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Diversity indices are frequently applied in the form of ratios of absolute diversity to the maximum diversity possible. Regardless of whether the maximum diversity is defined to be limited by the number of species or by the number of individuals present, the resultant indices can be shown to possess mathematically undesirable qualities. All such indices, including equitability as measured by Pilou's J and J', Lloyd and Ghelardi's ratios, redundancy, and the scaled diversity formulations of Fager, are inappropriate for most ecological applications.
Robert K. Peet (Sat,) studied this question.