Fluctuation Scaling, Calibration of Dispersion, and Detection of Differences

Fluctuation scaling describes the relationship between the mean and standard deviation of a set of measurements. An example is Horwitz scaling, which has been reported from interlaboratory studies. Horwitz and similar studies have reported simple exponential and segmented scaling laws with exponents (α) typically between 0.85 (Horwitz) and 1 when not operating near a detection limit. When approaching a detection limit, the exponents change and approach an apparently Gaussian (α = 0) model. This behavior is often presented as a property of interlaboratory studies, which makes controlled replication to understand the behavior costly to perform. To assess the contribution of instrumentation to larger scale fluctuation scaling, we measured the behavior of two inductively coupled plasma atomic emission spectrometry (ICP-AES) systems, in two laboratories measuring thulium using two emission lines. The standard deviation universally increased with the uncalibrated signal, indicating the system was heteroscedastic. The response from all lines and both instruments was consistent with a single exponential dispersion model having parameters α = 1.09 and β = 0.0035. No evidence of Horwitz scaling was found, and there was no evidence of Poisson noise limiting behavior. The "Gaussian" component was a consequence of background subtraction for all lines and both instruments. The observation of a simple exponential dispersion model in the data allows for the definition of a difference detection limit (DDL) with universal applicability to systems following known dispersion. The DDL is the minimum separation between two points along a dispersion model required to claim they are different according to a particular statistical test. The DDL scales transparently with the mean and works at any location in a response function.