The ‘b’s’ Knees: Logistic Stage-Transition Rules as Percentile Conventions on a Standardised Time Scale

Recently published:

• Levinson, D. (2026) The ‘b’s’ Knees: Logistic Stage-Transition Rules as Percentile Conventions on a Standardised Time Scale. Findings, June 1, 2026. [doi] [Data/code]

Logistic diffusion is often described as passing through early growth, rapid growth, and maturity, but the dates separating these stages, the knees of the curve, are defined in different ways across fields. This note brings several familiar stage-transition rules for the standard logistic curve into one common form. Each symmetric analytic rule can be written as t0 +/- c/b, where b is the logistic growth-rate parameter and t0 is the inflection date. Equivalently, each rule implies a fixed percentile window of the asymptote K. The paper shows that common stage-transition conventions differ by the implied percentile window, and provides a synthetic Python example comparing analytic rules with a free segmented-regression benchmark.

Keywords: logistic growth, diffusion of innovations, stage transitions, sigmoid curves, technology diffusion, inflection point, percentile conventions, segmented regression