# The Relation between Evenness and Diversity

## 1. Introduction

## 2. Theoretical Background

_{GS}). It turns out that all standard complexity measures have the same numbers equivalent [6]:

^{q}D). There are other true diversities (Gregorius, personal communication) but they are not yet used by ecologists.

## 3. Evenness, Richness, Diversity: Which Two Are Independent?

#### 3.1. Diversity Cannot Be Decomposed into Independent Richness and Evenness Components

^{H}= S∙EF

_{0, 1}

_{0, 1}stands for an undetermined “evenness factor”, and the subscripts mean that diversity of orders 0 and 1 are involved. This can be solved for EF

_{0, 1}

_{0, 1}= e

^{H}/S =

^{1}D/S.

_{0, 1}, cannot be independent, and furthermore EF

_{0, 1}is less than or equal to unity so it cannot be the numbers equivalent of anything. If these two components were really independent, then the value of one component would put no mathematical constraints on the value of the other (they would form a Cartesian product space). This is the case when we decompose gamma diversity into independent alpha and beta components. Knowing alpha (and only alpha, not gamma) tells us nothing at all about beta, and vice-versa. Yet if we try to decompose diversity e

^{H}into evenness EF

_{0, 1}and richness S, the value of S does constrain the possible values of EF

_{0, 1}, and vice versa. For example, if S = 2, we know that e

^{H}must be between 1 (its minimum possible value) and 2 (its maximum possible value when S = 2). This implies that EF

_{0, 1}can only range from 0.5 to 1, else EF

_{0, 1}∙S will be lower than the lowest possible value for total diversity. Similarly, if S = 20, we can infer that EF

_{0, 1}falls into the interval [1/20, 1]. This shows that the range of EF

_{0, 1}(and more generally, EF

_{0, q}) is determined by the value of S. Since the value of S mathematically constrains the range of values of EF

_{0, 1}, EF

_{0, 1}is not independent of S. Evenness and richness cannot be considered as orthogonal components of diversity. For most species abundance distributions the dependence of evenness on richness is observed to be strong [5].

#### 3.2. Derivation of Evenness and Inequality Measures from the Partitioning Theorem

^{q}D∙X.

^{q}D is always less than or equal to S, X must always be greater than or equal to unity, with equality only if the community is perfectly even (because then and only then is S =

^{q}D). X therefore satisfies the theorem’s requirement that the components are both valid numbers equivalents. The partitioning equation can now be solved for X. If the diversity measure

^{q}D is the diversity of order 1 (the exponential of Shannon entropy) we get

^{H}X

^{H}=

^{0}D/

^{1}D.

_{0, 1}= e

^{H}/S. If use diversity of order 2, the inverse Simpson concentration, X is

^{2}D =

^{0}D/

^{2}D

_{0, 2}(“E

_{1/D}” in [3]). X may therefore be considered a measure of inequality (“unevenness”). We will call it the “inequality factor of orders zero and q” and write it as IF

_{0, q}. The general expression for the inequality factor of orders 0 and q, with q > 0, is

_{0, q }of Section 2 turns out to be the reciprocal of IF

_{0, q}:

^{q}D. Suppose we are told only that the value of

^{q}D is 20. We can infer that S must be greater than or equal to 20 (since

^{0}D> =

^{q}D for all q > 0). Yet this only tells us that IF

_{0, q }must be greater than or equal to unity, which we already knew since this is always the case. If we were told instead that

^{q}D was 40, we would still only be able to say that IF

_{0, q}must be greater than or equal to unity. Knowing

^{q}D gives us no information about IF

_{0, q}. The same sort of argument applies if we are told the value of IF

_{0, q}. Knowing that IF

_{0, q}equals 5 tells us nothing about the value of

^{q}D. Therefore the diversity of order q (q > 0) and the inequality factor are mathematically independent (they form a Cartesian product space). The partitioning theorem has given us the unique decomposition of species richness into independent diversity and inequality components. The evenness factor EF

_{0, q}, which is the reciprocal of IF

_{0, q}, must therefore also be independent of diversity, since monotonic transformations like taking the reciprocal do not affect independence.

**Figure 1.**Inequality and evenness factors. In Row A, all communities are maximally even, so their inequality factors are all unity. Their evenness factors are also all unity. In Row B, communities all have an inequality factor of 2 and an evenness factor of ½. In Row C, the communities show maximal inequality; their inequality factors equal the number of species. Evenness factors are their reciprocals.

## 4. IF_{0, q} and EF_{0, q} Satisfy Common Requirements for Evenness and Inequality Measures

_{0, q}and EF

_{0, q}follow directly from the mathematics of diversity, which makes them easy to interpret and which guarantees their logical consistency. Smith and Wilson [3] presented a list of properties ecologists require in an evenness measure, along with properties that are desirable but not essential. Using numerical examples, they show that EF

_{0, 2}(their “E

_{1/D}”) satisfies all required properties, and most of the desirable but inessential properties. They did not consider EF

_{0, 1}or other orders of EF

_{0, q}, but it is possible to prove that all EF

_{0, q}satisfy properties “close to” the four properties required by Smith and Wilson. The proofs will be presented elsewhere [16]. We have to reinterpret one of Smith and Wilson’s properties, though. Their most essential property was “independence from S”, which they tested by checking if a measure was unchanged when an assemblage was replaced by N copies of itself, each copy with different species. This property is called “replication invariance” in the economics literature. Our EF

_{0, q}and IF

_{0, q}are indeed replication invariant, but as we have shown, they are not independent of richness. Smith and Wilson wrongly equated replication invariance with independence. Replication invariance is not sufficient to guarantee independence; one must also check the limits of each component to make sure they are not constrained by the other component.

_{0, q}satisfies this important property, and IF

_{0, q}satisfies the corresponding version for inequality measures.

_{1}− ε/S, p

_{2}− ε/S,..., p

_{S-1}– ε/S, ε], where the S-th species is the rarest and has relative abundance ε. In the limit as ε approaches zero, the evenness of this community is required to continuously approach the evenness of the community with the S-1 remaining species with relative abundances [p

_{1}, p

_{2}, ..p

_{S-1}]. Routledge [17] notes that this continuity requirement is inconsistent with some of the other common requirements for evenness measures, and it seems unnatural to require it of a concept that depends in an essential way on the discontinuous variable S. The lack of continuity does make these measures difficult to estimate; see Section 8.3 for a suggested solution.

## 5. Interpretation of Evenness and Inequality Factors

#### 5.1. Interpretation in Terms of Proportion of Dominant Species

^{1}D as the “number of common species”, and interpreted

^{2}D as the “number of abundant species”. While these interpretations are not rigorous, they do provide insight into the meaning of the evenness and inequality factors. If

^{2}D is the number of abundant or dominant species, then the evenness factor

^{2}D/S is, roughly speaking, the proportion of dominant species in the community. If there are some species that are abundant in a community,

^{2}D only “sees” these, and hardly takes into account the rare species in the community. In the idealized communities of Figure 1, this interpretation is exact.

^{2}D and

^{1}D differ in the sharpness of the cut-off between dominant and rare species.

^{2}D has a fairly sharp cutoff if there are some species that are much more abundant than the rest.

^{1}D does not have a sharp cutoff and counts average species as well as dominant ones; it is always greater than or equal to

^{2}D, so EF

_{0, 1}is always greater than or equal to EF

_{0, 2}. The measure 1-EF

_{0, q}gives the proportion of rare species in the community, roughly speaking.

#### 5.2. Interpretation in Terms of Equivalent Maximally Uneven Communities

_{0, q}, IF

_{0, q}, and all their monotonic transformations can be converted into a common, easily interpretable scale.

_{0, 1}has the value 0.143. How uneven is the community? Any two communities with this evenness value are equivalent with respect to the aspect of evenness measured by EF

_{0, 1}. One community whose evenness is easy to visualize is the maximally uneven community of T species, with virtually its entire population concentrated in one species, and all other species represented by vanishingly small populations. The evenness EF

_{0, 1}of such a community is e

^{H}/S = 1/T, because e

^{H}approaches unity when virtually all the population belongs to one species. If EF

_{0, 1}of a community is 0.143, then the maximally uneven community with the same value of EF

_{0, 1}is

_{0, 1}is even easier to interpret using this method. IF

_{0, 1}(and IF

_{0, q}generally) is exactly the number of species in the maximally uneven community equivalent to the community of interest with respect to inequality. Any monotonic transformation of IF

_{0, q}will have the same equivalent maximally uneven community.

_{0, q}(which is also the size of the equivalent maximally unequal community) for various idealized communities. For communities such as these, whose species are all either equally dominant or vanishingly rare, the value of q is irrelevant and IF

_{0, q}has the same value for any q.

#### 5.3. Graphic Interpretation

_{0, q}and EF

_{0, q}, have simple graphical interpretations based on the “diversity profile”. A diversity profile is a graph of diversity

^{q}D versus q, for nonnegative values of q. Diversity profiles are perfectly flat when the community is perfectly even, and the profiles become more steeply decreasing as the community becomes more uneven. The evenness and inequality measures derived here are really measures of how steeply the diversity profile decreases. The inequality factor IF

_{0, 1}compares

^{0}D with

^{1}D, so it is just the ratio of the two distances shown in Figure 2a. Similarly the inequality factor IF

_{0, 2}is the ratio of the two distances shown in Figure 2b.

**Figure 2.**Inequality factors in relation to the diversity profile. The y-axis is diversity of order q, and the x-axis is the order q. IF

_{0,1}is the ratio of the two distances shown in blue on the left. IF

_{0,2}is the ratio of the two distances shown in blue on the left.

#### 5.4. Interpretation in Terms of Mean Deviation from Equiprobability

_{0, 1}clearer, we can rewrite it in terms of the individual species frequencies. Suppose a community of S species has N individuals. (The final result will turn out to be independent of N.) If the community were perfectly even, every individual would belong to a species that had frequency 1/S. In an uneven community, some individuals will belong to species whose frequencies are higher than 1/S, and some individuals would belong to species whose frequencies are lower than 1/S. The factor p

_{i}/(1/S) measures the proportional deviation from perfect evenness for Species i. For example, if p

_{i}is 0.4 and 1/S is 0.2, the species is twice as abundant (0.4/0.2) as it would have been in a perfectly even community. What is the average, over all the individuals in the community, of this “inequality factor”? The most appropriate average when products are involved is the geometric mean. We take the factors for each individual, multiply them all together, and then take the Nth root to get the (geometric) mean inequality factor for the community. For example, suppose N = 10 and there are 8 individuals of Species A, 1 individual of Species B, and 1 individual of Species C. This is a very uneven community. If it were perfectly even, each species would have frequency 1/S = 1/3. Each individual of Species A has an inequality factor of (8/10)/(1/3) = 2.4. Each individual of Species B has an inequality factor of (1/10)/(1/3) = 0.3. Each individual of Species C has this same inequality factor, 0.3. The product of all of their inequality factors (one factor for each individual) is

^{8}∙ (0.3)

^{1}∙ (0.3)

^{1}=99.07.

^{1/10}= 1.58. 1.58 is the single number that could replace all the individual inequality factors in Equation 2 and still give the same final product. It is exactly our IF

_{0, 1}. The reciprocal, 0.63, is exactly our evenness factor EF

_{0, 1}. The inequality factor IF

_{0, 1}is the geometric mean of the inequality factors of the individuals in the community, where an individual’s inequality factor is just the factor by which the individual’s species exceeds or undershoots the frequency that each species would have if the community were perfectly even.

_{0, 2}. The reciprocal is 0.505, the evenness EF

_{0, 2}.

## 6. Monotonic Transformations of the Evenness and Inequality Factors

#### 6.1. Motivation

_{0, q}has a minimum value of unity for a perfectly even community. It might sometimes be useful to say that a community with no inequality has an inequality of zero. This would agree more closely with common language. Many biologists and economists have therefore preferred inequality measures whose minimum value is zero. Several monotonic transformations of IF

_{0, q}have minimum values of zero and maximum values of infinity, and preserve the important mathematical properties of the base measures.

#### 6.2. Theil Entropy Inequality Measure

_{i}/N while in ecology it is a probability p

_{i}. In this section we will take the viewpoint of economics and assume the population is finite with size N, and each species has abundance N

_{i}. This is a mere formality since the end results are independent of N and depend only on the ratios N

_{i}/N, which are just the p

_{i}of ecologists.

_{0, 1}.

#### 6.3. Logarithmic Transformations of General IF_{0, q}

_{0, q}will transform it into a measure with a minimum value of zero (for a perfectly even community) and no upper limit (increasingly large when inequality is large). This gives some commonly used measures of evenness and inequality in ecology. In the preceding paragraph we mentioned

_{0, 1}) = ln(S/e

^{H}) = ln(S) – H.

_{0, 1}yields another commonly used measure:

_{0, 1}) = ln(e

^{H/S}) = H – ln(S).

_{0, 1}since EF

_{0, 1}is the reciprocal of IF

_{0, 1}. Since EF

_{0, 1}is replication invariant, this transformation is also replication-invariant.

#### 6.4. Deformed Logarithmic Transformations

^{1-q}– 1) /(1 − q) ≡ ln

_{q}(X)

_{0, q}(the reciprocal of IF

_{0, q}) we obtain a measure that is (–q) times the very important generalized entropy inequality index of economics GEI [22]:

_{0, 1}or −ln EF

_{0, 1}, since EF

_{0, 1}and IF

_{0, 1}are reciprocals. This limit is exactly TEI, the Theil entropy inequality measure.

^{q-1}– 1]/[q(q − 1)]. For q = 1 the maximum possible value for fixed S is lnS.

## 7. Relative Inequality and Evenness

#### 7.1. Motivation

_{0, q}) than the communities to its left, since the dominant species in the community form a smaller proportion of the total number of species. If species were households and abundance was wealth, the distribution of wealth among these households is much less equal than the distribution of wealth in the communities to the left of it in Figure 1. Yet all of these communities are maximally unequal, given their number of species. It is impossible for the leftmost community, with just two species, to show as much inequality as the rightmost community with its sixteen species. This is a necessary feature of inequality measures that preserve the Lorentz partial order [4].

_{0, 1}for this forest is 3.55, meaning that it has the same inequality as a maximally uneven forest of 3.55 species. The inequality factor IF

_{0, 2}, which puts more emphasis on the dominant species, equals 3.84 species. The evenness factors are interpreted as the proportion of dominant species in the community, and are EF

_{0, 1}= 0.28, and EF

_{0, 2}= 0.26 (about a quarter of the species are dominant, which is right). Since the community has four species, its maximum possible inequality factor is 4.00, and its minimum possible evenness is 0.25. The community is close to its maximum possible inequality and its minimum possible evenness.

_{0, 1}= 5.72, and IF

_{0, 2}= 14.70. The evenness factors are EF

_{0, 1}= 0.175, and EF

_{0, 2}= 0.07. This correctly shows that the proportion of dominant species in Barro Colorado Island (0.07 according to EF

_{0, 2}) is actually smaller than the proportion of dominant species in the Jack Pine Forest (0.26). In this sense the unevenness and inequality of Barro Colorado Island are greater than the unevenness and inequality of the Jack Pine forest.

#### 7.2. Linear Transformations of Evenness and Inequality Factors

_{0, q}and evenness factor EF

_{0, q}onto the unit interval using the linear transformation (x − x

_{min})/(x

_{max}− x

_{min}). Evenness EF

_{0, q}has a minimum value of 1/S and a maximum value of 1.0. The transformation (x − x

_{min})/(x

_{max}− x

_{min}) of EF

_{0, q}onto the unit interval give a relative evenness index, RE

_{0, q}:

_{0, q}:

_{0, q}≡ (IF

_{0, q}– 1)/(S – 1).

**Figure 3.**Intermediate evenness. Community A is a maximally uneven four-species community. Community C is perfectly even. Community B is exactly intermediate in evenness and inequality.

#### 7.3. Transformations of Logarithms of Evenness and Inequality Factors

_{0, q}and EF

_{0, q}before transforming them. Since IF

_{0, q}and EF

_{0, q}are reciprocals of each other, their logarithms show a simple linear relationship (ln EF

_{0, q}= −ln IF

_{0, q}). The linear transformation (x − x

_{min})/(x

_{max}− x

_{min}), when applied to these logarithms, preserves this linear relationship, producing relative logarithmic evenness and inequality measures, RLE

_{0, q}and RLI

_{0, q}, that are complements of each other. The natural logarithm of EF

_{0, q}ranges from –ln S to 0, so the linear transformation of ln EF

_{0, q}onto the unit interval is:

_{0, q}≡ (ln EF

_{0, q}+ ln S) / (ln S)

= (ln

^{q}D – ln S + ln S) / ln S

= ln

^{q}D / ln S.

^{1}D = S

^{J}where J is this relative logarithmic evenness index. The same exponential relationship between

^{q}D, S, and RLE

_{0, q}holds for all q.

_{0, q}ranges from 0 to ln S, so the linear transformation onto the unit interval is:

_{0, q}≡ (ln IF

_{0, q})/(ln S)

= (ln S – ln

^{q}D)/ ln S

= 1 − RLE

_{0, q}

**Figure 5.**Complementarity of relative logarithmic evenness RLE

_{0, 1}and relative logarithmic inequality RLI

_{0, 1}. RLE

_{0, 1}and RLI

_{0, 1}and their complements for Community A, Community B, and Community C from Figure 3. This shows that RLI

_{0, 1}and RLE

_{0, 1}are complements of each other. Compare Figure 4.

#### 7.4. Slope of Chord of Renyi Spectrum

^{q}D vs q, this replication m times would cause the profile to rise everywhere by the same amount, ln m. The shape of the profile of the logarithm of

^{q}D is therefore replication invariant, which makes it a useful tool in diversity analysis. The logarithm of the diversity profile is known in statistics as the Renyi entropy spectrum of the community [24].

^{q}D – ln S)/q.

_{0, q}, divided by q. This can be converted to a measure of relative inequality by the usual linear transformation (x − x

_{min})/(x

_{max}− x

_{min}). The slope for a maximally uneven community of S species is –(ln S)/q, because a maximally uneven community has ln

^{q}D = ln 1 = 0 for q > 0. The slope of the chord therefore could range from 0 (perfectly even community) to −(ln S)/q (maximally uneven community). It can be transformed onto the unit interval by dividing by −(ln S)/q:

^{q}D – ln S)/q]/[–(ln S)/q]

= (ln S – ln

^{q}D)/ln S

= RLI

_{0, q}.

_{0, q}derived in the previous section. The complement of this is a relative measure of evenness:

^{q}D)/ln S

= ln

^{q}D/ln S

= RLE

_{0, q}.

_{0, q}derived in the previous section. When q = 1 this is Pielou’s evenness measure J again. Her formula and its generalizations to higher-order q have this simple graphical interpretation in terms of the slope of the chord of the Renyi spectrum from x = 0 to x = q.

#### 7.5. Relative Evenness Measures Cannot and Should Not Be Replication Invariant

_{0, q}(Equation 5) and relative inequality RLI

_{0, q}(Equation 6) provide more intuitive results than the raw evenness and inequality factors for our Jack Pine and Barro Colorado Island forest example. For the Jack Pine forest, relative logarithmic inequality RLI

_{0, 2}is 97%, accurately showing that this community is almost maximally uneven for a four-species community. Its relative logarithmic evenness RLE

_{0, 2}is 3%, correctly showing that evenness is close to the minimum possible for a four-species community. For Barro Colorado Island, relative logarithmic inequality RLI

_{0, 2}is 47%, far lower than the 95% of the Jack Pine forest. Its relative logarithmic evenness RLE

_{0, 2}is 53%, a reasonably moderate value, showing that the Barro Colorado Island rain forest is far more even, given its richness, than the Michigan Jack Pine forest.

## 8. Statistical Concerns

**Figure 6.**Sensitivity of evenness measures to ultra-rare species for a variety of measures. The initial community consists of two species, each with abundance 100,000. From left to right, vanishingly rare species are added, one at a time. Each vanishingly rare species consists of one individual.

#### 8.2. Partitioning Higher-Order Diversity Measures

^{0}D of course) into independent diversity and inequality components. This dooms us from the start if we are worried about statistical reliability of our measures, since it is based directly on the difficult-to-estimate value of S. Many authors have trued to avoid this problem by partitioning not

^{0}D but

^{1}D, which can be accurately estimated from small samples [25].

^{1}D can be partitioned into any higher-order diversity and an independent “inequality” component. The most logical higher-order diversity to use is

^{2}D, so that

^{1}D =

^{2}D*IF

_{1, 2}.

_{1, 2}=

^{1}D/

^{2}D,

_{1, 2}=

^{2}D/

^{1}D.

_{1, 2}.

^{1}D =

^{2}D and the evenness is unity, as it should be. If the assemblage is extremely uneven, so that the diversity profile was very steep around q=0, then the profile would also be nearly horizontal for q ≥ 1. This would cause

^{1}D and

^{2}D to be nearly equal, and the evenness would again be close to unity, even though this assemblage is maximally uneven. These measures are therefore non-monotonic with respect to increasing evenness. For example, the highly uneven assemblage whose species frequencies are [0.999, 0.001] has an “evenness” EF

_{1, 2}equal to 0.9941, close to unity. The more even assemblage (according to the Principle of Transfers) with frequencies [0.9, 0.1] has a lower “evenness”; EF

_{1, 2}= 0.881.

^{1}D, the minimum possible value for the evenness factor

^{2}D/

^{1}D is 1/

^{1}D, since the minimum possible value of

^{2}D is 1. The maximum possible value is unity for the perfectly even community (since then

^{2}D =

^{1}D = S). Using the standard transformation (x − x

_{min})/(x

_{max}− x

_{min}), we obtain the higher-order relative evenness RE

_{1, 2}:

_{1, 2}= [EF

_{1, 2}– (1/

^{1}D)]/[1 − 1/1D)]

= (

^{2}D − 1)/(

^{1}D − 1).

_{1, 2}had been proposed by Alatalo [26].

_{0, 1}instead of EF

_{0, 1}itself. We do the same thing here, transforming ln EF

_{1, 2}into a relative measure using (x − x

_{min})/(x

_{max}− x

_{min}):

_{1, 2}= [ln EF

_{1, 2}– ln (1/

^{1}D)]/[ln 1 – ln (1/

^{1}D)]

= [ln

^{2}D – ln

^{1}D + ln

^{1}D)]/[ln

^{1}D]

= [ln

^{2}D]/[ln

^{1}D].

_{0, 1}.

_{1, 2}and RLE

_{1, 2 }are apparently not affected by the problems of the absolute evenness EF

_{1, 2}. The relative evenness RE

_{1, 2}of [0.999, 0.001] is 0.25, and the relative evenness RE

_{1, 2}of [0.9, 0.1] is 0.57. The relative logarithmic evenness RLE

_{1, 2}of [0.999, 0.001] is 0.25 while the RLE

_{1, 2}of [0.9, 0.1] is 0.61. These measures correctly show that relative evenness of the second community is greater than the relative evenness of the first community. However, to my knowledge, the relation between these measures and the Lorentz partial ordering is not yet known.

#### 8.3. An Estimable Evenness Measure

_{1}/N)

_{1}is the number of singleton species in the sample (the number of species represented by exactly one individual in the sample). This estimate is most accurate when f

_{1}is large. If we wanted to estimate the richness of a community at 95% coverage, we would keep sampling until C = 0.95, and then measure the evenness of the sample. More accurate would be to make a sample that exceeds 95% coverage, and repeatedly rarefy it down to 95% coverage, averaging the evenness of each rarefied sample.

## 9. Discussion

#### 9.1. Relative versus Absolute Evenness and Inequality

_{0, q}) but endorses EF

_{0, 1}. His Table 1 lists values for these measures when applied to a community in which half the species have a large relative abundance X, and the other half have a vanishingly small relative abundance. He gives several such communities, each with different richness. He argues that these communities should all have an evenness of 0.50 independent of their richness, and he notes that EF

_{0, 1}does give 0.5 for all these communities while J increases sharply with richness. However, his arguments confuse absolute and relative evenness, and the apparent defects of J are precisely what are needed in a measure of relative evenness (the amount of evenness relative to the range of evenness possible for the given richness). J and EF

_{0, 1}are looking at exactly the same thing from different but equally valid viewpoints.

_{0, 1}, and EF

_{0, 1}for the maximally uneven community, the maximally even community, and the intermediate community considered by Alatalo [26]. J correctly gives unity for all completely even communities, regardless of richness. It also correctly gives zero for the maximally uneven communities, independent of richness, as a relative measure must do. Note that IF

_{0, 1}and EF

_{0, 1}are not independent of richness for the maximally uneven community, so they are clearly not relative measures of evenness or inequality. They are giving the absolute evenness and inequality. When richness is greater, the maximally unequal community shows more absolute inequality (less absolute evenness) than when richness is low.

**Table 1.**Relative and absolute evenness and inequality. Relative logarithmic evenness J is always zero when community is maximally uneven, and is always unity when community is perfectly even. The intermediate communities of Alatalo [26] are not really intermediate in inequality or evenness when richness is high; they are actually closer to the completely even community. See Figure 7.

Maximally uneven | Intermediate according | Completely even | |||||||
---|---|---|---|---|---|---|---|---|---|

to Alatalo [26] | |||||||||

RichnesS | J | IF_{0,1} | EF_{0,1} or 1/IF_{0,1} | J | IF_{0,1} | EF_{0,1} or 1/IF_{0,1} | J | IF_{0,1} | EF_{0,1} or 1/IF_{0,1} |

4 | 0 | 4 | 0.25 | 0.5 | 2 | 0.5 | 1 | 1 | 1 |

8 | 0 | 8 | 0.125 | 0.67 | 2 | 0.5 | 1 | 1 | 1 |

16 | 0 | 16 | 0.0625 | 0.75 | 2 | 0.5 | 1 | 1 | 1 |

512 | 0 | 512 | 0.002 | 0.89 | 2 | 0.5 | 1 | 1 | 1 |

**Figure 7.**

**Changes in evenness and inequality in equal steps**. Each step involves transfer of half the abundance of the community. When richness S is high, Alatalo’s [26] intermediate communities are closer to the perfectly even community than to the center. This explains why J seems to vary with richness in his example.

#### 9.2. An Alternative Evenness Concept

## 10. Conclusion

## Acknowledgements

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**MDPI and ACS Style**

Jost, L.
The Relation between Evenness and Diversity. *Diversity* **2010**, *2*, 207-232.
https://doi.org/10.3390/d2020207

**AMA Style**

Jost L.
The Relation between Evenness and Diversity. *Diversity*. 2010; 2(2):207-232.
https://doi.org/10.3390/d2020207

**Chicago/Turabian Style**

Jost, Lou.
2010. "The Relation between Evenness and Diversity" *Diversity* 2, no. 2: 207-232.
https://doi.org/10.3390/d2020207