Author: John M. Cimbala, Penn State University
Latest revision: 30 January 2008
useful standard distributions and PDFs besides the Gaussian PDF. These include the
binomial, chi-squared, exponential, gamma, lognormal, Poisson, student’s
, uniform, and Weibull PDFs.
We discuss some of these in this learning module, although not in as much detail as for the Gaussian
is defined as
a PDF that becomes Gaussian when the x-axis is plotted as a log scale
Lognormal PDFs often appear in air quality measurements, e.g., the size distribution of particles. It is
also useful for some life and durability analyses of components and equipment or instruments.
When the PDF is plotted as usual (linear
scale), it is skewed towards the left (lower values), and has a
very long tail to the right (higher values). This is shown on the first plot below.
However, when the PDF is plotted with a logarithmic
scale, all else being equal, it is no longer skewed,
but becomes symmetric. In fact, it’s bell shape is identical to that of a Gaussian or normal PDF. This is
shown on the second plot below.
Another way to plot lognormal PDFs is to first convert the
values to log
) or ln(
), and then plot
abscissa scale. Either way, the PDF again looks like a standard Gaussian PDF, as
To calculate statistics with a lognormal PDF, we substitute either log
) or ln(
) as our variable instead
itself. For example, if the data are for particle diameter
in units of microns (
m), we let our
statistics variable be
m)] instead of
itself. All statistics are then based on