5
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.
0
2
4
6
8
10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Lognormal Distribution
z
Density
Lognormal
MAD
IQR
Recommended to OMB by Committee on Statistics: 07-17-2013
Reviewed and approved by OMB: 07-18-2013
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