The Normal and Lognormal Distributions are Asymptotes of the Johnson SB Distribution

Abstract

P-794 Introduction: In environmental exposure data analysis one often makes an a priori assumption that when data are symmetrical a Normal Distribution (ND) should be chosen and if positively skewed that a 2-parameter lognormal distribution (LN2) is appropriate for computing statistics. It is not generally recognized that both the ND and LN2 are asymptotes of the same distribution, the Johnson SB Distribution. The SB distribution provides a smooth continuum of distributions connecting the ND and the LN2. Method: We write the SB Distribution as follows for a variable x, where Xmin < x < Xmax. Y = log [(x - Xmin) / (Xmax-x)] Where Y is normally distributed between − infinity and + infinity with mean: and standard deviation Φ. Using the approximation log (1 + w) goes to w in the limit as w goes to zero, we show that in the limit as Xmax goes to + infinity and Xmin goes to zero the distribution is equivalent to LN2; In the limit as Xmax goes to infinity and Xmin goes to -Xmax, the distribution is equivalent to a ND. Thus there is a continuum between them for intermediate finite values of Xmax and Xmin. Results: Table 1: Demonstration that an SB Distribution approaches a Normal Distribution for Xmin = - Xmax in the limit as Xmax becomes large. The Normal Distribution has a mean of: = 10 and Φ = 1, where z is a standard normal deviate.TABLE 1: Demonstration That an SB Distribution Approaches a Normal Distribution for Xmin = −Xmax in the limit as Xmax becomes large. The Normal Distribution has a Mean of: = 10 and Φ = 1, Where z is a Standard Normal DeviateConclusion: The LN2 and ND are simply different asymptotes of the SB Distribution. It is therefore proposed that the SB distribution is an optimal distribution for fitting environmental data because by its very nature it must always fit any data set better than a ND, a LN2 or even a 3-parameter lognormal distribution which is also a member of the family.