📚 Pre-U OCR Biology: Quick Reference Handbook of Formulas and Theorems | Pre-U OCR 生物：公式定理速查手册
This handbook compiles the essential quantitative relationships, equations, and statistical tests required for the Pre-U OCR Biology specification. Each section presents the formula or theorem in a clear format, accompanied by worked applications and common pitfalls. Use it as a revision aid to master the numerical and analytical skills assessed in both written papers and practical investigations.
本手册汇集了 Pre-U OCR 生物学必需的重要定量关系、方程式和统计检验方法。每节都以清晰的格式呈现公式或定理，并附有应用示例和常见陷阱。可将它作为复习工具，掌握笔试和实验探究中考核的计算和分析技能。
1. Microscopy: Magnification and Actual Size | 显微镜：放大率与实际大小
The total magnification of a compound light microscope is the product of the eyepiece lens magnification and the objective lens magnification.
复合光学显微镜的总放大率是目镜放大率与物镜放大率的乘积。
Total Magnification = Eyepiece Magnification × Objective Magnification
To calculate the actual size of a specimen from a micrograph or drawing, use the magnification triangle. The formula requires consistency of units; convert all measurements to micrometres (µm) or millimetres (mm) as appropriate, remembering that 1 mm = 1000 µm.
要根据显微照片或绘图计算样本的实际大小，需要使用放大率公式。计算时单位必须一致；将所有量度转换为微米 (µm) 或毫米 (mm)，记住 1 mm = 1000 µm。
Actual Size = Image Size ÷ Magnification
When using a stage micrometer and eyepiece graticule, calibrate the graticule divisions at each objective magnification. Record the number of micrometres per graticule unit and apply this conversion to any measurement taken.
当使用镜台测微尺和目镜测微尺时，需在每个物镜放大率下校准目镜测微尺刻度。记录每个刻度单位对应的微米数，并将此换算用于所有测量。
2. Chromatography: Rf Value | 色谱法：Rf 值
The retention factor (Rf) is used to identify components separated by paper or thin‑layer chromatography. It is a ratio and therefore has no units.
比移值 (Rf) 用于识别纸色谱或薄层色谱分离后的组分。它是一个比值，没有单位。
Rf = Distance moved by substance ÷ Distance moved by solvent front
Measure distances from the origin (pencil line) to the centre of the spot and to the furthest point reached by the solvent. Report Rf values to two decimal places. The value is characteristic for a given substance under identical stationary and mobile phase conditions, but may vary with temperature and solvent saturation.
测量距离时，应从原点（铅笔线）量至斑点中心，以及量至溶剂前沿的最远点。Rf 值通常保留两位小数。在固定相和流动相条件相同时，该值对特定物质具有特征性，但会随温度和溶剂饱和度的变化而改变。
3. Gas Exchange: Respiratory Quotient (RQ) | 气体交换：呼吸商
The respiratory quotient indicates which substrate is being respired by a whole organism, a tissue, or seeds. It is derived from the volumes of gases exchanged measured using a respirometer.
呼吸商能够显示生物体、组织或种子正在呼吸的底物类型。它通过使用呼吸计测得气体交换体积后计算得出。
RQ = Volume of CO₂ produced ÷ Volume of O₂ consumed
Typical RQ values: carbohydrate = 1.0, lipid ≈ 0.7, protein ≈ 0.8–0.9. Values above 1.0 suggest anaerobic respiration or a shift to organic acid metabolism. For germinating seeds, an RQ change may indicate switching between lipid and carbohydrate reserves. Always ensure temperature and pressure are constant during respirometer readings.
常见的 RQ 值：碳水化合物为 1.0，脂质约 0.7，蛋白质约 0.8–0.9。RQ 值高于 1.0 提示存在无氧呼吸或转向有机酸代谢。对萌发种子而言，RQ 值变化可能意味着从脂质储备转向碳水化合物。读取呼吸计数据时，务必确保温度和压力恒定。
4. Genetics: Hardy–Weinberg Equilibrium | 遗传学：哈代-温伯格平衡
The Hardy–Weinberg principle predicts allele and genotype frequencies in a non‑evolving population. It serves as a null hypothesis for detecting evolutionary change.
哈代-温伯格原理可预测非进化群体中的等位基因频率和基因型频率，是检测进化变动的零假设工具。
p + q = 1
p² + 2pq + q² = 1
Where p = frequency of the dominant allele, q = frequency of the recessive allele. Genotype frequencies: homozygous dominant = p², heterozygous = 2pq, homozygous recessive = q². The five conditions required are: no mutation, random mating, no gene flow (migration), extremely large population size, and no natural selection. If observed proportions deviate significantly from the expected, a chi‑squared test is used to assess statistical significance.
其中 p = 显性等位基因频率，q = 隐性等位基因频率。基因型频率：纯合显性 = p²，杂合 = 2pq，纯合隐性 = q²。该原理成立需满足五个条件：无突变、随机交配、无基因流动（迁移）、群体极大、无自然选择。若实际观测比例与预期值偏差显著，则用卡方检验评估其统计学显著性。
5. Chi‑squared (χ²) Test | 卡方检验
The chi‑squared test compares observed results with expected frequencies to determine whether any deviation is due to chance or a significant factor. It is widely used in genetics, ecology, and behavioural studies.
卡方检验用于比较观测结果与预期频率，以判断偏差是由偶然因素造成还是存在显著因子。该检验广泛用于遗传学、生态学和行为学研究。
χ² = Σ (O – E)² / E
O = observed value, E = expected value. Calculate (O–E)² / E for each category, then sum all categories. The degrees of freedom (df) depend on the test: goodness‑of‑fit df = number of categories – 1; genetic cross df = number of phenotypic classes – 1; contingency table df = (rows – 1) × (columns – 1). Compare the calculated χ² value against the critical value at p = 0.05 in statistical tables. If χ² > critical value, reject the null hypothesis; the difference is statistically significant.
O 为观测值，E 为预期值。对每一类别计算 (O–E)² / E，再将所有类别求和。自由度 (df) 因检验类型而异：拟合优度检验 df = 类别数 – 1；遗传杂交 df = 表型种类数 – 1；列联表 df = (行数 – 1) × (列数 – 1)。将计算所得 χ² 值与统计表中 p = 0.05 时的临界值比较。若 χ² > 临界值，则拒绝零假设；差异具有统计学意义。
6. Estimating Population Size: Lincoln Index | 种群数量估计：林肯指数
The Lincoln index provides an estimate of mobile animal population size using capture‑mark‑release‑recapture data. It assumes a closed population with random mixing and no mark‑induced mortality or loss.
林肯指数利用捕捉-标记-释放-重捕的数据来估算活动性动物种群数量。该方法假设种群封闭、个体随机混合，且标记不影响死亡或丢失。
N = (M × C) / R
N = estimated total population, M = number of individuals marked in first capture, C = total number captured in second sample, R = number of marked individuals recaptured in the second sample. To improve accuracy, repeat the procedure multiple times and take an average. Calculate 95% confidence limits where possible using the formula for variance. Assumptions include: no births, deaths, immigration, or emigration between samples; marks are not lost; marking does not affect survival. If R is very small, the estimate becomes unreliable.
N = 估计种群总数，M = 首次捕获并标记的个体数，C = 第二次样本捕获总数，R = 第二次样本中重捕的标记个体数。为提高准确性，可多次重复该过程并取平均值。如有可能，使用方差公式计算 95% 置信区间。前提假设包括：两次取样间无出生、死亡、迁入或迁出；标记不丢失；标记不影响存活。若 R 很小，估计值将不可靠。
7. Biodiversity: Simpson’s Diversity Index | 生物多样性：辛普森多样性指数
Simpson’s index measures the probability that two randomly selected individuals from a community belong to different species. High values indicate high diversity.
辛普森指数衡量从群落中随机抽取两个个体属于不同物种的概率。数值越高，表明多样性越高。
D = 1 – Σ (nᵢ / N)²
nᵢ = number of individuals of species i, N = total number of individuals of all species. An alternative form, Simpson’s diversity index, is D = 1 – (Σ nᵢ(nᵢ – 1)) / (N(N – 1)), which reduces bias for small sample sizes. Always specify which version is being used. The index can be used to compare different habitats or to monitor changes over time. A community dominated by a single species yields an index near zero.
nᵢ = 物种 i 的个体数，N = 所有物种的总个体数。另一种形式，即辛普森多样性指数，为 D = 1 – (Σ nᵢ(nᵢ – 1)) / (N(N – 1))，这样可减少小样本偏差。务必说明使用的是何种版本。该指数可用于比较不同生境或监测随时间的变化。由单一物种主导的群落，其指数接近于零。
8. Productivity and Energy Transfer | 生产力与能量传递
Ecological productivity quantifies the energy fixed by organisms. Gross primary productivity (GPP) is the total light energy converted by photosynthesis, while net primary productivity (NPP) is the energy remaining after plant respiration.
生态生产力衡量生物体固定的能量。总初级生产力 (GPP) 是光合作用转化的总光能，而净初级生产力 (NPP) 则是植物呼吸作用后剩余的能量。
NPP = GPP – R
R = respiratory losses. Units are typically kJ m⁻² yr⁻¹ or g m⁻² yr⁻¹. For consumers, the efficiency of energy transfer between trophic levels is calculated as:
R = 呼吸消耗。单位通常为 kJ m⁻² yr⁻¹ 或 g m⁻² yr⁻¹。对消费者而言，营养级间的能量传递效率计算如下：
Ecological Efficiency = (Energy available to next trophic level ÷ Energy consumed) × 100%
This efficiency rarely exceeds 20% and is often around 10%. Reasons for energy loss include respiration, excretion, and incomplete digestion. Net secondary productivity can similarly be expressed as NSP = ingested energy – (respiration + waste). Use calibrated calorimetry or biomass measurements to obtain energy values.
该效率极少超过 20%，通常在 10% 左右。能量损失的原因包括呼吸、排泄和不完全消化。净次级生产力可类似表示为 NSP = 摄食能量 – (呼吸 + 废物)。通过量热法校准或生物量测量来获取能量值。
9. Water Potential in Plants | 植物水势
Water potential (Ψ) determines the direction of water movement across plant cell membranes. Water moves from regions of higher (less negative) to lower (more negative) water potential.
水势 (Ψ) 决定水分穿过植物细胞膜的移动方向。水总是从水势较高（负值较小）的区域流向水势较低（负值较大）的区域。
Ψ = Ψₛ + Ψₚ
Ψₛ is solute potential (also called osmotic potential) arising from dissolved solutes, and it is always negative or zero. For dilute solutions at constant temperature, solute potential can be calculated using the van’t Hoff relation:
Ψₛ 是由溶解溶质引起的溶质势（也称渗透势），其值总是负数或零。对于恒温下的稀溶液，可用范特霍夫关系式计算溶质势：
Ψₛ = –iCRT
i = ionization constant (1 for sucrose, 2 for NaCl fully dissociated), C = molar concentration (mol dm⁻³), R = pressure constant (0.00831 kPa m³ mol⁻¹ K⁻¹ or 8.314 J mol⁻¹ K⁻¹), T = absolute temperature (K). Ψₚ is pressure potential, which can be positive (turgor pressure) or zero. At plasmolysis, Ψₚ = 0 and Ψ = Ψₛ. For practical measurements, use a calibration curve of Ψₛ against known sucrose concentrations with incipient plasmolysis or changes in mass of plant tissue.
i = 解离常数（蔗糖为 1，完全解离的 NaCl 为 2），C = 摩尔浓度 (mol dm⁻³)，R = 压力常数 (0.00831 kPa m³ mol⁻¹ K⁻¹ 或 8.314 J mol⁻¹ K⁻¹)，T = 绝对温度 (K)。Ψₚ 为压力势，可为正值（膨压）或零。在质壁分离时，Ψₚ = 0，Ψ = Ψₛ。实验测量时，可利用已知蔗糖浓度下组织发生初始质壁分离或质量变化的数据，建立 Ψₛ 标准曲线。
10. Statistical Tests: t‑test and Spearman’s Rank Correlation | 统计检验：t 检验与 Spearman 秩相关
The Student’s t‑test compares the means of two sets of normally distributed, continuous data to determine whether a significant difference exists. For independent samples with equal variance, the formula is:
学生 t 检验用于比较两组正态分布连续数据的平均值，以判断是否存在显著差异。对于方差相等的独立样本，公式为：
t = (x̄₁ – x̄₂) / √(sₚ² (1/n₁ + 1/n₂))
where sₚ² = ((n₁ – 1)s₁² + (n₂ – 1)s₂²) / (n₁ + n₂ – 2). x̄₁, x̄₂ are sample means; s₁², s₂² are sample variances; n₁, n₂ are sample sizes. Calculate degrees of freedom as df = n₁ + n₂ – 2. Compare t with the critical value at p = 0.05. If |t| > critical value, reject the null hypothesis that the means are equal.
其中 sₚ² = ((n₁ – 1)s₁² + (n₂ – 1)s₂²) / (n₁ + n₂ – 2)。x̄₁、x̄₂ 为样本均值；s₁²、s₂² 为样本方差；n₁、n₂ 为样本大小。自由度 df = n₁ + n₂ – 2。将 t 值与 p = 0.05 时的临界值比较。若 |t| > 临界值，则拒绝均值相等的零假设。
Spearman’s rank correlation coefficient (rₛ) tests the strength and direction of association between two ranked variables. It does not require normality.
Spearman 秩相关系数 (rₛ) 检验两个排序变量之间关联的强度和方向。它不需要数据服从正态分布。
rₛ = 1 – (6 Σd²) / (n(n² – 1))
d = difference in ranks for each matched pair; n = number of pairs. The value of rₛ ranges from –1 (perfect negative correlation) to +1 (perfect positive correlation). Look up the critical value for n in a Spearman’s rank table. If |rₛ| > critical value, the correlation is statistically significant.
d = 每对匹配数据的秩次差；n = 配对数量。rₛ 的取值范围从 –1（完全负相关）到 +1（完全正相关）。查阅 Spearman 秩相关临界值表，若 |rₛ| > 临界值，则相关性具有统计学意义。
11. Quantitative Practical Skills: Dilutions and Standard Curves | 定量实验技能：稀释与标准曲线
Serial dilutions are used to prepare a range of known concentrations for constructing a calibration curve. The dilution formula simplifies the calculation of volumes and concentrations.
连续稀释常用于配制一系列已知浓度的溶液，以构建标准曲线。稀释公式可简化体积和浓度的计算。
C₁V₁ = C₂V₂
C₁ and C₂ are initial and final concentrations; V₁ and V₂ are initial and final volumes. Ensure units match (e.g., mol dm⁻³ and cm³, or g dm⁻³ and dm³). For a 1 in 10 serial dilution, transfer 1 cm³ of stock to 9 cm³ of diluent and mix thoroughly.
C₁ 和 C₂ 为初始和最终浓度；V₁ 和 V₂ 为初始和最终体积。确保单位一致（如 mol dm⁻³ 与 cm³ 搭配，或 g dm⁻³ 与 dm³ 搭配）。进行 10 倍系列稀释时，移取 1 cm³ 母液至 9 cm³ 稀释液中并充分混合。
In colorimetry, the absorbance of a solution is often directly proportional to its concentration over a limited range (Beer–Lambert law). Plot absorbance against known concentrations to generate a standard curve; use the line of best fit to interpolate unknown concentrations. Never extrapolate beyond the range of measured standards.
在比色分析中，溶液吸光度常与浓度在有限范围内呈正比（比尔-朗伯定律）。以吸光度对已知浓度作图得到标准曲线；利用最佳拟合线内插计算未知浓度。切勿将结果外推至标准溶液测定范围之外。
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