📚 Pre-U Edexcel Biology: Formula & Theorem Quick Reference Handbook | Pre-U Edexcel 生物：公式定理速查手册
This handbook compiles the essential formulae, statistical tests, and key quantitative relationships that feature consistently across the Pre-U Edexcel Biology specification. Mastering these equations is not merely about memorisation; it is about understanding when to apply each tool, interpreting the resulting values, and using them to support evidence-based conclusions in genetics, ecology, physiology, and biochemistry. Each entry below is presented with its formula, a brief explanation, and the typical context in which it appears in examination papers.
本手册汇集了Pre-U Edexcel生物学课程中反复出现的核心公式、统计检验及关键定量关系。掌握这些方程不仅需要记忆，更要求理解何时应用每种工具、如何解读所得数值，并运用它们来支撑遗传学、生态学、生理学及生物化学中基于证据的结论。以下每个条目均配有公式、简要说明以及其在考试中常见的应用情境。
1. Hardy-Weinberg Equilibrium | 哈代-温伯格平衡
Hardy-Weinberg equations allow you to calculate allele and genotype frequencies in a population that is not evolving. The two fundamental equations are p + q = 1 and p² + 2pq + q² = 1, where p is the frequency of the dominant allele and q is the frequency of the recessive allele. You will typically be given the frequency of the homozygous recessive genotype, and from that you derive q, then p, and then the carrier frequency (2pq).
哈代-温伯格方程用于计算未进化种群中的等位基因频率和基因型频率。两个基本方程为 p + q = 1 和 p² + 2pq + q² = 1，其中 p 为显性等位基因的频率，q 为隐性等位基因的频率。题目通常会给出隐性纯合子的基因型频率，由此可推导出 q，再得到 p，进而求出携带者频率 (2pq)。
p + q = 1
p² + 2pq + q² = 1
Remember to state the assumptions of the Hardy-Weinberg principle every time you apply it: large population, random mating, no mutation, no migration, and no natural selection. In many Pre-U questions, one of these assumptions is deliberately violated, and you are asked to discuss the consequences.
每次应用时请记住陈述哈代-温伯格定律的假设条件：大种群、随机交配、无突变、无迁移、无自然选择。在许多Pre-U试题中，其中一项假设会被刻意打破，要求你讨论其后果。
2. Chi-Squared (χ²) Test | 卡方(χ²)检验
The chi-squared test is used to determine whether there is a significant difference between observed and expected frequencies in categorical data. In biology, it is most commonly applied to genetic crosses and ecological distribution studies. The formula is:
卡方检验用于判断分类数据中观察值与期望值之间是否存在显著差异。在生物学中，它最常用于遗传杂交实验和生态分布研究。公式为：
χ² = Σ (O – E)² / E
O represents the observed frequency, E the expected frequency, and Σ means ‘sum of’. After calculating the χ² value, you compare it against a critical value from a χ² distribution table at a given probability level (usually p = 0.05) and the appropriate degrees of freedom (number of categories minus 1). If χ² calculated is greater than the critical value, you reject the null hypothesis.
O 代表观察频率，E 代表期望频率，Σ 表示“总和”。计算出 χ² 值后，需将其与给定概率水平（通常 p = 0.05）和相应自由度（类别数减 1）下的 χ² 分布表中的临界值进行比较。若计算的 χ² 大于临界值，则拒绝原假设。
3. Student’s t-Test | 学生t检验
Student’s t-test compares the means of two sets of normally distributed data to see if they are significantly different from each other. The version most frequently required in Pre-U Edexcel Biology is the unpaired (independent) t-test, used when comparing two separate groups. The formula given in the exam booklet is:
学生t检验用于比较两组正态分布数据的均值是否存在显著差异。Pre-U Edexcel生物学中最常要求的是非配对（独立）t检验，用于比较两个独立的组。考试手册中给出的公式为：
t = (x̄₁ – x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Here, x̄₁ and x̄₂ are the sample means, s₁² and s₂² are the sample variances, and n₁ and n₂ are the sample sizes. After computing t, you calculate the degrees of freedom (often n₁ + n₂ – 2) and find the critical t value. If the calculated t exceeds the critical value at p = 0.05, the difference is statistically significant. Always frame your conclusion in biological terms, not just ‘reject null hypothesis’.
其中 x̄₁ 和 x̄₂ 为样本均值，s₁² 和 s₂² 为样本方差，n₁ 和 n₂ 为样本大小。计算 t 值后，需计算自由度（通常为 n₁ + n₂ – 2）并查找临界 t 值。若计算出的 t 值大于 p = 0.05 时的临界值，则差异具有统计学意义。务必用生物学语言表述结论，而非仅仅“拒绝原假设”。
4. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rank correlation (rₛ) tests the strength and direction of association between two ranked variables. It is applicable when data are not normally distributed or when investigating a monotonic relationship. The standard formula is:
斯皮尔曼等级相关系数 (rₛ) 用于检验两个秩次变量之间关联的强度和方向。它适用于数据非正态分布或研究单调关系时。标准公式为：
rₛ = 1 – (6 Σ d²) / (n (n² – 1))
d is the difference between the ranks of each pair of observations, and n is the number of paired observations. The resulting coefficient ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). You must then compare the absolute value of rₛ with a critical value at a chosen significance level. Do not forget to state a null hypothesis of no correlation before beginning.
d 为每对观测值秩次之差，n 为配对观测值的数量。所得系数范围从 -1（完全负相关）到 +1（完全正相关）。必须将 rₛ 的绝对值与所选显著性水平下的临界值进行比较。开始计算前，别忘了先陈述无相关性的原假设。
5. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s Diversity Index (D) quantifies the biodiversity of a habitat by considering both species richness and evenness. The version commonly used in Pre-U Edexcel is:
辛普森多样性指数 (D) 通过综合考虑物种丰富度和均匀度来量化栖息地的生物多样性。Pre-U Edexcel 中常用的版本为：
D = 1 – Σ (n/N)²
n is the number of individuals of a particular species, and N is the total number of individuals of all species. The value of D falls between 0 and 1; the closer the index is to 1, the greater the diversity. An alternative form using n(n-1)/N(N-1) is sometimes presented for small samples, but the above formula is standard. This index is extremely useful when comparing the impact of environmental change on different communities.
n 为某一特定物种的个体数，N 为所有物种的个体总数。D 值介于 0 到 1 之间；指数越接近 1，多样性越高。有时会看到另一种形式 n(n-1)/N(N-1) 用于小样本，但上述公式是标准的。该指数在比较环境变化对不同群落的影响时极为有用。
6. Lincoln Index (Mark-Release-Recapture) | 林肯指数（标记-释放-再捕获）
The Lincoln Index provides an estimate of population size for mobile organisms. The method involves capturing a sample (M), marking them, releasing them back, and later capturing a second sample (C) and counting how many are marked (R). The population estimate N is given by:
林肯指数用于估算移动生物种群的大小。该方法包括捕获一个样本 (M)、标记、释放，之后再捕获第二个样本 (C)，并计数其中已标记的个体数 (R)。种群估计值 N 由下式给出：
N = (M × C) / R
This formula assumes that the marked individuals mix randomly with the unmarked individuals, that marking does not affect survival or catchability, and that the population is closed with no significant births, deaths, or migration during the study. In exam scenarios you are often asked to critique these assumptions or to calculate the reliability of the estimate.
此公式假设标记个体与未标记个体随机混合，标记不影响生存或捕获率，且种群封闭，在研究期间没有显著的出生、死亡或迁入迁出。考试中常要求你评价这些假设或计算估计值的可靠性。
7. Respiratory Quotient (RQ) | 呼吸商 (RQ)
Respiratory quotient is a measure of the ratio of carbon dioxide produced to oxygen consumed during respiration. It provides insight into the respiratory substrate being utilised. The equation is:
呼吸商是衡量呼吸过程中产生的二氧化碳与消耗的氧气之比的一个指标。它可揭示正在被利用的呼吸底物。方程为：
RQ = CO₂ produced / O₂ consumed
Pure carbohydrate respiration yields an RQ of 1.0, lipid respiration gives approximately 0.7, and protein respiration around 0.9. Values greater than 1.0 suggest anaerobic respiration is also occurring. In spirometer or respirometer experiments, you are typically required to calculate RQ from volume changes and then deduce the metabolic fuel. Always ensure units are consistent (e.g., both in cm³).
纯碳水化合物呼吸产生的 RQ 为 1.0，脂质呼吸约为 0.7，蛋白质呼吸约为 0.9。RQ 值大于 1.0 提示同时存在无氧呼吸。在肺活量计或呼吸计实验中，通常要求你根据体积变化计算 RQ，并推断代谢燃料。务必确保单位一致（例如，均以 cm³ 计）。
8. Enzyme Rate Calculations | 酶促反应速率计算
Enzyme activity is often quantified as the initial rate of reaction to avoid complications from product inhibition or substrate depletion. The foundational expression is:
酶活性通常以反应的初始速率来量化，以避免产物抑制或底物耗竭带来的复杂影响。基础表达式为：
Rate = 1 / time taken to reach a defined end point
Alternatively, if you measure product concentration over time, rate = change in concentration / change in time (Δ[P]/Δt). In investigations involving the Michaelis-Menten model, you are not required to derive the equation algebraically, but you must be able to calculate Vₘₐₓ and Kₘ from a given graph and understand that Kₘ is the substrate concentration at half Vₘₐₓ, reflecting enzyme-substrate affinity. Be prepared to plot Lineweaver-Burk graphs by calculating 1/V and 1/[S] if specified, though the raw formula is not typically memorised.
或者，如果测量产物浓度随时间的变化，速率 = 浓度变化 / 时间变化 (Δ[P]/Δt)。在涉及米氏模型的研究中，你不需要代数推导方程，但必须能从给定的曲线图中计算出 Vₘₐₓ 和 Kₘ，并理解 Kₘ 是半 Vₘₐₓ 时的底物浓度，反映了酶与底物的亲和力。准备好通过计算 1/V 和 1/[S] 来绘制 Lineweaver-Burk 曲线（如有要求），但原始公式通常无需背诵。
9. Serial Dilutions & Concentration Calculations | 连续稀释与浓度计算
Serial dilutions are fundamental in preparing calibration curves for colorimetry, in antibiotic sensitivity testing, and in microbiological counting. The key relationship is:
连续稀释在比色法校准曲线制备、抗生素敏感性测试以及微生物计数中具有基础地位。关键关系式为：
C₁ × V₁ = C₂ × V₂
where C is concentration and V is volume. A 1 in 10 serial dilution reduces the concentration by a factor of 10 at each step. To calculate the dilution factor after multiple steps, multiply the individual dilution factors. When counting cells using a haemocytometer, average count per square × dilution factor × volume factor converts raw counts into cells per cm³. Practice integrating these calculations with logarithms when plotting exponential growth or death of microorganisms.
其中 C 为浓度，V 为体积。1:10 的连续稀释每一步将浓度降低为原来的十分之一。要计算多步后的稀释倍数，需将每步的稀释因子相乘。使用血球计数板计数细胞时，用每个方格的平均计数 × 稀释倍数 × 体积因子，可将原始计数转换为每 cm³ 的细胞数。练习在绘制微生物指数生长或死亡曲线时将这些计算与对数相结合。
10. Surface Area to Volume Ratio (SA:V) | 表面积与体积比 (SA:V)
The surface area to volume ratio is a critical concept explaining why cells are microscopic, why organisms have specialised exchange surfaces, and how metabolic rate relates to body size. The ratio is calculated simply as:
表面积与体积比是解释为何细胞微小、为何生物体具有特化交换表面以及代谢率与体型大小之间关系的关键概念。该比率简单计算如下：
SA:V = surface area / volume
For a cube of side length a, SA = 6a² and V = a³, giving SA:V = 6/a. As an organism increases in size, its SA:V decreases, reducing the efficiency of diffusion. In exams, you may be asked to calculate SA:V for spheres, cubes, or composite shapes and then link the result to adaptations such as alveolar structure, root hair cells, or the circulatory system. Always express the ratio in simplest form (e.g., 3:1 rather than 300:100).
对于边长为 a 的立方体，表面积 = 6a²，体积 = a³，得出 SA:V = 6/a。随着生物体尺寸增大，SA:V 下降，扩散效率降低。考试中可能要求你计算球体、立方体或复合形状的 SA:V，然后将结果与肺泡结构、根毛细胞或循环系统等适应性联系起来。务必以最简形式表示比率（例如 3:1 而非 300:100）。
11. Genetic Probability & Punnett Squares | 遗传概率与庞尼特方格
Predicting genotypic and phenotypic ratios in monohybrid and dihybrid crosses relies on the multiplication and addition rules of probability. The product rule states that the probability of two independent events both occurring is the product of their individual probabilities. The sum rule adds probabilities for mutually exclusive events. In a typical cross, you set up a Punnett square and determine the ratio, but a mathematical check can always be done: for a monohybrid cross Bb × Bb, the probability of homozygous recessive is (½ from one parent) × (½ from the other) = ¼.
预测单基因和双基因杂交中的基因型比例和表现型比例依赖于概率的乘法和加法规则。乘法规则指出，两个独立事件同时发生的概率等于各自概率的乘积。加法规则用于互斥事件的概率相加。在典型杂交中，你画出庞尼特方格并确定比率，但总可以进行数学验证：对于单基因杂交 Bb × Bb，隐性纯合子的概率为 (来自亲本一的½) × (来自亲本二的½) = ¼。
For dihybrid crosses, the 9:3:3:1 ratio emerges from the product of two independent 3:1 ratios. You may also encounter the chi-squared test to confirm whether observed offspring ratios fit expected Mendelian ratios. Ensure you can transition fluidly between Punnett squares and probability calculations.
对于双基因杂交，9:3:3:1 比率来源于两个独立 3:1 比率的乘积。你还会遇到运用卡方检验来确认子代观察比例是否符合预期孟德尔比率的情况。确保能够在庞尼特方格和概率计算之间流畅转换。
12. Water Potential (Ψ) Components | 水势 (Ψ) 组成
Water potential governs the movement of water across plant cell membranes and is expressed as the sum of solute potential (Ψₛ) and pressure potential (Ψₚ). The core equation is:
水势控制水分子跨越植物细胞膜的运动，它表示为溶质势 (Ψₛ) 与压力势 (Ψₚ) 之和。核心方程为：
Ψ = Ψₛ + Ψₚ
Ψₛ is always negative or zero in biological systems due to dissolved solutes, and it can be calculated using Ψₛ = -iCRT, where i is the ionisation constant, C is the molar concentration, R is the pressure constant, and T is the temperature in Kelvin. However, at Pre-U level you are more frequently asked to interpret graphs of Ψₛ against cell volume or to predict the direction of water movement between two solutions with known Ψ values. Water moves from a region of higher (less negative) water potential to a region of lower (more negative) water potential.
在生物系统中，由于溶解的溶质，Ψₛ 总是为负值或零，并可用 Ψₛ = -iCRT 来计算，其中 i 为电离常数，C 为摩尔浓度，R 为压力常数，T 为开氏温度。但在Pre-U阶段，更多见的是要求你解读 Ψₛ 随细胞体积变化的曲线图，或预测具有已知 Ψ 值的两个溶液之间的水分运动方向。水从水势较高（负值较小）的区域向水势较低（负值较大）的区域移动。
Common examination applications include turgid plant cells (Ψ = 0, Ψₚ = positive, Ψₛ = negative), plasmolyzed cells, and the explanation of water uptake from the soil. Always state the equation before explaining the scenario.
常见的考试应用包括膨胀植物细胞 (Ψ = 0, Ψₚ 为正值, Ψₛ 为负值)、质壁分离细胞以及从土壤中吸收水分的解释。在解释情境之前，务必先陈述该方程。
Published by TutorHao | Biology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导