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 (σ 2 /〈 x 〉 = 1) for a Poisson-distributed process with variance σ 2 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....