📚 Year 13 CCEA Biology: Formula & Theorem Quick Reference Handbook | Year 13 CCEA 生物：公式定理速查手册
This quick reference handbook consolidates the essential mathematical formulas and biological theorems required for Year 13 CCEA Biology. Mastering these equations is crucial for quantitative analysis in genetics, ecology, physiology, and data handling. Each entry includes the formula, symbol definitions, and practical tips for use in examinations.
本速查手册汇总了 CCEA 生物学 Year 13 所需的关键数学公式和生物学定理。掌握这些公式对于遗传学、生态学、生理学和数据处理中的定量分析至关重要。每个条目都包括公式、符号定义以及考试应用技巧。
1. Magnification and Size Calculations | 放大率与尺寸计算
M = I ÷ A
The magnification (M) of a biological drawing or micrograph equals the image size (I) divided by the actual specimen size (A). Both I and A must be expressed in the same units, typically micrometres (µm) or millimetres (mm).
生物图或显微照片的放大率 (M) 等于图像尺寸 (I) 除以实际标本尺寸 (A)。I 和 A 必须采用相同单位，通常为微米 (µm) 或毫米 (mm)。
A = I ÷ M and I = M × A
Rearrange the basic formula to calculate either actual size or image size. Remember to convert units: 1 mm = 1000 µm. Always show unit conversions clearly in your working.
可依据基本公式变形计算实际尺寸或图像尺寸。记得转换单位：1 mm = 1000 µm。解题时必须清晰展示单位转换步骤。
For scale bars, measure the bar length in the image, determine its stated actual length, and use the magnification formula to verify or find the magnification.
对于比例尺，先测量图中比例尺的长度，确定其标注的实际长度，再代入放大率公式验证或求取放大率。
2. Hardy-Weinberg Principle | 哈代-温伯格定理
p + q = 1 and p² + 2pq + q² = 1
The Hardy-Weinberg principle predicts allele and genotype frequencies in a large, randomly mating population that is not subject to mutation, gene flow or natural selection. Here p = frequency of the dominant allele, q = frequency of the recessive allele.
哈代-温伯格定理预测理想大群体（随机交配、无突变、无基因流、无自然选择）中的等位基因频率和基因型频率。其中 p 为显性等位基因频率，q 为隐性等位基因频率。
p² represents the homozygous dominant genotype frequency, 2pq the heterozygous frequency, and q² the homozygous recessive frequency. You can calculate allele frequencies from observed recessive phenotypes and then test for equilibrium.
p² 代表显性纯合子基因型频率，2pq 代表杂合子频率，q² 代表隐性纯合子频率。可从观察到的隐性表现型频率推算等位基因频率，进而检验群体是否平衡。
This principle is used to estimate carrier frequencies for genetic conditions and to detect evolutionary change when observed genotype frequencies differ significantly from expected.
该定理常用于估算遗传病携带者频率，并在观察基因型频率显著偏离预期值时检测进化改变。
3. Chi-Squared (χ²) Test | 卡方 (χ²) 检验
χ² = ∑ (O − E)² / E
The chi-squared test compares observed (O) and expected (E) categorical data to determine if any deviation is statistically significant. Calculate E based on the null hypothesis, then subtract E from each O, square the difference, divide by E, and sum across all categories.
卡方检验比较观察频数 (O) 与期望频数 (E) 的类别数据，以判断偏差是否具有统计学显著性。根据原假设算出期望值，再将每个观察值与期望值的差平方后除以期望值，最后对所有类别求和。
Degrees of freedom (df) usually equal the number of categories minus 1 for simple goodness-of-fit tests. Compare your calculated χ² to the critical value at p = 0.05 and the appropriate df; if χ² > critical value, reject the null hypothesis.
简单拟合优度检验的自由度 (df) 通常等于类别数减 1。将计算所得的 χ² 值与 p=0.05 及相应自由度下的临界值比较；若 χ² 大于临界值，则拒绝原假设。
Common uses in CCEA include testing Mendelian ratios, population genetics, and ecological distribution patterns.
在 CCEA 考试中常用于检验孟德尔比率、群体遗传学以及生态分布模式。
4. Net Primary Production (NPP) | 净初级生产力
NPP = GPP − R
Net primary production is the energy or biomass that remains in producers after respiratory losses (R) are subtracted from gross primary production (GPP). It represents the energy available to primary consumers in an ecosystem.
净初级生产力是总初级生产力 (GPP) 减去生产者呼吸消耗 (R) 后剩余的能量或生物量，代表生态系统中可供初级消费者利用的能量。
NPP is usually expressed in units of energy per area per year (kJ m⁻² yr⁻¹) or biomass per area per year (g m⁻² yr⁻¹). You may be required to calculate missing values in simple food chain energy budgets.
NPP 通常以每面积每年的能量 (kJ m⁻² yr⁻¹) 或生物量 (g m⁻² yr⁻¹) 表示。考试可能要求补全简单食物链能流预算中的缺失值。
High NPP indicates productive ecosystems such as tropical rainforests; low NPP is typical of desert or arctic environments.
高 NPP 指示生产力强的生态系统，如热带雨林；低 NPP 常见于荒漠或极地环境。
5. Respiratory Quotient (RQ) | 呼吸商
RQ = CO₂ produced / O₂ consumed
The respiratory quotient is the ratio of the volume of carbon dioxide released to the volume of oxygen absorbed during respiration. It gives insight into the metabolic substrate being oxidised.
呼吸商是呼吸作用中释放的二氧化碳体积与吸收的氧气体积之比，可揭示被氧化的代谢底物类型。
Typical RQ values: carbohydrate ≈ 1.0, lipid ≈ 0.7, protein ≈ 0.9. Mixed diets yield intermediate values. In anaerobic respiration, RQ becomes very large or even immeasurable owing to lack of O₂ consumption.
典型 RQ 值：碳水化合物约为 1.0，脂质约为 0.7，蛋白质约为 0.9。混合饮食会产生中间值。在无氧呼吸中，因不消耗氧气，RQ 会变得极大甚至无法测量。
Use RQ data from respirometer investigations or provided tables to deduce the likely respiratory substrate. Always record gas volumes at the same temperature and pressure.
利用呼吸计实验或所给表格中的 RQ 数据可推断最可能的呼吸底物。记录气体体积时务必确保温度和压强一致。
6. Ecological Efficiency | 生态效率
Efficiency (%) = (Energy in higher trophic level / Energy in lower trophic level) × 100
Ecological efficiency measures the percentage of energy transferred from one trophic level to the next. It is commonly calculated between producers and primary consumers, or between primary and secondary consumers.
生态效率衡量能量从一个营养级传递到下一营养级的百分比，通常在生产者与初级消费者之间或初级与次级消费者之间计算。
Typical transfer efficiencies range from 5% to 20%, with most energy lost through respiration, excretion, and uneaten material. You may also calculate production efficiency as (Net production / Assimilation) × 100.
典型传递效率在 5 % 至 20 % 之间，大部分能量因呼吸、排泄和未取食物料而损耗。也可计算生产效率 = (净生产量 / 同化量) × 100。
In CCEA exams you might be given energy flow diagrams and asked to compute missing percentages or draw pyramid-based conclusions about ecosystem stability.
在 CCEA 考试中，可能会给出能量流动示意图，要求计算缺失的百分比或根据能量金字塔推断生态系统稳定性。
7. Lincoln Index (Mark–Release–Recapture) | 林肯指数（标记-释放-重捕法）
N = (M × C) / R
The Lincoln Index estimates the total population size (N) of mobile organisms. M is the number initially captured, marked and released; C is the total number captured in a second sample; R is the number of marked individuals recaptured in that second sample.
林肯指数用于估算移动生物的总种群数量 (N)。M 为首次捕获、标记并释放的个体数；C 为第二次样本中捕获的总数；R 为第二次捕获中带有标记的个体数。
Assumptions include: marks are not lost, marked individuals mix randomly with the population, no significant births, deaths, immigration or emigration between samples, and marking does not affect survival or catchability.
该方法假设：标记不会脱落、标记个体与种群随机混合、两次取样间无大量出生、死亡、迁入或迁出，且标记不影响存活率或可捕性。
Calculate N and round to a sensible number of organisms. When the recapture rate R is very small, reliability decreases, so consider ethical and practical limitations.
算出 N 并四舍五入至合理的个体数。当重捕率 R 极低时，估算可靠性下降，需考虑伦理与实际限制。
8. Standard Deviation and Standard Error | 标准差与标准误
s = √[ ∑(x − x̄)² / (n − 1) ]
The sample standard deviation (s) measures the spread of data points around the sample mean (x̄). Each deviation (x − x̄) is squared, summed, divided by n−1 (degrees of freedom), and then square rooted.
样本标准差 (s) 衡量数据点围绕样本均值 (x̄) 的离散程度。将每个偏差 (x − x̄) 平方后求和，除以自由度 n−1，再开平方即得。
SE = s / √n
The standard error (SE) estimates the precision of the sample mean as an estimate of the true population mean. A smaller SE indicates a more reliable mean.
标准误 (SE) 评估样本均值作为总体均值估计值的精确度。SE 越小表示均值越可靠。
In practical work, plot mean values with error bars of ±1 SE to visually compare data sets; non-overlapping bars often suggest significant difference.
在实践工作中，用 ±1 SE 误差棒绘制均值图可直观比较数据集；不重叠的误差棒通常提示存在显著差异。
9. Student’s t-Test | 学生氏 t 检验
t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
The unpaired (independent) t-test compares the means of two unrelated samples to determine if they are significantly different. x̄₁ and x̄₂ are the sample means, s₁² and s₂² the sample variances, and n₁ and n₂ the sample sizes.
不配对（独立）t 检验比较两个不相关样本的均值，判断差异是否显著。x̄₁ 和 x̄₂ 为样本均值，s₁² 和 s₂² 为样本方差，n₁ 和 n₂ 为样本大小。
Calculate the t-statistic and compare it with the critical t-value from a statistical table, using the appropriate degrees of freedom (often approximated by (n₁ − 1) + (n₂ − 1) or Welch’s correction). If t-calculated > t-critical, reject the null hypothesis.
计算 t 统计量并与统计表中相应自由度（通常近似为 (n₁−1)+(n₂−1) 或 Welch 校正）下的临界 t 值比较。若 t 计算值大于临界值，则拒绝原假设。
Common applications in CCEA practicals include comparing plant growth under two light intensities or enzyme activity at two pH levels.
CCEA 实践中常见应用包括比较两种光照强度下的植物生长或两种 pH 下的酶活性。
10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数
rₛ = 1 − [ 6 ∑d² ] / [ n (n² − 1) ]
Spearman’s rank correlation tests the strength and direction of association between two ordinal or non-normally distributed variables. Rank each set of data, find the difference (d) between paired ranks, square these differences, and sum them.
斯皮尔曼秩相关检验两个定序或非正态分布变量之间关联的强度和方向。将每组数据排秩，计算配对秩次差 (d)，将其平方并求和，然后代入公式。
The coefficient rₛ ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation); 0 indicates no monotonic association. Compare the calculated rₛ with a critical value at p = 0.05 for the given sample size n.
系数 rₛ 取值从 −1（完全负相关）到 +1（完全正相关）；0 表示无单调关联。将计算所得的 rₛ 与给定样本量 n 下 p=0.05 的临界值比较。
Ideal for CCEA fieldwork correlations, e.g. light intensity vs. distribution of shade plants, or soil moisture vs. species richness.
非常适合 CCEA 野外调查相关性分析，如光照强度与阴生植物分布的关系，或土壤湿度与物种丰富度的关系。
11. Population Growth Models | 种群增长模型
Exponential: dN/dt = rN
In unlimited environments, populations grow exponentially: the rate of change in population size (dN/dt) is the product of the intrinsic rate of increase (r) and the current population size (N). This produces a J-shaped curve.
在无限环境中，种群呈指数增长：种群大小变化率 (dN/dt) 等于内禀增长率 (r) 与当前种群大小 (N) 的乘积，形成 J 形曲线。
Logistic: dN/dt = rN [ (K − N) / K ]
When resources are limited, growth follows a logistic model. Here K is the carrying capacity of the environment. As N approaches K, the growth rate slows, producing an S-shaped (sigmoid) curve.
当资源有限时，增长遵循逻辑斯蒂模型。K 为环境容纳量。随着 N 接近 K，增长率下降，形成 S 形（Sigmoid）曲线。
Understanding these models helps interpret yeast population curves or bacterial growth in nutrient broth experiments commonly analysed in CCEA papers.
理解这些模型有助于解读 CCEA 试卷中常涉及的酵母种群曲线或营养肉汤中细菌生长实验。
12. Water Potential (Ψ) | 水势
Ψ = Ψₛ + Ψₚ
Water potential (Ψ) of a plant cell is the sum of the solute (osmotic) potential (Ψₛ) and the pressure potential (Ψₚ). Water moves from regions of higher (less negative) water potential to regions of lower (more negative) water potential.
植物细胞的水势 (Ψ) 是溶质势（渗透势，Ψₛ）与压力势 (Ψₚ) 之和。水分从水势较高（负值较大）的区域向水势较低（负值较小）的区域移动。
Ψₛ = − i C R T
The solute potential of a solution can be estimated with the van’t Hoff relation: i is the ionisation constant (e.g. 1 for sucrose, 2 for NaCl), C is the molar concentration (mol dm⁻³), R is the pressure constant (0.00831 kPa dm³ mol⁻¹ K⁻¹), and T is the absolute temperature in kelvin.
溶液溶质势可用范特霍夫公式估算：i 为电离常数（如蔗糖为 1，NaCl 为 2），C 为摩尔浓度 (mol dm⁻³)，R 为压力常数 (0.00831 kPa dm³ mol⁻¹ K⁻¹)，T 为绝对温度 (K)。
At incipient plasmolysis, Ψₚ = 0, so Ψ = Ψₛ. You may be asked to calculate water potential values from given temperature and concentration data, or to predict the direction of water movement.
在初始质壁分离时，Ψₚ = 0，故 Ψ = Ψₛ。考试可能要求根据给定的温度和浓度数据计算水势值，或预测水分运动方向。
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