📚 SQA Biology Formula & Theorem Quick Reference | SQA生物公式定理速查手册
This quick-reference handbook brings together the essential formulae and key theorems you need for Year 12 SQA Biology. Each entry is presented with a clear explanatory paragraph in English followed by the equivalent in Chinese, covering microscopy, genetics, population ecology, energy transfer and statistical testing. Use it as a portable revision tool to reinforce quantitative skills and conceptual understanding.
本手册汇总了SQA Year 12生物学所需的核心公式与关键定理。每个条目均以英文说明段落开头，随后给出对应的中文解释，涵盖显微镜使用、遗传学、种群生态、能量传递和统计检验等内容。可作为便携式复习工具，帮助强化定量技能与概念理解。
1. Microscopy Calculations | 显微镜计算
The relationship between magnification, image size and actual size is fundamental to all microscope work. The formula is simply magnification = I ÷ A, where I is the measured image length and A is the true object length. Units must be the same before division. A related calculation is the conversion involved when using a graticule: actual size = stage micrometer divisions spanned × known distance per division ÷ eyepiece graticule divisions.
放大倍数、图像大小和实际大小之间的关系是所有显微镜工作的基础。公式为：放大倍数 = 图像长度 I ÷ 实际长度 A，两者单位必须一致。使用目镜测微尺时，还需进行校准：实际大小 = 跨越的镜台测微尺格数 × 每格已知距离 ÷ 目镜测微尺格数。
Magnification = Image size ÷ Actual size
2. Chi-squared Test (χ²) | 卡方检验（χ²）
The chi-squared test determines whether observed categorical data differ significantly from expected outcomes. You compute χ² = Σ (O − E)² ÷ E for each category, where O is observed frequency and E is expected frequency. The resulting statistic is compared against a critical value in a chi-squared distribution table, using degrees of freedom (number of categories − 1). If χ² exceeds the critical value at p = 0.05, the difference is statistically significant.
卡方检验用于判断分类数据的观测值与预期值之间是否存在显著差异。计算公式为 χ² = Σ (O − E)² ÷ E，其中 O 为观测频数，E 为预期频数。将所得统计量与卡方分布表中的临界值比较，自由度 = 类别数 − 1。若 χ² 大于 p = 0.05 时的临界值，则表明差异具有统计学意义。
χ² = Σ [(O − E)² ÷ E]
3. Hardy–Weinberg Equilibrium | 哈迪–温伯格平衡
For a large, randomly mating population with no selection, mutation or gene flow, allele and genotype frequencies remain constant. For a gene with two alleles A and a (frequencies p and q), p + q = 1. The genotype frequencies are predicted by p² (AA), 2pq (Aa) and q² (aa). The theorem is used to detect microevolution: if observed genotype frequencies deviate from Hardy–Weinberg expectations, one of the equilibrium assumptions may be violated.
在一个足够大、随机交配、没有选择、突变和基因流动的种群中，等位基因频率和基因型频率将保持恒定。对于具有两个等位基因 A 和 a 的基因（频率分别为 p 和 q），p + q = 1。预期基因型频率为：AA = p²，Aa = 2pq，aa = q²。该定理用于检测微进化：若实际基因型频率偏离哈迪–温伯格预期，则可能违反了一个或多个平衡假设。
p + q = 1 | p² + 2pq + q² = 1
4. Population Growth Models | 种群增长模型
Exponential growth occurs when resources are unlimited: dN/dt = rN, where N is population size and r is the intrinsic growth rate. This gives Nt = N0 ert. In nature, resources impose carrying capacity K, leading to logistic growth: dN/dt = rN (K − N) ÷ K. The per capita growth rate declines as N approaches K, producing an S-shaped sigmoid curve.
在资源无限时，种群呈指数增长：dN/dt = rN，其中 N 为种群数量，r 为内禀增长率，由此得出 Nt = N0 ert。自然界中，资源限制设定了环境容纳量 K，种群呈现逻辑斯谛增长：dN/dt = rN (K − N) ÷ K。随着 N 接近 K，每个个体的增长率下降，形成 S 型生长曲线。
Exponential: Nt = N0 ert | Logistic: dN/dt = rN (K − N) ÷ K
5. Mark–Release–Recapture (Lincoln Index) | 标记–释放–再捕捉（林肯指数）
Used to estimate mobile animal population size (N). A sample of M individuals is caught, marked and released. Later, a second sample of C individuals is captured; let R be the number that are already marked. The estimate is N = (M × C) ÷ R. This assumes the marked individuals mix randomly, marks are not lost, and no births, deaths or migration affect the population proportion between samples.
此法用于估算移动动物种群的大小（N）。先捕捉 M 个个体，标记后释放。随后再捕获 C 个个体，计其中已标记的为 R。估算公式为 N = (M × C) ÷ R。假设前提包括：标记个体与未标记个体随机混合，标记不丢失，且两次取样间没有出生、死亡或迁入迁出影响比例。
N = (M × C) ÷ R
6. Probability and Genetic Ratios | 概率与遗传比率
Mendelian inheritance follows the product rule (multiply independent probabilities) and the sum rule (add mutually exclusive probabilities). For a dihybrid cross AaBb × AaBb, the probability of aabb offspring is ¼ × ¼ = 1/16. The binomial theorem can also be used: the probability of exactly k successes in n trials is C(n, k) × pk × (1−p)n−k. These principles underpin pedigree analysis and risk assessment.
孟德尔遗传遵循乘法法则（独立事件概率相乘）和加法法则（互斥事件概率相加）。对于双杂合子杂交 AaBb × AaBb，出现 aabb 后代的概率为 ¼ × ¼ = 1/16。二项式定理可用于计算：n 次独立试验中恰有 k 次成功的概率为 C(n, k) × pk × (1−p)n−k。这些原理是系谱分析和遗传风险评估的基础。
Product rule: P(A and B) = P(A) × P(B) | Sum rule: P(A or B) = P(A) + P(B)
7. Cellular Respiration and ATP Yield | 细胞呼吸与ATP产量
Aerobic respiration produces a theoretical maximum of 38 ATP per glucose in prokaryotes, and typically 36 in eukaryotes due to mitochondrial transport costs. Glycolysis yields 2 ATP and 2 NADH; the link reaction and Krebs cycle produce 2 ATP, 6 NADH and 2 FADH₂ per glucose; the electron transport chain generates ~34 ATP. Anaerobic respiration yields only 2 ATP from glycolysis.
有氧呼吸在原核细胞中每分子葡萄糖理论上最多产生 38 个 ATP，在真核细胞中由于线粒体转运消耗，通常为 36 个 ATP。糖酵解净产 2 ATP 和 2 NADH；连接反应与三羧酸循环每葡萄糖分子产生 2 ATP、6 NADH 和 2 FADH₂；电子传递链则产生约 34 ATP。无氧呼吸仅在糖酵解阶段产生 2 ATP。
Aerobic (eukaryote): C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ~36 ATP
8. Photosynthetic Efficiency | 光合作用效率
The efficiency of photosynthesis can be expressed as the percentage of light energy converted into chemical energy in biomass. Gross primary productivity (GPP) is total photosynthetic energy fixed; net primary productivity (NPP) = GPP − respiratory loss (R). Photosynthetic efficiency = (GPP ÷ total light energy incident on the plant) × 100. Crop breeders aim to improve this efficiency by increasing light capture and CO₂ fixation.
光合作用效率可用光能转化为生物质中化学能的百分比表示。总初级生产力（GPP）为固定的光合总能量；净初级生产力（NPP） = GPP − 呼吸消耗（R）。光合效率 = （GPP ÷ 照射到植物上的总光能）× 100。作物育种通过提高光捕获和二氧化碳固定来提升此效率。
NPP = GPP − R | Efficiency = (GPP ÷ Incident light energy) × 100%
9. Ecological Energy Transfer | 生态能量传递
Energy transfer between trophic levels is roughly 10% efficient on average (Lindeman efficiency). The efficiency is calculated as (energy ingested at trophic level n+1 ÷ energy ingested at level n) × 100, or by using biomass. Pyramid of energy is always upright. The low efficiency constrains food chain length and explains why large predators are rare.
营养级之间的能量传递效率平均约为 10%（林德曼效率）。计算公式为：（n+1 营养级摄入的能量 ÷ n 营养级摄入的能量）× 100，也可用生物量计算。能量金字塔始终呈正立形。低传递效率限制了食物链长度，并解释了为什么大型捕食者数量稀少。
Transfer efficiency = (Energy at level n+1 ÷ Energy at level n) × 100%
10. Water Potential and Osmosis | 水分势与渗透
Water moves from a region of higher water potential (ψ) to a region of lower water potential. The water potential of pure water at atmospheric pressure is zero. In plant cells, ψ = ψs + ψp, where ψs is solute (osmotic) potential and ψp is pressure potential. Plasmolysis occurs when ψ of the external solution is lower than that of the cell, causing water loss and cell membrane detachment from the cell wall.
水分总是从水分势（ψ）较高的区域向较低的区域移动。纯水在大气压下的水分势为零。植物细胞中，ψ = ψs + ψp，其中 ψs 为溶质（渗透）势，ψp 为压力势。当外界溶液的水分势低于细胞时，会发生质壁分离，即细胞失水，细胞膜与细胞壁脱离。
ψ = ψs + ψp
11. Gene Frequencies and Evolutionary Change | 基因频率与进化变化
Evolution is defined as changes in allele frequencies within a gene pool over time. The frequency of an allele is calculated as (number of copies of that allele ÷ total number of all alleles) × 100. Mechanisms changing frequencies include natural selection, genetic drift, gene flow and mutation. The Hardy–Weinberg theorem provides a null model; any departure indicates that evolution is occurring.
进化被定义为基因库中等位基因频率随时间的变化。某等位基因的频率 = （该等位基因拷贝数 ÷ 所有等位基因总数）× 100。改变频率的机制包括自然选择、遗传漂变、基因流动和突变。哈迪–温伯格定理提供了一个零模型；任何偏离都表明进化正在发生。
Allele frequency = (Number of allele copies ÷ Total alleles) × 100%
12. Statistical Measures in Biology | 生物学中的统计度量
Standard deviation (s) quantifies the spread of data around the mean: s = √[Σ(x − x̄)² ÷ (n−1)]. Standard error of the mean is s ÷ √n and indicates the precision of the mean estimate. Correlation coefficients (r) measure linear association between two variables, with values ranging from −1 to +1. Student’s t‑test compares two sample means; the formula t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂) assumes equal variances.
标准差（s）衡量数据围绕均值的离散程度：s = √[Σ(x − x̄)² ÷ (n−1)]。均值的标准误为 s ÷ √n，表示均值估计的精确度。相关系数（r）度量两个变量间的线性关联，取值范围为 −1 到 +1。学生 t 检验用于比较两个样本均值；在方差齐性假设下，公式为 t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂)。
Standard deviation s = √[Σ(x − x̄)² ÷ (n−1)] | Standard error = s ÷ √n
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