On the diversity-based measures of equalness and evenness

While the notion of evenness is generally undisputed, its opposite, unevenness, is vague. Conventionally, evenness is understood to decrease, and thus unevenness to increase, as the effective number of types of positive frequency tends towards one while the number of types is held constant. Complete unevenness can thus be approached but never realized for more than one type. In order to arrive at explicit states of minimum evenness for any number of types, equalness is introduced as a complementary concept of evenness. Realization of the new concept turns out to require a change in orientation from unevenness to unequalness. Decreasing evenness in the sense of increasing unequalness entails an increase in inequality among type frequencies. Unequalness can effectively be envisioned as the diversity in the distribution of step-heights in a frequency distribution ranked in decreasing order (frequency profile). Application of appropriate measures of diversity reveals that maximum unequalness (minimum equalness) is assumed for linearly decreasing frequency profiles (stepladders). A consistent diversity-based measure of equalness is obtained by normalizing the step-height diversity with respect to the number of types, with the result that this measure equals one for complete equalness/evenness (uniformity) and zero for complete unequalness (as in stepladders). It turns out that the new equalness measure is highly sensitive to characteristics of frequency profiles that are commonly associated with the evenness notion but are poorly reflected by the conventional evenness measures as a consequence of the realization of their minimum for monomorphism.