Sample Size-Based Indication of Normality in Lognormally Distributed Populations

Occupational and environmental hygiene sampling strategies are usually dictated by factors that limit sample sizes to relatively small numbers. Often, parameters estimated from small sample sizes are then used to make further estimates of the occurrence of extreme events, which are governed by the underlying exposure distribution. We investigated the limitations superimposed by the number of samples in distinguishing an asymmetric (Lognormal) distribution through the rejection of a hypothesized symmetric (Normal) distribution. Sets of 5 to 250 synthetic samples from underlying Lognormal distributions with unit median were generated for 24 separate geometric standard deviations (GSDs), ranging from 1.25 to 7.00. Each simulated combination was repeated in blocks of 200 and each block was repeated tenfold. The synthetic samples were then tested for goodness of fit for Normality by using the Shapiro and Wilk's W Test. Results indicated that the number of samples required to distinguish between Normal and Lognormal distributions was inversely related to GSD. When GSD = 1.25, 169 samples were required for 90 percent distinction at alpha = 0.05. The criteria for success for GSD of 2.00 and 4.00 were 25 and 15 samples, respectively. These results led to the conclusion that the general inability to distinguish an underlying distribution may impose serious difficulties in the estimation of extreme events associated with occupational and environmental hygiene-related sampling.