﻿ Equation: Lorentzian

# Equation: Lorentzian

## Introduction

A Lorentzian distribution is bell shaped, but has much wider tails than does a 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 are 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 "Lorentzian" equation from the "Gaussian" family of equations.

## Model (Lorentzian distribution)

Y=Amplitude/(1+((X-Center)/Width)^2)

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

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

Width is a measure of the width of the distribution, in the same units as X. This is not identical to a standard deviation, but has the same general meaning.

## Model (sum of two Lorentzian distributions)

One=Amplitude1/(1+((X-Center1)/Width1)^2)

Two=Amplitude2/(1+((X-Center2)/Width2)^2)

Y=One + Two

Amplitude1 and Amplitude2 are the heights of the center of the distribution in Y units.

Center1 and Center2 are the X values at the center of the two distributions.

Width1 and Width2 are measures of the widths of the distributions, in the same units as X.

Prism is not very smart about assigning initial values to the parameters. If you have trouble getting this model to fit, try fussing with the initial parameter values.