# Shannon index

The

**Shannon index**, also known as the**Shannon-Weaver Index**and sometimes referred to as the Shannon-Wiener Index [*Krebs, Charles (1989) "Ecological Methodology." HarperCollins, New York.*] ), $H^\{prime\}$, is one of several diversity indices used to measure diversity incategorical data . It is simply theInformation entropy of the distribution, treating species as symbols and their relative population sizes as the probability.This article treats its use in the measurement of

biodiversity . The advantage of this index is that it takes into account the number of species and the evenness of the species. The index is increased either by having additional unique species, or by having a greaterspecies evenness .The "Shannon-Weaver" name is a misnomer; apparently some biologists jumped to the conclusion that

Warren Weaver , author of an influential preface to the book form [*cite book|last=Weaver|first=W.|coauthors=C.E. Shannon|title=The Mathematical Theory of Communication|publisher=University of Illinois|location=Urbana, Illinois|date=1949*] of Claude Shannon's 1948 paper [*cite journal|last=Shannon|first=C.E.|title=A mathematical theory of communication|journal=Bell System Technical Journal|volume=27|month=July and October|pages=379–423 and 623–656|year=1948*] foundinginformation theory , was a cofounder of this theory. Weaver did play a crucial role in the rapid postwar development of information theory in a different way, however; as an influential early administrator of the Rockefeller Foundation, he ensured that the first information theorists received generous research grants.Norbert Wiener had no hand in the index either, although his influential popularisation ofcybernetics was often conflated with information theory in the 1950s.**Definitions***$n\_i$ The number of individuals in species i; the abundance of species i.

*$S$ The number of species. Also calledspecies richness .

*$N$ The total number of all individuals

*$p\_i$ The relative abundance of each species, calculated as the proportion of individuals of a given species to the total number of individuals in the community: $n\_iover\; N$**Computing the index**:$H^prime\; =\; -sum\_\{i=1\}^S\; p\_i\; ln\; p\_i$

It can be shown that for any given number of species, there is a maximum possible $H^prime$, $H\_max=ln\; S$ which occurs when all species are present in equal numbers.

**Proof that maximum evenness maximizes the index**The following will prove that any given population will have a maximum Shannon Index if and only if each species represented is composed of the same number of individuals.

Expanding the index:

:$H^prime\; =\; -sum\_\{i=1\}^S\; \{n\_iover\; N\}\; ln\; \{n\_iover\; N\}$

:$N\; H^prime\; =\; -sum\_\{i=1\}^S\; n\_i\; left\; (\; ln\; n\_i\; -\; ln\; N\; ight\; )=\; -sum\_\{i=1\}^S\; n\_i\; ln\; n\_i\; +\; ln\; N\; sum\_\{i=1\}^S\; n\_i$

:$N\; H^prime\; -\; N\; ln\; N\; =\; -sum\_\{i=1\}^S\; n\_i\; ln\; n\_i$

Now, let's define $H\_s\; =\; -sum\_\{i=1\}^S\; n\_i\; ln\; n\_i$ Clearly, since $N$ is a positive constant for a given population size, and $Nln\; N$ is also a constant, then maximizing $H\_s$ is equivalent to maximizing $H^prime$.

**trategy**Let's split an arbitrarily sized population into two groups, with each group receiving an arbitrary number of individuals and an arbitrary number of species. Now, within each group, each species has the same number of individuals as any other species in that group, but the number of individuals per species in the first group may be different from the number of individuals per species in the second group.

Now, if it can be proven that $H\_s$ is maximized when the number of individuals per species in the first group matches the number of individuals per species in the second group, then it has been proved that the population has a maximum index only when each species in the population is evenly represented. $H\_s$ doesn't depend on the total population. So $H\_s$ may be built by simply adding the indices of two sub-populations. Since the population size is arbitrary, this proves that if you have two species (the smallest number that can be considered two groups), their index is maximized if they are present in equal numbers. So the rules of

mathematical induction have been satisfied.**Proof**Now, divide the species into two groups. Within each group, the population is evenly distributed among the species present.

*$k$ The number of individuals in the second group.

*$p$ The number of species in the second group.

*$n\_\{i2\}\; =\; k/p$ Number of individuals in each species in the second group.

*$N-k$ The number of individuals in the first group.

*$S-p$ The species in the first group.

*$n\_\{i1\}\; =\; \{N-k\; over\; S-p\}$ The individuals in each species in the first group.:$H\_s\; =\; -sum\_\{i=1\}^\{S-p\}\; \{N-k\; over\; S-p\}\; ln\; \{N-k\; over\; S-p\}\; -\; sum\_\{i=1\}^p\; \{kover\; p\}\; ln\; \{k\; over\; p\}\; =\; -left\; (\; N-k\; ight\; )\; ln\; \{N-k\; over\; S-p\}\; -\; k\; ln\; \{kover\; p\}.$

To find out which value of $k$ will maximize $H\_s$, we must find the value of $k$ which satisfies the equation:

:$\{dover\; dk\},\; H\_s=0.$

Differentiating,

:$ln\; \{\; N-k\; over\; S-p\}\; +\; (N-k)\{1\; over\; N-k\}\; -\; ln\; \{kover\; p\}\; -\; k\; \{1\; over\; k\}\; =\; 0,$

:$ln\; \{N-kover\; S-p\}\; =\; ln\; \{k\; over\; p\}$

Exponentiating:

:$\{N-kover\; S-p\}\; =\; \{k\; over\; p\}\; =\; \{pN\; over\; S\}.$

Now by applying the definitions of $N\_\{i1\}$ and $N\_\{i2\}$, we get

:$N\_\{i1\}\; =\; N\_\{i2\}\; =\; \{Nover\; S\}.$

**Result**Now we have accomplished the proof that the Shannon index is maximized when each species is present in equal numbers (see

#strategy ). But what is the index in that case? Well, $n\_i\; =\; \{Nover\; S\}$, so $p\_i\; =\; \{1over\; S\}$ Therefore::$H\_max\; =\; -\; sum\_\{i=1\}^S\; \{1over\; S\}\; ln\; \{1over\; S\}\; =\; ln\; S.$

**Notes****ee also***

biodiversity

*Species richness

*Information theory

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**Shannon-Index**— Der Shannon Index (häufig auch als Shannon Wiener oder Shannon Weaver Index bezeichnet [1][2]) ist eine mathematische Größe, die in der Biometrie für die Beschreibung der Diversität (vgl. Biodiversität) eingesetzt wird. Er beschreibt die Vielfalt … Deutsch Wikipedia**Shannon index**— /ʃænən ˈɪndɛks/ (say shanuhn indeks) noun an index of biodiversity which registers both the number of species and the evenness of distribution of species. {named after Claude Shannon, 1916–2001, US mathematician} … Australian English dictionary**Shannon**— steht für: Shannon (Name) ohne Nennung des Vornamens häufig Claude Elwood Shannon (1916–2001), US amerikanischer Informationstheoretiker Shannon (Einheit), Maßeinheit für den Informationsgehalt einer Nachricht Shannon Index für die Beschreibung… … Deutsch Wikipedia**Shannon (Einheit)**— Shannon (Sh) ist die nach dem amerikanischen Mathematiker und Begründer der Informationstheorie Claude Elwood Shannon benannte Einheit für den Informationsgehalt einer Nachricht. Inhaltsverzeichnis 1 Definition 2 Beispiele 3 Normung … Deutsch Wikipedia**Shannon**— The name Shannon originated in Ireland and is directly linked to the River Shannon, the island s longest river. The name is a combination of Irish sean old, wise and abhainn river . Shannon is now used as a first name for both males and females.… … Wikipedia**Shannon Tobin**— Shannon John Tobin (born March 12, 1985) is a Canadian politician running for the Progressive Canadian Party.Tobin was born in Happy Valley Goose Bay, Newfoundland and Labrador. On January 7th 2008, he announced that he is running in the riding… … Wikipedia**Shannon Esra**— (real name Shannon Esrechowitz ) is a South African actress.Early careerEsra had her first taste of acting at age 4, narrating a nursery school Purim concert. She later did impersonations of Elvis Presley at local talent contests.Esra had her… … Wikipedia**Shannon O'Donnell**— Shannon Kay O Donnell is the weeknight 10 PM meteorologist for KING 5 News on KONG and cable station NWCN. BiographyEarly life weather careerShannon was born in Redmond, Washington and attended Redmond High School and she graduated as a… … Wikipedia**Shannon Keith**— is an American animal rights lawyer, activist, and documentary director/producer. [ [http://www.animal rights lawyer.com Shannon Keith s animal rights legal practice] ] In 2006, after three years of filming, Keith released a documentary film… … Wikipedia**Shannon Bool**— (born 1972) is an artist based in Berlin.Bool was born on Vancouver Island. Her work focuses on the relevance of the ornamental in contemporary culture. [ [http://www.fkv.de/frontend en/archiv ausstellungen detail.php?id=58 FrankfurterKunstverein … Wikipedia