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Applying the standard weighted mean formula, sumᵢ nᵢ sigma^{-2ᵢ} / sumᵢ sigma^{-2ᵢ}, to determine the weighted mean of data, nᵢ, drawn from a Poisson distribution, will, on average, underestimate the true mean by ~1 for all true mean values larger than ~3 when the common assumption is made that the error of the ith observation is sigmaᵢ = max (sqrtnᵢ, 1). This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman's chi-square statistic, chi²₍ equiv sumᵢ (nᵢ-yᵢ) ² / max (nᵢ, 1), to compare Poisson-distributed data with model values, yᵢ, will typically predict a total number of counts that underestimates the true total by about 1 count per bin. Based on my finding that the weighted mean of data drawn from a Poisson distribution can be determined using the formula sumᵢ [nᵢ + min (nᵢ, 1) (nᵢ+1) ^-1] / sumᵢ (nᵢ+1) ^-1, I propose that a new chi-square statistic, chi²gamma equiv sumᵢ nᵢ + min (nᵢ, 1) - yᵢ² / nᵢ + 1, should always be used to analyze Poisson-distributed data in preference to the modified Neyman's chi-square statistic. I demonstrate the power and usefulness of chi-square-gamma minimization by using two statistical fitting techniques and five chi-square statistics to analyze simulated X-ray power-law 15-channel spectra with large and small counts per bin. I show that chi-square-gamma minimization with the Levenberg-Marquardt or Powell's method can produce excellent results (mean slope errors <=3%) with spectra having as few as 25 total counts.
Kenneth J. Mighell (Thu,) studied this question.