Effect of normal standard errors on lognormal distributions

Abstract

Simulated ‘empirical’ data sets were generated to assess the effect of a normally distributed blank on the apparent frequency distribution of a constituent whose concentrations are actually lognormally distributed. When the standard deviation of the blank ( s blank) is relatively small, on the order of 10–50% of the geometric mean, the major consequence is a significant downturn in the cumulative probability plot at the low end of the concentration scale. With larger standard deviations, the upper portion of the plot is also substantially affected; concentrations for a given cumulative frequency can be significantly higher than those which actually exist. However, regression of the data above the detection limit (defined as 2 times s blank) can yield reasonably good estimates of the actual distribution parameters even when the standard deviation is comparable to the actual geometric mean. At higher standard deviations, regression analysis yields substantially biased results. A simple moments method is suggested for estimating the parameters (geometric mean and standard geometric deviation) of the actual lognormal distribution when s blank is known, as it should be under most experimental protocols. Results from the simulated data sets indicate that this method yields very good estimates of the parameters even when s blank is significantly larger than the geometric mean. Mineral aerosol concentrations from American Samoa are presented to illustrate the effects in a real data set and the ability of the moments method to predict the actual distribution.