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Geographic shape compactness is of concern in such contexts as urban morphology, political districting, and accuracy of enumeration unit values. The purpose of the present paper is to compare, categorize, and evaluate the various methods suggested for measuring compactness. Review of suggested compactness indices resulted in the identification of four categories, based upon: 1) perimeter-area measurement, 2) single parameters of related circles, 3) direct comparison to a standard shape, and 4) dispersion of elements of a shape's area. In all, eleven indices were calculated for a sample of U.S. counties and their frequency distributions were compared. Additionally, similarities and differences in the units of high, median, and low compactness were identified. Within each category, indices exhibited considerable similarity while significant differences were evident from category to category. These differences suggested variation in accuracy with which the indices measure compactness. It was hypothesized that measures based upon dispersion of elements of a shape's area would provide the most accurate measure because they consider the shape as a whole while the other indices do not. Determination of the accuracy with which each index measures compactness supported this hypothesis. Indices based upon direct comparison to a standard shape were found to be of similar accuracy and, therefore, also judged to be suitable compactness measures. Those measures based upon perimeter-area measurement and single parameters of related circles were determined to be considerably less accurate measures of compactness.
Alan M. MacEachren (Mon,) studied this question.