Statistical modelling of Poisson/log-normal data

In statistical data fitting, self consistency is checked by examining the closeness of the quantity chi(2)/NDF to 1, where chi(2) is the sum of squares of data minus fit divided by standard deviation, and NDF is the number of data minus the number of fit parameters. In order to calculate chi(2) one needs an expression for the standard deviation. In this note several alternative expressions for the standard deviation of data distributed according to a Poisson/log-normal distribution are proposed and evaluated by Monte Carlo simulation. Two preferred alternatives are identified. The use of replicate data to obtain uncertainty is problematic for a small number of replicates. A method to correct this problem is proposed. The log-normal approximation is good for sufficiently positive data. A modification of the log-normal approximation is proposed, which allows it to be used to test the hypothesis that the true value is zero.