OMB Meeting Book - January 8, 2015

55

LOGNORMAL DISTRIBUTION Now consider the standard lognormal distribution with log mean 0 and log standard deviation 1. The unlogged mean is 1.6487 and the unlogged median is 1.0, with an unlogged standard deviation of 2.1612. Based on 100,000 realizations of samples of size 24, the estimated mean standard deviation (‘s’) is 1.899, the equivalent estimate based on MAD is 0.8904, and the equivalent estimate based on the IQR is 1.062. The biases in the MAD- and IQR-based estimates are substantial. The standard errors of the statistics (i.e., standard deviations of the sampling distributions) are 1.054 for s, 0.8904 for the MAD-based estimate and 0.3756 for the IQR-based estimate. The sample standard deviation s is imprecise, but unbiased. The MAD- and IQR-based estimates are precise, but heavily biased. Use of ‘robust’ estimators for the standard deviation when the underlying distribution is lognormal (i.e., heavily skewed) results in estimates which are only ½ of the correct value.

Lognormal Distribution

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Lognormal MAD IQR

Density

0

2

4

6

8

10

z

5

24

Recommended to OMB by Committee on Statistics: 07-17-2013 Reviewed and approved by OMB: 07-18-2013