📚 Edexcel Maths 4.2 Biodiversity: Simpson's Index and Statistical Measures | Edexcel 数学 4.2 生物多样性：辛普森指数与统计测量
Biodiversity data brings together counts, proportions, logarithms and sampling theory. In Edexcel Maths, the key skills are using Simpson's index, the Shannon-Wiener index, the Lincoln index and standard deviation to turn field observations into rigorous numerical conclusions.
生物多样性数据涉及计数、比例、对数与抽样理论。在 Edexcel 数学中，核心技能是运用辛普森指数、香农-维纳指数、林肯指数和标准差，将野外观察转化为严谨的数值结论。
1. Why Biodiversity Needs Mathematical Tools | 为什么生物多样性需要数学工具
Biodiversity cannot be described by a single count. A meadow and a conifer plantation might both contain 200 individual organisms, yet the meadow can be far more diverse because its individuals are spread across many species. Mathematical indices summarise this difference so that habitats can be compared fairly.
生物多样性不能仅靠一个总数来描述。一片草地和一片针叶林地可能都有 200 个生物个体，但草地的多样性可能远高于针叶林地，因为个体分布在更多物种中。数学指数可以概括这种差异，使不同栖息地能够被公平比较。
2. Species Richness and Evenness | 物种丰富度与均匀度
Species richness is the number of different species present. Species evenness measures how equally individuals are shared among those species. Two sites with the same richness can differ greatly in evenness, so both ideas must be considered before an index is calculated.
物种丰富度是指出现的不同物种数量。物种均匀度衡量个体在这些物种之间的分配是否平均。两个丰富度相同的地点，均匀度可能差异很大，因此计算指数前必须同时考虑这两个概念。
- Richness: the total number of species.
- Evenness: whether one species dominates the sample.
- Example: counts 50, 50, 50, 50 show high evenness; counts 197, 1, 1, 1 show low evenness.
丰富度是物种总数；均匀度反映是否由某一物种占绝对优势。例如，计数为 50、50、50、50 时均匀度高；计数为 197、1、1、1 时均匀度低。
3. Simpson's Index of Diversity | 辛普森多样性指数
The Simpson index D is a standard metric that accounts for both richness and evenness. It is calculated by squaring each species' proportional abundance, summing these squares, and subtracting the total from 1. It is one of the most frequently examined formulas in this topic.
辛普森指数 D 是同时考虑丰富度和均匀度的标准指标。计算方法是先求每个物种相对多度的平方，再求和，最后用 1 减去该总和。这是本专题最常考到的公式之一。
D = 1 − Σ (nᵢ ÷ N)²
Here nᵢ is the number of individuals of species i, N is the total number of all individuals, and Σ means sum over all species. A higher D value indicates greater diversity.
其中 nᵢ 是第 i 个物种的个体数，N 是所有个体的总数，Σ 表示对所有物种求和。D 值越大，多样性越高。
4. Worked Example: Simpson's Index | 例题：辛普森指数计算
A pond sample contains four species: 20 mayfly nymphs, 15 caddisfly larvae, 10 shrimps and 5 worms. Find Simpson's index of diversity.
一个池塘样本包含四个物种：20 只蜉蝣若虫、15 只石蛾幼虫、10 只虾和 5 条蠕虫。求辛普森多样性指数。
| Species | Count nᵢ | nᵢ ÷ N | (nᵢ ÷ N)² |
|---|---|---|---|
| Mayfly nymph | 20 | 0.40 | 0.16 |
| Caddisfly larva | 15 | 0.30 | 0.09 |
| Shrimp | 10 | 0.20 | 0.04 |
| Worm | 5 | 0.10 | 0.01 |
| Total | N = 50 | Σ = 0.30 |
cates relatively high diversity because no single species dominates the sample completely.
D = 1 − 0.30 = 0.70。这表明多样性较高，因为没有单一物种完全占主导地位。
5. Shannon-Wiener Index | 香农-维纳指数
The Shannon-Wiener index H′ is an alternative diversity measure that uses natural logarithms. It gives more weight to rare species than Simpson's index does, so it is useful when small populations matter ecologically.
香农-维纳指数 H′ 是另一种使用自然对数的多样性测度。与辛普森指数相比，它给予稀有物种更大的权重，因此在稀有种群具有生态意义时非常有用。
H′ = −Σ pᵢ ln pᵢ
Here pᵢ is the proportion of species i, and ln is the natural logarithm. Higher H′ values indicate greater diversity.
其中 pᵢ 是第 i 个物种的比例，ln 是自然对数。H′ 值越大，多样性越高。
e −0.916, −1.204, −1.609 and −2.303. Multiplying by pᵢ and changing sign gives H′ ≈ 1.279.
用前一个样本计算，pᵢ 分别为 0.40、0.30、0.20 和 0.10。自然对数值为 −0.916、−1.204、−1.609 和 −2.303。乘以 pᵢ 后取相反数，得到 H′ ≈ 1.279。
6. Sampling and Randomisation | 抽样与随机化
Reliable diversity calculations start with unbiased sampling. Quadrats must be placed randomly rather than in visually interesting spots. The more quadrats you use, the closer your estimate of mean density is likely to be to the true population mean.
可靠的多样性计算始于无偏抽样。样方必须随机放置，而不是放在视觉上有趣的位置。使用的样方越多，你对平均密度的估计就越接近真实总体平均值。
Density = total individuals ÷ (number of quadrats × area of one quadrat)
rats of area 0.25 m² each, the estimated density is 120 ÷ (10 × 0.25) = 48 daisies per m².
如果在 10 个面积为 0
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