The index of dispersion as a metric of quanta - Unravelling the Fano factor

In statistics, the index of dispersion (or variance-to-mean ratio) is unity (σ²/x = 1) for a Poisson-distributed process with variance σ² for a variable x that manifests as unit increments. Where x is a measure of some phenomenon, the index takes on a value proportional to the quanta that constitute the phenomenon. That outcome might thus be anticipated to apply for an enormously wide variety of applied measurements of quantum phenomena. However, in a photon-energy proportional radiation detector, a set of M witnessed Poisson-distributed measurements {W₁, W₂,... W_M} scaled so that the ideal expectation value of the quantum is unity, is generally observed to give σ²/W < 1 because of detector losses as broadly indicated by Fano [Phys. Rev. (1947), 72, 26]. In other cases where there is spectral dispersion, σ²/W > 1. Here these situations are examined analytically, in Monte Carlo simulations, and experimentally. The efforts reveal a powerful metric of quanta broadly associated with such measurements, where the extension has been made to polychromatic and lossy situations. In doing so, the index of dispersion's variously established yet curiously overlooked role as a metric of underlying quanta is indicated. The work's X-ray aspects have very diverse utility and have begun to find applications in radiography and tomography, where the ability to extract spectral information from conventional intensity detectors enables a superior level of material and source characterization.The need for high-quality assessments of data motivates appreciation of shot noise as a data resource. The index of dispersion is a critical but often overlooked metric in this regard. When properly understood and carefully used it reveals the energies of quanta, in both monochromatic and polychromatic situations. To accomplish this, and in due course its wider application, its conceptual origins and quantitative and physical bounds are identified. Situations are examined that involve large numbers of quanta, but in which individual quanta are below measurement noise threshold. Practical applications are extremely diverse, so relevant considerations and how to use the metric are shown in broad contexts.