📚 Quick Reference Handbook: Formulas and Theorems for Year 13 SQA Biology | SQA 生物公式定理速查手册
This article compiles all essential formulas, theorems and statistical tools required for Year 13 SQA Biology. Mastering these equations will help you solve numerical problems, interpret experimental data and deepen your understanding of key biological concepts such as population genetics, energy flow and respiration.
本文汇编 SQA 高年级生物所需的核心公式、定理与统计工具。掌握这些等式将帮助你解决计算题、解读实验数据，并加深对种群遗传学、能量流动和呼吸作用等关键生物学概念的理解。
1. Hardy-Weinberg Principle | 哈代-温伯格定律
The Hardy-Weinberg principle states that allele and genotype frequencies in a large, randomly mating population remain constant from generation to generation provided there is no mutation, gene flow, genetic drift or natural selection.
哈代-温伯格定律指出，在一个较大的随机交配种群中，只要没有突变、基因流动、遗传漂变和自然选择，等位基因频率和基因型频率就会在世代间保持恒定。
For a gene with two alleles, let p = frequency of the dominant allele A and q = frequency of the recessive allele a. The sum of the allele frequencies must equal 1.
对于一个具有两个等位基因的基因，设 p = 显性等位基因 A 的频率，q = 隐性等位基因 a 的频率。等位基因频率之和必定为 1。
p + q = 1
The expected genotype frequencies under random mating are then p² (homozygous dominant AA), 2pq (heterozygous Aa) and q² (homozygous recessive aa). The equation linking them is:
随机交配下的预期基因型频率分别为 p²（显性纯合子 AA）、2pq（杂合子 Aa）和 q²（隐性纯合子 aa）。它们之间的关系为：
p² + 2pq + q² = 1
If the observed genotype frequencies differ significantly from these expectations, the population may be evolving. The chi-squared test is often used to compare observed and expected numbers.
如果观察到的基因型频率与上述预期有显著差异，则该种群可能正在进化。通常使用卡方检验来比较观察数与期望数。
2. Chi-Squared (χ²) Test | 卡方检验
The chi-squared test determines whether there is a significant difference between observed (O) and expected (E) frequencies in categorical data. The test statistic is given by:
卡方检验用于判断分类数据中观察频数（O）与期望频数（E）之间是否存在显著差异。其检验统计量为：
χ² = Σ (O – E)² / E
A larger χ² value indicates a greater discrepancy between observed and expected results. The calculated χ² is compared against a critical value from the chi-squared distribution table at the chosen significance level (typically 0.05) and appropriate degrees of freedom (df).
χ² 值越大，表明观察值与期望值之间的差异越大。将计算出的 χ² 与所选显著性水平（通常为 0.05）和相应自由度（df）下的临界值进行比较。
Degrees of freedom for a single-category test = number of categories – 1. For example, in a Hardy-Weinberg analysis with three genotypic classes, df = 2.
单因素检验的自由度 = 类别数 – 1。例如，在具有三种基因型的哈代-温伯格分析中，自由度为 2。
If χ²calc > χ²crit, the null hypothesis is rejected, meaning the difference is statistically significant.
若 χ²calc > χ²crit，则拒绝原假设，表明差异具有统计学意义。
3. Standard Error and Confidence Intervals | 标准误差与置信区间
The standard error (SE) estimates how far a sample mean is likely to be from the true population mean. It is calculated from the sample standard deviation (s) and sample size (n).
标准误差（SE）用于估计样本均值与真实总体均值之间的可能偏差，由样本标准差（s）和样本容量（n）计算得出。
SE = s / √n
When the sample size is large (n ≥ 30), the 95% confidence interval for the population mean is approximately:
当样本容量较大（n ≥ 30）时，总体均值的 95% 置信区间近似为：
95% CI = sample mean ± 1.96 × SE
If two sets of data are compared and their 95% confidence intervals do not overlap, it suggests a significant difference between the means at the 5% level.
若两组数据的 95% 置信区间不重叠，则表明在 5% 的显著性水平上，两均值之间存在显著差异。
4. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient indicates the type of substrate being respired. It is the ratio of the volume of carbon dioxide produced to the volume of oxygen consumed over a given time.
呼吸商可用于判断呼吸作用消耗的底物类型，它是单位时间内产生的二氧化碳体积与消耗的氧气体积之比。
RQ = Volume of CO₂ produced / Volume of O₂ consumed
Typical RQ values vary with the substrate:
不同底物的典型 RQ 值如下：
| Substrate / 底物 | Respiratory Quotient (RQ) / 呼吸商 |
|---|---|
| Carbohydrate / 碳水化合物 | 1.0 |
| Fat / 脂肪 | 0.7 |
| Protein / 蛋白质 | 0.9 |
An RQ close to 1.0 suggests carbohydrate respiration; values near 0.7 indicate fat metabolism. Mixed diets give intermediate RQs.
RQ 接近 1.0 表明以碳水化合物呼吸为主；接近 0.7 则提示脂肪代谢；混合膳食对应的 RQ 介于两者之间。
5. Net Primary Productivity (NPP) | 净初级生产力
Net primary productivity is the rate at which plants store chemical energy in their biomass after accounting for energy lost through respiration. It is the energy available to the next trophic level.
净初级生产力是指植物在扣除呼吸作用消耗的能量后，将化学能固定在生物量中的速率，也是可供下一营养级利用的能量。
NPP = GPP – R
where GPP is gross primary productivity (total light energy converted into chemical energy by photosynthesis) and R is the energy used in plant respiration.
式中，GPP 为总初级生产力（光合作用将光能转化为化学能的总量），R 为植物呼吸作用消耗的能量。
NPP is typically expressed in units such as kJ m⁻² yr⁻¹ or g biomass m⁻² yr⁻¹. The efficiency of photosynthesis can be calculated as (GPP / light energy absorbed) × 100.
NPP 通常以 kJ m⁻² yr⁻¹ 或 g 生物量 m⁻² yr⁻¹ 为单位表示。光合作用效率可按 (GPP / 吸收的光能) × 100 计算。
6. Energy Transfer Efficiency | 能量传递效率
When energy flows from one trophic level to the next, a large proportion is lost as heat, uneaten parts and waste. The efficiency of energy transfer is expressed as a percentage.
能量从一个营养级流向下一营养级时，大部分以热量、未食部分和排泄物的形式散失。能量传递效率用百分数表示。
Efficiency (%) = (Energy in higher trophic level / Energy in lower trophic level) × 100
In most ecosystems, the transfer efficiency averages about 10%, though it can range from 5% to 20%. This low efficiency limits the length of food chains.
大多数生态系统的能量传递效率约为 10%，但可在 5% – 20% 之间变化。这种低效率限制了食物链的长度。
Ecologists often use ecological pyramids to represent energy, biomass or numbers. The efficiency calculation helps explain why top predators are fewer in number.
生态学家常用生态金字塔表示能量、生物量或数量。效率计算有助于解释为什么顶级捕食者数量稀少。
7. Genetic Probability Rules | 遗传概率法则
Monohybrid and dihybrid crosses rely on the basic rules of probability. The product rule states that the probability of two independent events both occurring is the product of their individual probabilities.
单基因杂交和双基因杂交依赖概率的基本法则。乘积法则指出，两个独立事件同时发生的概率等于它们各自概率的乘积。
P(A and B) = P(A) × P(B)
The sum rule applies to mutually exclusive events: the probability that either of two mutually exclusive events occurs is the sum of their individual probabilities.
加和法则适用于互斥事件：两个互斥事件之一发生的概率等于它们各自概率的和。
P(A or B) = P(A) + P(B)
Punnett squares apply these rules systematically. For example, in a cross between two heterozygous individuals (Aa × Aa), the probability of a homozygous recessive offspring (aa) is q² = ¼.
庞纳特方格系统地应用了这些法则。例如，两个杂合子（Aa × Aa）杂交，得到隐性纯合子后代（aa）的概率为 q² = ¼。
8. Magnification and Actual Size | 放大率与实际尺寸
When using a light microscope or an electron micrograph, it is essential to relate image size, actual size and magnification.
使用光学显微镜或电子显微照片时，必须建立图像尺寸、实际尺寸与放大率之间的关系。
Magnification = Image size / Actual size
To find the actual size of a specimen, rearrange the formula: Actual size = Image size / Magnification. Ensure both measurements use the same units (e.g. micrometres).
计算样本的实际尺寸时，可变换公式：实际尺寸 = 图像尺寸 / 放大率。确保两个测量值使用相同的单位（例如微米）。
The magnitude of a cell or organelle is often expressed as the ‘order of magnitude’—a power of ten indicating its scale relative to another object.
细胞或细胞器的大小常用“数量级”表示——即以 10 的幂次来表达其相对于另一物体的尺度。
9. Percentage Change and Rate Calculations | 百分比变化与速率计算
Many SQA Biology questions require you to quantify how a variable changes over time or in response to a treatment. The percentage change formula is:
许多 SQA 生物试题要求你量化某个变量随时间或处理而发生的变化。百分比变化的计算公式为：
Percentage change (%) = ((Final value – Initial value) / Initial value) × 100
A positive percentage indicates an increase; a negative percentage indicates a decrease.
正百分比表示增加，负百分比表示减少。
To describe the rate of a biological process, use the rate formula:
描述生物过程的速率时，使用速率公式：
Rate = Change in quantity / Time taken
Applications include enzyme reaction rates (e.g. cm³ O₂ produced per minute), population growth rate, and the rate of water uptake by a plant.
应用实例包括酶反应速率（如每分钟产生的 O₂ 体积 cm³）、种群增长速率和植物的吸水速率。
10. Population Growth Rate | 种群增长率
Populations change size because of births, deaths, immigration and emigration. The simple population growth rate over a period can be expressed as:
种群大小因出生、死亡、迁入和迁出而变化。一段时间内的简单种群增长率可表示为：
Population growth rate = (Births – Deaths) / Initial population size
For a closed population (no migration), this simplifies to (b – d) × 100%, where b = birth rate and d = death rate.
对于封闭种群（无迁移），可简化为 (b – d) × 100%，其中 b = 出生率，d = 死亡率。
When resources are unlimited, populations can grow exponentially, described by the equation N(t) = N₀ × 2^(t/T) (where T is the doubling time). In reality, limiting factors produce logistic (S-shaped) growth that plateaus at carrying capacity.
当资源不受限制时，种群可呈指数增长，公式为 N(t) = N₀ × 2^(t/T)（T 为倍增时间）。实际上，限制因素会产生逻辑斯蒂（S 形）增长，最终在环境容纳量处达到平稳。
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