IB & Edexcel Biology: Essential Formula Handbook | IB与爱德思生物必备公式手册

📚 IB & Edexcel Biology: Essential Formula Handbook | IB与爱德思生物必备公式手册

In both IB and Edexcel A‑level Biology, quantitative skills are essential. From calculating magnification to applying the Hardy–Weinberg principle, students must be familiar with a set of key formulae. This handbook compiles the most important equations used in these syllabi, with clear explanations and worked examples.

在IB和爱德思A‑level生物课程中，定量分析能力至关重要。从计算放大倍数到应用哈代–温伯格定律，学生必须熟练掌握一系列关键公式。本手册汇总了这些课程中最常用的公式，并配以清晰的解释与示例。

1. Magnification and Actual Size | 放大倍数与实际大小

The magnification formula connects the size of an image (e.g. a photomicrograph) to the real size of the specimen. It is expressed as M = I ÷ A, where M is magnification, I is image size and A is actual size. To find actual size, rearrange to A = I ÷ M. Always convert all measurements to the same unit (typically micrometres, µm) before calculating.

放大倍数公式将图像大小与标本实际大小联系在一起。表示为 M = I ÷ A，M 为放大倍数，I 为图像尺寸，A 为实际尺寸。求实际大小时可变形为 A = I ÷ M。计算前务必将所有测量值换算为同一单位（通常为微米 µm）。

Magnification (M) = Image size (I) ÷ Actual size (A)

2. Surface Area to Volume Ratio | 表面积与体积比

As an object increases in size, its volume grows faster than its surface area. The surface area to volume ratio (SA:V) is calculated by dividing the total surface area by the total volume. A large SA:V ratio facilitates efficient diffusion; this explains why cells are microscopic and why organisms have specialised exchange surfaces.

物体尺寸增大时，体积的增长远快于表面积。表面积与体积比（SA:V）用总表面积除以总体积来计算。较高的 SA:V 比有利于高效扩散，这解释了为什么细胞是微观的，以及生物为何演化出特化的交换表面。

SA:V ratio = Surface Area ÷ Volume

3. Percentage Change and Error | 百分比变化与误差

Percentage change quantifies how much a variable has increased or decreased relative to its original value: (final value − original value) ÷ original value × 100%. Percentage error estimates the accuracy of a measurement: (uncertainty ÷ measured value) × 100%. In IB Biology, students are frequently asked to calculate percentage difference between experimental and theoretical values.

百分比变化可量化变量相对于初始值的增减程度：（终值 − 初值）÷ 初值 × 100%。百分比误差用于估计测量的准确度：（不确定度 ÷ 测量值）× 100%。在IB生物中常要求计算实验值与理论值之间的百分比差异。

Percentage change = (Final − Initial) ÷ Initial × 100%

Percentage error = (Uncertainty ÷ Measured value) × 100%

4. Cardiac Output and Ventilation Rates | 心输出量与肺通气量

Cardiac output (CO) is the volume of blood pumped by the heart per minute: CO = heart rate (HR) × stroke volume (SV). Similarly, pulmonary ventilation (minute volume) represents the volume of air moved into the lungs each minute: Pulmonary ventilation = tidal volume (TV) × breathing rate (BR). Both formulae link rate and volume to describe the overall performance of the circulatory and respiratory systems.

心输出量（CO）是心脏每分钟泵出的血液量：CO = 心率（HR）× 每搏输出量（SV）。类似地，肺通气量（每分通气量）表示每分钟进出肺部的气体量：肺通气量 = 潮气量（TV）× 呼吸频率（BR）。这两个公式将速率与容积联系起来，反映循环和呼吸系统的整体效能。

Cardiac output = Heart rate × Stroke volume

Pulmonary ventilation = Tidal volume × Breathing rate

5. Genetic Probability: Hardy–Weinberg Equilibrium | 遗传概率：哈代–温伯格平衡

The Hardy–Weinberg principle describes allele and genotype frequencies in a non‑evolving population. For a gene with two alleles, if p is the frequency of the dominant allele and q is the frequency of the recessive allele, then p + q = 1. The expected genotype frequencies are given by p² + 2pq + q² = 1, where p² = homozygous dominant, 2pq = heterozygous, and q² = homozygous recessive. These equations are used to estimate carrier frequencies or predict changes when evolution occurs.

哈代–温伯格定律描述了理想群体中等位基因与基因型频率。对于一个有两个等位基因的基因，若 p 为显性等位基因频率，q 为隐性等位基因频率，则 p + q = 1。预期基因型频率为 p² + 2pq + q² = 1，其中 p² 为显性纯合子，2pq 为杂合子，q² 为隐性纯合子。这些方程常用于估算携带者频率或判断演化是否发生。

p + q = 1

p² + 2pq + q² = 1

6. Chi‑Squared Test | 卡方检验

The chi‑squared (χ²) test determines whether there is a significant difference between observed and expected frequencies. It is widely used in genetics and ecology. The formula is χ² = Σ ((O − E)² ÷ E), where O = observed value, E = expected value. The calculated χ² is compared against a critical value from a table, using the appropriate degrees of freedom (df = number of categories − 1). If χ² > critical value, the null hypothesis is rejected.

卡方（χ²）检验用于判断观测频率与预期频率之间是否存在显著差异，广泛应用于遗传学与生态学。公式为 χ² = Σ ((O − E)² ÷ E)，O 为观测值，E 为预期值。计算得到的 χ² 与查表所得的临界值比较，自由度 df = 类别数 − 1。若 χ² 大于临界值，则拒绝原假设。

χ² = Σ ((O − E)² ÷ E)

7. Simpson’s Diversity Index | 辛普森多样性指数

Biodiversity can be measured using Simpson’s Diversity Index. In IB Biology, the index is commonly calculated as D = N(N − 1) ÷ Σ n(n − 1), where N is the total number of organisms and n is the number of individuals of a particular species. A high D indicates high diversity. In Edexcel A (Salters‑Nuffield) the formula is often given as D = 1 − Σ (n/N)², which produces a value between 0 and 1; the closer to 1, the higher the diversity. Check the specification for your particular course.

生物多样性可通过辛普森多样性指数衡量。在IB生物中，指数通常计算为 D = N(N − 1) ÷ Σ n(n − 1)，N 为总个体数，n 为某一物种的个体数。D 值越高，多样性越大。在Edexcel A (Salters‑Nuffield) 课程中，公式常以 D = 1 − Σ (n/N)² 给出，结果介于0至1之间，越接近1多样性越高。请核对所在考纲的具体要求。

D = N(N − 1) ÷ Σ n(n − 1) (IB)

D = 1 − Σ (n/N)² (Edexcel A)

8. Water Potential | 水势

Water potential (ψ) governs the direction of water movement across plant cell membranes. It is the sum of solute potential (ψₛ) and pressure potential (ψₚ): ψ = ψₛ + ψₚ. Pure water at atmospheric pressure has a water potential of 0 MPa. Dissolving solutes lowers ψₛ (makes it more negative), while turgor pressure raises ψₚ (makes it less negative). Water always moves from regions of higher water potential to lower water potential.

水势（ψ）决定水分跨越植物细胞膜的方向。它是溶质势（ψₛ）与压力势（ψₚ）之和：ψ = ψₛ + ψₚ。在标准大气压下纯水的水势为0 MPa。溶解溶质会降低 ψₛ（更负），而膨压会提高 ψₚ（较不正）。水分总是从高水势区域流向低水势区域。

ψ = ψₛ + ψₚ

9. Population Growth Rate | 种群增长率

Population growth can be described by the rate of natural increase: Growth rate = (Births − Deaths) ÷ Initial population × 100%. In exponential models, the intrinsic growth rate (r) is used: dN/dt = rN (not usually required to calculate but useful for understanding). For bacterial growth, the number of cells after n generations is Nₜ = N₀ × 2ⁿ, where N₀ is the initial number. These models help explain population dynamics and carrying capacity.

种群增长可用自然增长率描述：增长率 = （出生数 − 死亡数）÷ 初始种群数量 × 100%。在指数模型中，采用内禀增长率（r）：dN/dt = rN（通常不要求计算，但有助于理解）。对于细菌生长，n 代后的细胞数为 Nₜ = N₀ × 2ⁿ，N₀ 为初始数量。这些模型有助于解释种群动态与环境容纳量。

Growth rate = (Births − Deaths) ÷ Initial N × 100%

Nₜ = N₀ × 2ⁿ

10. Standard Deviation | 标准差

Standard deviation measures the spread of data around the mean. For a sample, it is calculated using s = √( Σ(x − x̄)² ÷ (n − 1) ), where x is each data point, x̄ is the sample mean, and n is the number of observations. A smaller standard deviation indicates data are closely clustered around the mean; a larger one shows greater variability. This statistic is essential when comparing two data sets and when using the t‑test.

标准差衡量数据围绕平均值的离散程度。对于样本，计算公式为 s = √( Σ(x − x̄)² ÷ (n − 1) )，x 为各数据点，x

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IB & Edexcel Biology: Essential Formula Handbook | IB与爱德思生物必备公式手册