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

David Levinson
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David Levinson: TransportLab, School of Civil Engineering, University of Sydney

Working Papers from University of Minnesota: Nexus Research Group

Abstract: 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. The main point is that each symmetric analytic rule can be written as t_0 c/b , where b is the logistic growth-rate parameter and t_0 is the inflection date. Equivalently, each rule implies a fixed percentile window of the asymptote K . In particular, the rule based on zeros of the third-derivative implies 21.13% to 78.87% of K , and the tangent-at-inflection rule implies 11.92% to 88.08% of K . A small synthetic example also shows how a free three-segment linear regression compares with analytic conventions. This paper clarifies conventions that are often conceived of as conceptually distinct.

Keywords: transportation; public transport (search for similar items in EconPapers)
JEL-codes: R40 (search for similar items in EconPapers)
Date: 2026
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https://doi.org/10.32866/001c.162369 Published version landing page, 2026 (text/html)

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Persistent link: https://EconPapers.repec.org/RePEc:nex:wpaper:paper-2026-16

DOI: 10.32866/001c.162369

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