A review on measurement of Alpha diversity in biology

Abstract

Diversity studies make a central theme of study of biodiversity. Several measures of diversity have been proposed by different authors, but Shannon's, Simpson's and Brillouin's indices are the most widely used ones. At the same time, these are the least understood indices. In the present review, some of the commonly used diversity indices have been discussed with specific examples. Special emphasis has been laid to explain the derivation of diversity indices from basic concepts.

Full text

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*Corresponding author : akthukral.gndu@gmail.com
Date of receipt : 14.12.2016, Date of acceptance : 09.02.2017
Agric Res J 54 (1) : 1-10, March 2017
DOI No. 10.5958/2395-146X.2017.00001.1
Loss of biodiversity is a global issue and an elusive
hazard. United Nations’ Earth Summit held at The Rio
de Janeiro in 1992, described biodiversity as “The variability
among living organisms from all sources, including interalia
terrestrial, marine and aquatic systems and the ecological
complexes of which they are a part: this includes diversity
within species, between species and the ecosystems”. The
totality of genes, species and ecosystems of an area constitutes
the biodiversity. Biodiversity can be studied at three dierent
levels of organization (Wikipedia, 2017a):
1. Diversity of genes within a species: Genetic diversity,
2. Diversity among the species: Species diversity,
3. Diversity at ecosystem and landscape levels: Ecosystem
diversity.
With respect to the habitat, measurement of diversity can
be done at the three levels, Alpha, Beta and Gamma diversity
(Whittaker, 1972). Alpha diversity is the within habitat or
intra-community diversity. It has two components – species
richness and evenness. Beta diversity is the between habitat
or inter community diversity. It measures change in species
composition along a gradient. Gamma diversity encompasses
diversity at the landscape level.
Measurement of diversity of species is one of the most
important characteristics of phytosociology. The quantitative
characterization of communities can be done using diversity
indices. A diversity index takes into consideration the number
of species, the number of individuals of dierent species in
a sample, or habitat or a community. Change of diversity
from one community to another, or communities along an
environmental gradient, or pooled communities or at the
landscape level constitutes beta and gamma diversities. Based
on certain assumptions, several indices have been proposed
by dierent authors for measurement of diversity. Some of
the authors who have contributed to diversity studies are
Baumgartner (2005), Greig-Smith (1978), Henderson (2003),
Hulbert (1969), Jost (2006a), Keylock (2005), McIntosh
(1967), Mee et al. (2002), Magurran (1988), Parker (1979),
Ponce-Hernandez (2004), Routledge (1977), Smith (1986),
Southwood and Henderson (2000), Thukral (2010), Thukral
et al. (2006), Wilson and Mohler (1983) and many others.
Diversity indices are constrained by changes in species
composition, inux of new species with time, out ux of
species, immigration and emigration of species, dormant or
hidden species, propagules of species etc. Model assumptions,
variations in sampling techniques, time of sampling,
experimental errors during sampling add to variations in
results. The present paper describes the indices commonly
used for the quantitative measurement of species diversity at
the community and ecosystem levels.
Alpha diversity
Diversity at habitat level is the most widely used
component in the characterization of communities. It has two
components viz. species richness and equitability indices.
Heterogeneity indices combine both these components.
Species richness quanties the number of individuals per
unit area or per sample (e.g., quadrat), and species content of
the area. Species richness indices are also known as variety
indices, are higher for species rich communities. Equitability
indices give the relative abundance of dierent species
of a community in terms of their evenness of distribution.
A community, in which the species have equal number of
individuals of dierent species, will have a higher evenness
index. On the other hand, a community dominated by one or
few species in terms of the number of individuals, will have
a lower evenness index.
Species richness
As per Peet (1974), species richness is dened by the
function:
E(S) = f (k, N)
where, E(S) = Expected value for number of species,
N = Number of individuals,
A REVIEW ON MEASUREMENT OF ALPHA DIVERSITY IN BIOLOGY
Ashwani K Thukral*
Department of Botanical and Environmental Sciences
Guru Nanak Dev University, Amritsar-143005, Punjab
ABSTRACT
Diversity studies make a central theme of study of biodiversity. Several measures of diversity have been proposed
by dierent authors, but Shannon’s, Simpson’s and Brillouin’s indices are the most widely used ones. At the same
time, these are the least understood indices. In the present review, some of the commonly used diversity indices
have been discussed with specic examples. Special emphasis has been laid to explain the derivation of diversity
indices from basic concepts.
Key words: Biodiversity, Brillouin’s index, Chao’s index, Shannon’s index, Simpson’s index
Review Paper

2
k = Richness index.
Some of the commonly used richness indices are given
below:
Species richness index
Total number of species (S) present in the community
is called species richness index. Several indices have been
developed by dierent workers to determine diversity of
species in a habitat.
Biodiversity index: The ratio of number of species (S) to
number of individuals (N) in the sample is called biodiversity
index. This index is commonly used for characterization of
forest communities (Woodwell, 1967).
Menhinick’s index: Similar to biodiversity index, this index
is dened as the ratio of number of species to square root of
number of individuals in the sample.
Margalef’s index: It is dened as
Odum’s index: Odums’s index (Odum et al., 1960) is similar
to the Margalef’s index.
Berger – Parker dominance index: Berger-Parker index is
the ratio of number of individuals of most abundant species
(Nmax) to the total number of individuals of all the species
(Ntot) in the sample.
Fisher’s a: This index is based upon the logarithmic
distribution of number of individuals of dierent species.
where, S is the total number of species and N is the total
number of individuals in the sample. The value of Fisher’s a
is computed by iteration.
Average number of species per log cycle of
importance
Whittaker (1972) dened log cycle of importance EWI as:
where, S1 is the number of individuals of the most common
species and Sn is the number of individuals of least common
species, and S is the number of species.
Whittaker’s second richness index is based on dividing the
number of species with 4 times of the geometric standard
deviation (σgm ) of the sample.
where, pi is the proportion of the species (ni / n), and pgm is the
geometric mean of the pi values.
Estimation of total number of species
The number of species present in a sample will depend
upon the size of the sample. Larger the sample size, higher
the species count. Maximum no. of species in a community
may be estimated using a species accumulation curve or
Chao’s method.
Species accumulation curve: Species accumulation curve
is a curve plotted between the cumulative numbers of species
against the sampling eort (Fig. 1). The sampling eort may
be expressed in terms of number of samples, sample size,
or the number of individuals in the sample. The slope of
the curve decreases with the sampling eort and tends to a
limiting value.
The curve follows Michaelis–Menten’s equation and
transcribes a hyperbola as per the following equation:
where S(n) = Number of species foe a sample size
Smax = Maximum no. of species in the community,
n = number of samples or maximum sample size studied,
B = Constant.
The equation may be written in the form:
The y-intercept of the double reciprocal plot between
1/S(n) and 1/n will provide the value of 1/Smax .
R
S
N
biod =
R
S
N
MEN =
R
S
In N
MAR =
−1
R
S
In N
Odum =
DI
N
N
BP
tot
=max
S=




aa
In 1+
N
EIn In
W1 =−
S
S Sn
( )
1
EW2 =
S
gm
4
σ
σ
gm
i gm
p P
S
=−S( )In In 2
Fig. 1. Species accumulation curve
S n
S n
B n
( )
.
max
=
+
1 1
S n
B
S n S( ) .
max max
= +

3
Chao’s index
Chao’s index (Chao 1984, 1987) for estimation of
species richness is given by the equation:
S(max) Chao = Sobs + (a² + b²)
where, Smax = Maximum no. of species,
Sobs = Number of species observed in dierent samples,
a = Singletons (Number of species represented by one
individual each),
b = Doubletons (Number of species represented by two
individuals each).
An example for computation of Smax by Chao’s method is
given in Table 1. Let there be 3 quadrats (Q1 through Q3) and
a total of 5 species observed (A through E).
Table 1. An example to calculate Smax by Chao’s method
Sample Species
Sp1 Sp2 Sp3 Sp4 Sp5
I 1* 0 4 5 0
II 0 2** 3 1 1*
III 0 0 3 4 0
Sobs = 5, a = 2 singletons ( marked by *),
b = 1 doubleton ( marked by **), Smax = 5+(4+2) = 11
Shannon – Wiener’s index
One of the most widely used diversity indices is
Shannon – Wiener index proposed by Claude Shannon
in 1948. Shannon’s index was originally developed for
communication systems and is based on information theory
that any message can be transmitted using a binary code
(Shannon, 1948). Most of the popular texts only mention the
formula for Shannon’s index without explaining the logic
behind it. In this paper Shannon’s index is explained from its
basics for purpose of understanding by biologists.
Just like in decimal system consisting of 10 digits (viz.,
0,1,2,3,…9), a number can be written as a combination of
two binary digits called “bits”, a term derived from rst two
and last two letters of the term “binary digits”, viz., 0 and
1 (Table 2). Similar to binary number system, a message in
a text can be converted to a binary code (Table 3). As for
example, if a text contains only two letters, A and B, these can
be coded as 0 and 1. The binary coding of a text containing
4 letters, 8 letters or more is given in the table. It is seen
that the number of bits required for coding a text containing
2, 4, 8 or 16 letters will be 1, 2, 3 and 4 respectively, and
so on. The number of bits required to transmit a message is
called as the information content. It is generally represented
by the Greek letter ‘H’ (‘ Eta’, generally italicised), which is
similar to the English alphabet H. If a message consists of 16
alphabets, each occurring only once, the information content
per alphabet will be H/ number of alphabets.
Table 4 describes the conversion of information content
of a message consisting of ‘m’ number of alphabets into
Shannon’s information measure, generally written with a
Table 2. Decimal and binary systems
Decimal number Binary number
0 0
1 1
2 1 0
3 1 1
4 1 0 0
5 1 0 1
6 1 1 0
7 1 1 1
8 1 0 0 0
9 1 0 0 1
10 1 0 1 0
prime sign on it, viz. H. It seen from the rst two columns of
Table 2 that the message length (no. of alphabets) is related to
the information content (H) as a power function, i.e., m = 2H.
Vice-versa, H can be dened as a logarithmic function, viz., H
= Log2 m, as given in column 3. Probability (pi) of each letter
in a message consisting m alphabets, each alphabet occurring
only once, will be 1/m. Information content, H, therefore can
be written as a function of probability (columns 5 and 6).
It may be seen that the negative sign in column number 6
appears as a result of conversion of log of reciprocal of a
number and is not introduces to make the value of H positive,
as is generally misunderstood by the students. Column 7
gives information content of each letter in the message.
As for example, the total information content of a message
consisting 16 alphabets (one letter per alphabet) will be 4 bits
(column 2), and the information content per alphabet will
be 4/16 = 0.25 bit. This will be the same that we get from
column 7 of Table 4.
In actual practice, however, dierent letters in a text
occur with dierent frequencies. Table (5) gives an example
of a text consisting of ve alphabets occurring with dierent
frequencies (colums 1-4). The probability of each letter in
the text will be reciprocal of total number of letters, 16 in
this case. Columns 7 and 8 give the information content of
the message.
Units of information
Since the Shannon’s index as given above (generally
written as H’) uses binary numbers, the logarithmic function
uses 2 as the base. The units of information of H’are bits or
shannons. However, the logarithms to dierent bases are
inter-convertible. Information presented using log10 is given
in hartleys or decits. The units of information content using
loge (log natural or ln) are nats (natural digits). Thus, we
have,
H' (shannons or bits) = – Σ pi log2 pi
H' (nats) = – Σ pi loge pi
H' (hartleys or decits) = – Σ pi log10 pi

4
Table 3. Information content of a message as the number of bits required to transmit a message
1 2 3 4 5
Message length
(m)
Letters Binary code Information content
(H)
Information
content per letter
No. of alphabets I bit II bit III bit IV bit No. of bits required
for message
No. of bits required
for each letter
2 A 0 1 ½
B 1 ½
4 A 0 0 2 2/4
B 0 1 2/4
C 1 0 2/4
D 1 1 2/4
8 A 0 0 0 3 3/8
B 0 0 1 3/8
C 0 1 0 3/8
D 0 1 1 3/8
E 1 0 0 3/8
F 1 0 1 3/8
G 1 1 0 3/8
H 1 1 1 3/8
16 A 0 0 0 0 4 4/16
B 0 0 0 1 4/16
C 0 0 1 0 4/16
D 0 0 1 1 4/16
E 0 1 0 0 4/16
F 0 1 0 1 4/16
G 0 1 1 0 4/16
H 0 1 1 1 4/16
I 1 0 0 0 4/16
J 1 0 0 1 4/16
K 1 0 1 0 4/16
L 1 0 1 1 4/16
M 1 1 0 0 4/16
N 1 1 0 1 4/16
O 1 1 1 0 4/16
P 1 1 1 1 4/16
Because log2 tables or software are generally not
available, Shannon’s index is calculated either in nats or in
decits. Dierent units of diversity may be inter-converted as
follows:
Bits = 1.4427 x Nats
Bits = 3.3219 x Decits
Nats = 2.3026 x Decits
The units of the diversity index must be mentioned in the
calculations. Shannon’s information as explained above can
be used to dene the biological diversity of the communities.
In the text given above, message length can be equated with
the number of species in a community, an alphabet with
a species, and number of letters of an alphabet with the
frequency of the species. The Shannon’s index thus obtained
will be the diversity of the community as represented in the
sample. This index presumes that all the species in the sample
(or the quadrat) are represented in the proportions in a larger
community (Poole, 1974). Larger the information content
(H’), more heterogeneous the sample will be. For the analysis
of biological communities, Shannon’s index may be written
as
H' (nats) = – Σ loge
ni = Number of individuals of the i th species,
ni
N
ni
N

5
Table 4. Information content (H) of a message if each alphabet occurs only once
1 2 3 4 5 6 7
Message
length
Infor-
mation
content
Information
as log of
message
length
Probability
(pi) of each
letter in
message
Information as log
of reciprocal of
probability
Information as
negative log of
probability
Shannon’s
information of each
letter in message (bits)
No. of
alphabets
(m)
(H bits)
H = log2 m piH = log2H = – log2 p1Hi = – p1 log2 p1
2 1 H = log2 2 1/2 H = log2H = – log2Hi = – log2
4 2 H = log2 4 1/4 H = log2H = – log2Hi = – log2
8 3 H = log2 8 1/8 H = log2H = – log2Hi = – log2
16 4 H = log2 16 1/16 H = log2H = – log2Hi = – log2
1
1/4
1
1/8
1
1/6
1
1/2
1
2
1
2
1
4
1
16
1
16
1
2
1
4
1
2
1
2
1
4
1
4
1
4
1
8
1
8
1
8
1
16
1
16
1
16
1
p1
1
m
Table 5. Example of Shannon’s information content of a message if dierent letters occur with dierent frequencies
1 2 3 4 5 6 7 8
Message
length (m)
Alphabets Frequency Letters Probability of
each letter
Probability of
each alphabet
Shannon’s information
content of each alphabet
H
No. of
Alphabets
ni 1/m bits
5 A 8 A 1/16
8/16 = 1/2 Hi = – log21/2
A 1/16
A 1/16
A 1/16
A 1/16
A 1/16
A 1/16
A 1/16
B 4 B 1/16
4/16 = 1/4 Hi = – log21/2
B 1/16
B 1/16
B 1/16
C 2 C 1/16
2/16 = 1/8 Hi = – log23/8
C 1/16
D 1 D 1/16
1/16 Hi = – log2
1/4
E 1 E 1/16
1/16
1/4
Total
Frequency
16 Probability 1 H' = – Σ pi log2 pi1.875
1
16
1
16

6
N = Total number of individuals of all the species in the
sample.
Table 6 gives the computation of Shannon’s information
measure for plotting a graph. For a probability equal to 0,
since the limit of (0 log 0) is zero, the Shannon’s index at this
point will be zero. A typical graph for Shannon’s diversity
index diversity will be a concave curve (downward). A graph
of the Shannon’s information measure is plotted by taking
the probability values on the X-axis, and the H’ values on the
Y-axis (Fig. 2).
Table 6. Shannon-Wiener’s index for 2 species communities
having dierent probabilities
pi1-piH’
0 1 0 (Limit)
0.05 0.95 0.198515
0.1 0.9 0.325083
0.2 0.8 0.500402
0.3 0.7 0.610864
0.4 0.6 0.673012
0.5 0.5 0.693147
0.6 0.4 0.673012
0.7 0.3 0.610864
0.8 0.2 0.500402
0.9 0.1 0.325083
0.95 0.05 0.198515
1 0 0 (Limit)
Fig. 2. Graph for Shannon’s information
measure
Fig. 3. Simpson’s index for concentration
(abundance)
Fig. 6. Inverse Simpson’s index for diversity
The minimum value of H’ is 0 for one species community,
that is, when all the individuals in the sample belong to the
same species. The information content will be maximum (H’
= H’max = ln S), if all the species in the sample are represented
by equal number of individuals. More the heterogeneous a
sample is, larger the information content will be.
It would be pertinent to mention here that Shannon’s
index is variously refered to as Shannon-Weaver index
or Shannon-Wiener index. Spellerberg and Fedor (2003)
studied the historical perspective of the development of the
Shannon’s index. Shannon’s expression for entropy (H) was
rst published independently by Shannon in 1948. Shannon
and Weaver jointly authored a book 1949 “The Mathematical
Theory of Communication” published by the University of
Illinois. In the second part of the book Weaver developed
on the concept. Shannon had built the concept on the work
already done by Wiener (1939, 1948, 1949). The name
Shannon-Weaver came from the jointly written book and
the name Shannon-Wiener came from the several references
cited by Shannon in his work and the idea developed after
Wiener.
Simpson’s index
Another widely used index for community analysis is
Simpson’s index proposed by Edward Hugh Simpson (1949).
If there are k species consisting of n individuals, distributed
among dierent species as n1, n2, n3, …nk, then the
probability (p1) of the rst individual belonging to a species
will be
Fig. 4. Simpson’s index for diversity
Fig. 5. Relation between Simpson’s
concentration and diversity indices.
p
n
n
1
1
=

7
Probability (p1,2) that the second individual drawn from the
sample without replacement also belongs to the same species
will be
The sum of such probabilities for all the species is a measure
of the concentration (or abundance) (C) of the species.
If the sample size is quite large, then the probability (p1,2)
that the second individual drawn from the sample with
replacement also belongs to the same species (C’) will be
Simpson’s index for abundance (concentration) is a typical
convex curve as shown in Fig. 3. The maximum value of
Simpson’s concentration is one when all the individuals in
the sample belong to the same species. Gini-Simpson’s index
of diversity (D) is dened as
The graph for Gini-Simpson index of diversity is a
typical concave curve (Fig. 4). Simpson’s concentration and
diversity indices are inversely related to each other as shown
in Fig. 5. Another index of diversity is inverse Simpson’s
diversity index (D’) (Fig. 6):
Brillouin’s index
Let ni be the number of individuals of the ith species,
N be the total number of individuals of all the species (k)
present in the sample, then the Brillouin’s index (H), will be:
The units of information for Brillouin’s index will be
the same as those for Shannon-Wiener’s index. Minimum
value of H will be 0 for communities consisting of a single
species. This index is also widely computed for diversity
studies. Brillouin’s index is derived from the probability that
all the individuals drawn from a sample will belong to the
same species. Computation of Brillouin’s index is given in
Table 7. Let there be 3 species consisting of 10 individuals.
The probability of drawing an individual of species A, out
of 10 individuals is 1/10. The probability of drawing second
individual of species A, out of remaining 9 individuals will
be (1/10)(2/9). Given in the table are the steps to understand
the derivation of Brillouin’s index.
The negative logarithm of the equation thus derived (or
the logarithm of reciprocal of probability) may be used as
Brillouin’s information measure.
Good’s Series of Indices
Following generalized equation can be used to derive
some important indices:
where dierent values of m and n (0, 1, 2, 3) give dierent
indices. For example,
For species richness index (S), (m, n) = (0, 0)
For Simpson’s index (S pi2 ), (m, n) = (2, 0)
For Shannon’s index (H’), (m, n) = (1, 1)
Variance as a measure of diversity
Generally diversity is measured for discrete variables
which give the numbers of individuals of dierent species
in the community. Diversity indices can also be obtained
for all discrete data including non-living objects. Parkash
and Thukral (2010) proved that sample statistics such as
standard deviation, variance, standard error, coecient of
variation, quadratic mean, geometric mean, harmonic mean,
quadratic mean, power mean, exponential mean, log mean,
moments etc. may be used to derive information measures
which can be used as measures of concentration or diversity.
Two of the statistics frequently used to characterize a sample
are arithmetic mean and standard deviation. Given these
statistics we can dene the species abundance or diversity of
a community. Variance (σ2) of a discrete variable (X) with n
number of observations is dened as
In terms of community analysis, let the community
consist of n species, with the numbers individuals of each
species represented as xi. We can convert this equation into a
probability equation. We know that
Squaring both sides,
p
n
n
n
n
1 2
1 1
1
1
,.=
−
−





Cn n
n n
Simpson
i
=−
−








∑
11
1
( )
( )
C p n
n
Simpson i
i
= =
∑∑
2
2
2
D p
Gini Simpson i
−= − ∑
12
D
p
Inverse Simpson
' =∑
1
1
2
H nats n n
Brillouin
k
( ) ! !... !
=In
N!
n !n
1 2 3
H nats individual N n n
Brillouin
k
( / ) ! !... !
=
1
3
In
N!
n !n
1 2
H p p
Good i
m
i
n
= −
∑
( )In
σ
2
1
1
2 2
1
2
1
1
=
−
=
−
= ==
∑ ∑∑
( ) ( )x x
n
xnx
n
i
n
i i
i
n
i
n
p
x
x
i
i
i
i
n
=
=
∑
1
px
x
i
i
i
i
n
2
2
2
1
=
=
∑
( )

8
Table 7. Calculation of probability of drawing an individual from a community consisting 3 species and 10 individuals
Sp. No. of ind.
(ni)
Names of
ind.
No. of ind.
left
No. of ind. of sp. drawn Probability of drawing individuals upto the
i th individual
A 5 A1 10 1
(1st)
A2 9 2
(1st, 2nd)
A3 8 3
(1st, 2nd, 3rd)
A4 7 4
(1st, 2nd, 3rd, 4th)
A5 6 5
(1st, 2nd, 3rd, 4th, 5th)
B 3 B1 5 1
B2 4 2
B3 3 3
(5 ind of A and 3 of B)
C 2 C1 2 1
C2 1 2
(5 ind of A, 3 of B and 2 of C)
Total 10 All 10
1
10





1
10
2
9
.





1
10
2
9
3
8
. .






1
10
2
9
3
8
4
7
. . .





1
10
2
9
3
8
4
7
5
6
. . . .





1
10
2
9
3
8
4
7
5
6
1
5
. . . .









1
10
2
9
3
8
4
7
5
6
1
5
2
4
. . . . .









1
10
2
9
3
8
4
7
5
6
1
5
2
4
3
3
. . . . . .









1
10
2
9
3
8
4
7
5
6
1
5
2
4
3
3
1
2
. . . . . .













1
10
2
9
3
8
4
7
5
6
1
5
2
4
3
3
1
2
2
1
. . . . . . .













532
10!
1 2 3
! ! ! ! ! !
!
=
n n n
N
The authors proved that for a population or for a large
sample, the sum of squares of probabilities, i.e., Gini-
Simpson’s concentration (C’) will be given as
where M is the mean of the sample. Thus variance can be
used as a measure of heterogeneity of data for a continuous
variable such as length, height, weight, concentration etc.
The relation between the variance of a small sample (S2) and
a population or large sample (σ2) is,
Therefore, for a small sample, the Simpson’s index will be
Same way Simpson’s index can be found for standard error
(SE) and coecient of variation (CV).
Information Measures via Matrix Methods
Sarangal et al. (2012) proved that in several cases a
probability matrix can be produced by multiplication of
a column vector ƒ1 (p1) with a row vector ƒ2 (p2). Either
diagonal or non-diagonal elements, or both of these elements
of the probability matrix can be used to derive the information
measures which can be used as measures of diversity.
The probability matrix generated will be
In the above matrix if both the row and column matrices
comprise of pi elements, the sum of diagonal elements will be
produce Simpson’s concentration, which is Simpson’s index
of diversity.
Similarly, non-diagonal elements of this probability
matrix can also be used to derive new information measures
(I),
pM
nM
i
i
n
2
1
2 2
2
=
∑=+
σ
σ
2
21
=−S n
n
( )
p
S n
nM
nM
i
i
n
2
1
2
2
2
1
=
∑=
−+
( )
pSE n M
nM
i
i
n
2
1
2 2
2
1
=
∑=− +( )
I f p f p
i
n
i i
=
=
∑1
1
1 2 2
( ) ( )
C p p p
Simpson i i
i
n
i
i
n
'= =
= =
∑ ∑
1
2
1

9
which gives Gini-Simpson’s index
Similarly, in the matrix given above, if either the row
or the column vectors comprises of pi and the other vector
comprises of ln pi elements, the negative sum of diagonal
elements will be produce Shannon’s index of diversity.
H' (nats) = – Σ pi loge pi
Using matrix methods we can derive several other
measures of information which can be used as measures of
concentration or diversity.
Evenness indices
The evenness (equitability) of a sample implies equality
in the number of individuals of species (Pielou, 1975). Some
of these indices are given below. These are:
where S is the number of species, H’ is the Shannon-Wiener’s
index, Dmax and Dmin represent the maximum and minimum
values of Simpson’s reciprocal diversity index.
Eective number of species: The number of equally
common species in a sample is called eective number
of species. This index characterizes the true diversity of a
community (Jost 2006b).
Eective no. of species = e H'
For example if the Shannon’s index is 2 nats, this implies
that the true diversity is 7.39 species. For information in bits,
the eective number of species can be calculated with base
2, that is (2H').
Conclusion
Alpha diversity indices are extensively used in
characterisation of communities in biology (Chawla et al.
2008, 2012) and other research elds such as to describe
the vertical structure of forest ecosystems, succession
of communities (Petrere Jr. et al., 2004), rainfall data
(Bronikowski, 1996), language studies, water and energy
studies (Singh 2013) and many other research elds. Since
diversity indices are not well explained in the texts, their use
in literature has been reduced to just a convention. This paper
may help researchers to understand the fundamentals of
measurement of diversity and to apply their research results
for characterization of plant communities more justiably.
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