## Contents

## Definition

A **concentration index** is any Mathematical Expression (function) that converts a *distribution* of observed values into a *single number* expressing the prevalence (Concentration) of certain observations amongst the total set. Concentration indexes are closely associated with inequality and diversity indexes

## Usage

Concentration indexes find widespread use in (among others):

- studies of biological diversity where the objective is to identify the representation of different species (number count of individuals) in a given ecological region
- economic inequality, where the objective is to identify the distribution of income or other definitions of wealth amont participants in an economic system
- market competition, where the objective is to identify the market share enjoyed by firms operating in a given market
- different areas of Portfolio Management and Risk Management, where the objective is to identify and manage Concentration Risk

## Table of Concentration and Diversity Indexes

The following table aims to be a complete enumeration of the essentially distinct varieties of concentration and diversity indexes:

Index Name | Alternative Names | Domain | Remarks |
---|---|---|---|

Atkinson Index | |||

Berger-Parker Index | Special Case of the Concentration Ratio | ||

Concentration Ratio | |||

Generalized Entropy Index | Theil Index, Renyi Index or Entropy | Widely Used | |

Gini Index | |||

Gini-Simpson Index | Blau Index, Gibbs-Martin Index, Dominance | 1 - Simpson Index | |

Hannah Kay Index | |||

Hall-Tideman Index | Rosenbluth | Market Concentration | Related to the Gini Index |

Hoover Index | |||

Herfindahl-Hirschman Index | Hefindahl Index | Widely Used in Finance | 1 - Simpson |

Inverse Simpson Index | Inverse of the Simpon Index (Hence of the HHI) | ||

Kolm Index | |||

Menhinick Index | |||

Margalef Index | |||

Richness Index | Species Richness | Biodiversity | Simply the number of categories / taxa |

Simpson Index | Widely Used in Biodiversity | 1 - HHI | |

Shannon Index | Shannon-Weaver, Shannon-Wiener Index or Entropy | Widely Used | Special case of the Generalized Entropy Index |

## Spatial and Geographic Concentration Indexes

When spatial distribution (e.g., geographic location) is an important consideration the above general indexes may have reduced applicability. In this case a specialized type of Geographic Concentration Index might more applicable. Several geographic concentration indexes have been proposed:

- Generalization of the Gini index
- Generalization of the Herfindahl index
- Multigroup Thiel index
- Ellison and Glaeser index
- Maurel and Sedillot index
- Continuous indexes that use distances between locations

## Issues and Challenges

- The very widespread use of concentration indexes means that the same essential functions have been invented many times separately, have been given different names and are used in slightly different manner (inverse relationships, with different normalizations etc.).
- The reduction of Concentration Measurement to a single number is only one of the available quantitative tools to support an analysis. The Lorentz Curve is an example of a more complex approach to studying concentration.