Equation:: Log Gaussian distribution

Introduction

Data follow a Gaussian distribution when scatter is caused by the sum of many independent and equally weighted factors.

When scatter is caused by the product of many independent and equally weighted factors, data follow a log Gaussian distribution. When plotted on a linear X axis, this is skewed to the right (see below). When plotted on a logarithmic X axis, it looks like a bell-shaped Gaussian distribution.

Step-by-step

The data must be in the form of a frequency distribution on an XY table. The X values are the bin center and the Y values are the number of observations.

If you start with a column of data, and use Prism to create the frequency distribution, make sure that you set the graph type to "XY graph", with either points or histogram spikes. This ensures that Prism creates an XY results table with the bin centers entered as X values. If you pick a bar graph instead, Prism creates a column results table, creating row labels from the bin centers. This kind of table cannot be fit by nonlinear regression, as it has no X values.

Starting from the frequency distribution table, click Analyze, choose Nonlinear regression from the list of XY analyses, and then choose the "logGaussian" equation from the "Gaussian" family of equations.

Model

Y=Amplitude*exp(-0.5*(ln(X/Center)/Width)^2)

Amplitude is the height of the center of the distribution in Y units.

Center is the X value at the peak of the distribution.

Width is a measure of the width of the distribution, in the same units as X.