An Elementary Diversity Index Developed Using Taylor Series and Lagrange Multipliers
by
Donald E. Hooley
Bluffton College
Bluffton, OH

www.bluffton.edu/mat/seminar/

An Elementary 
Diversity Index
•	A Diversity Question
•	Properties of Diversity Indices
•	Shannon’s Diversity Index
•	Taylor Series Approximation
•	Lagrange Multipliers and Equiprobability
•	Results
•	Project Ideas
A Diversity Measurement Question
   Which classroom is more diverse?
•	Half Caucasian, Half African-American
•	1 Indian, 1 Mexican, 1 Russian, 1 French, 1 Laotian, 15 Caucasian

Diversity Indices
Berger-Parker  	Nmax/N
Margalef		(S-1)/ln N
McIntosh	
			U=

N     = 	number of individuals in total 		population
Nmax = 	number of individuals in most populous 	species
S      = 	total number of species	
Diversity Indices
Shannon		

Simpson		

pi =	proportion of species i in population
ni =	number of individuals of species i
N = 	number of individuals in total 	population
S =	total number of species in population
	
Minimal Diversity Property
Classroom Calculator Use
	All	   Some     Never
S1)      1	       0	    0
S2)    .20 	     .40 	  .40
S3)  	  0	       0	    1
Maximum Diversity
Flower Garden
	Tulips	      Roses	Mums
G 1)	   .80          .10	   .10
G 2)	   .30	        .40	   .30
G 3)	   .33	        .33	   .33
Equiprobability
Given population proportions pi, i = 1 to n, maximum diversity occurs when 
p1 = p2 = … = pn = 1/n
Maximum Diversity
(again)
Categories
	 A	 B	 C	 D
G 1)	.33	.33	.33
G 2)	.25	.25	.25	.25
Diversity Index Properties
Property 1
  When only one category is represented diversity equals 0.

Property 2   (Equiprobability)
  Maximum diversity occurs when each category is represented equally.

Property 3
  Diversity at equiprobability is greater when the number of categories is greater.

Shannon’s Diversity Index

pi = 	proportion of species i 	in population
S  = 	total number of species

Taylor Series Approximation
f(p) = - p ln p  near p = .5

so define

An Elementary Diversity Index

Minimal Diversity Property
If p1 = 1 then E = 0 

Note

Equiprobability Property
Maximize

subject to

Lagrange multipliers

     1 – 2pi =          for all i
thus pi = pj for all 

Increasing Categories Property

so if T > S
then Eq,T > EqS

Results
    Index	   Class 1	   Class 2
		         (.5,.5)           (.05,.05,.05,.05,.05,.75)		
		     .69	      .96
			
		     .50	      .43

Results
 Index	         Class1       Class 2
		                   (.5,.5)      (.05,.05,.05,.05,.05,.75)		
	     	           .69	           .96
			
		           .50	           .43

		           .80	         1.925

		           .35		  .71

Diversity Projects for Inside
•	Shoe type – male/female
•	Clothing type
•	Watches/glasses –student/fac.
•	Writing implement choice
•	Hair/Eye color
•	Lunch Beverage choice
•	Club membership
•	Opinions on issues
•	Discussion topics

Biological Diversity Projects
•	Flower type
•	Leaf shape
•	Species – lawn/woodlot
			  park/woods
			  garden/weedlot
•	Insect – lawn/log
   
   Sampling – select area
			       adjust size to pop.
			       equal organism count

Sociological Diversity Projects
•	Automobile color
•	Automobile type
•	Automobile license origin
		- student/faculty lots
		- campus/town lots
		- mall/grocery
•  Topics of discussion
		- town meeting/PTA
		- concert intermission/club

Bibliography
Abramson, Norman. 1963. Information Theory and Coding,  	New York: McGraw-Hill.
Ash, Robert. 1965. Information Theory, New York: John 	Wiley and Sons.
Gold, Harvey. 1977. Mathematical Modeling of Biological 	Systems – An Introductory Guidebook, New York: 	John Wiley and Sons.
Kolmes, Steven, and Mitchell, Kevin. 1991. Information 	Theory and Biological Diversity, UMAP Module 	705, Arlington, MA: Comap, Inc.
Magurran, Anne E. 1988. Ecological Diversity and Its 	Measurment, Princeton, NJ: Princeton University 	Press.

www.bluffton.edu/mat/seminar/

Email: hooleyd@bluffton.edu