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The Price equation partitions the change in the expected value of apopulation measure. The first component describes the partial change caused byaltered frequencies. The second component describes the partial change causedby altered measurements. In biology, frequency changes often associate with thedirect effect of natural selection. Measure changes reflect processes duringtransmission that alter trait values. More broadly, the two components describethe direct forces that change population composition and the altered frame ofreference that changes measured values. The classic Price equation is limitedto population statistics that can expressed as the expected value of a measure.Many statistics cannot be expressed as expected values, such as the harmonicmean and the family of rescaled diversity measures. We generalize the Priceequation to any population statistic that can be expressed as a function offrequencies and measurements. We obtain the generalized partition between thedirect forces that cause frequency change and the altered frame of referencethat changes measurements.

The generalized Price equation: forces that change population statistics