📚 IB CCEA Biology: Formula Summary Handbook | IB CCEA 生物：公式汇总手册
Mastering the key mathematical formulas in CCEA Biology is essential for success in both practical examinations and written questions. This handbook compiles the most frequently used equations, offering clear explanations and worked examples to help you apply them with confidence. From microscope calibration to population genetics, each section breaks down a formula into manageable steps.
掌握 CCEA 生物中的关键数学公式对于实验考试和笔试都至关重要。本手册汇总了最常用的方程，提供清晰的解释和计算示例，帮助你自信地应用它们。从显微镜校准到种群遗传学，每个部分都将公式拆解为易于掌握的步骤。
1. Microscope Calculations | 显微镜计算
The relationship between magnification, image size and actual size is fundamental in microscopy. Always remember the triangle: Magnification (M) equals Image size (I) divided by Actual size (A).
放大倍数、图像大小与实际大小之间的关系是显微镜学的基础。牢记这个三角关系：放大倍数（M）等于图像大小（I）除以实际大小（A）。
Magnification = Image size / Actual size (M = I / A)
. Ensure all measurements are in the same units; convert millimetres to micrometres by multiplying by 1000.
要计算样本的实际大小，可将公式变形为 A = I / M。确保所有测量单位一致；将毫米转换为微米时需乘以 1000。
2. Percentage Change and Rate | 百分比变化与速率
Percentage change is widely used to compare differences before and after a treatment. It is calculated by dividing the change in value by the original value, then multiplying by one hundred.
百分比变化广泛用于比较处理前后的差异。其计算方法是用数值变化量除以原始数值，再乘以一百。
Percentage Change = (Final Value – Initial Value) / Initial Value × 100%
The rate of a reaction measures how quickly a product is formed or a substrate is consumed. It is often expressed as change in quantity per unit time, for example cm³ of gas produced per minute.
反应速率衡量产物生成或底物消耗的快慢。它通常表示为每单位时间的变化量，例如每分钟产生气体的 cm³ 数。
Rate = Change in Quantity / Time Taken
3. Respiratory Quotient (RQ) | 呼吸商 (RQ)
The respiratory quotient reveals which respiratory substrate is being metabolised. It is the ratio of carbon dioxide produced to oxygen consumed over a given period.
呼吸商可揭示正在被代谢的呼吸底物。它是一定时间内产生的二氧化碳与消耗的氧气之比。
RQ = CO₂ Produced / O₂ Consumed
An RQ of 1.0 typically indicates carbohydrate respiration, while fat metabolism yields values around 0.7. Protein respiration gives an RQ of approximately 0.9. Always make sure gas volumes are corrected to standard temperature and pressure if required.
RQ 为 1.0 通常表示糖类呼吸，脂肪代谢产生的数值约为 0.7，而蛋白质呼吸的 RQ 约为 0.9。如有需要，务必确保气体体积已校正至标准温度和压力。
4. Photosynthesis Rate Calculations | 光合作用速率计算
The rate of photosynthesis can be estimated by measuring oxygen production or carbon dioxide uptake. Net photosynthesis equals gross photosynthesis minus respiration losses.
光合作用速率可通过测量氧气产生量或二氧化碳吸收量来估算。净光合作用等于总光合作用减去呼吸消耗。
Net Photosynthesis = Gross Photosynthesis – Respiration
When investigating light intensity, the rate is often expressed as 1 / time to reach a standard indicator change. For submerged plants, count bubbles or use a gas syringe to measure volume per minute.
当研究光强时，光合速率通常表示为达到标准指示剂变色的时间倒数。对于沉水植物，可计数气泡数或使用气体注射器测量每分钟的体积。
5. Population Sampling (Lincoln Index) | 种群取样（林肯指数）
The Lincoln Index estimates the size of a motile population using mark-release-recapture data. It assumes random mixing and no significant births, deaths or migration during the study.
林肯指数利用标记-释放-重捕数据估算行动中的种群大小。它假设个体随机混合，且研究期间无显著的出生、死亡或迁移。
N = (M × C) / R
Here N is the estimated total population, M the number marked initially, C the total caught in the second sample and R the number of marked individuals recaptured. This method is ideal for estimating fish populations in a lake.
其中 N 为估算的种群总数，M 为首次标记的个体数，C 为第二次捕获的总数，R 为重捕的标记个体数。此方法适用于估算湖泊中的鱼类种群。
6. Dilutions and Concentration | 稀释与浓度
Serial dilutions are routinely used to prepare standard solutions for calibration curves. The dilution equation links the concentration and volume of a stock solution to those of the diluted solution.
系列稀释常用于制备校准曲线的标准溶液。稀释方程将储备液与稀释溶液的浓度和体积联系起来。
C₁ V₁ = C₂ V₂
C₁ and V₁ represent the original concentration and volume, while C₂ and V₂ refer to the desired dilute solution. For a 1 in 10 dilution, simply mix 1 part stock with 9 parts solvent. Always label tubes clearly and use appropriate pipettes.
C₁ 和 V₁ 代表初始浓度和体积，C₂ 和 V₂ 指所需的稀释后溶液。进行 1:10 稀释时，只需将 1 份原液与 9 份溶剂混合。务必清晰标记试管并使用合适的移液器。
7. Chi-Squared Test | 卡方检验
The chi-squared (χ²) test determines whether the difference between observed and expected results is significant. It is essential for analysing genetic crosses or ecological distributions.
卡方（χ²）检验用于判断观测值与预期值之间的差异是否显著。它在分析遗传杂交或生态分布时不可或缺。
χ² = Σ (O – E)² / E
O stands for observed frequency and E for expected frequency. The calculated χ² value is compared against a critical value from the chi-squared distribution table using degrees of freedom (n – 1). If χ² > critical value, reject the null hypothesis.
O 代表观测频率，E 代表预期频率。计算出的 χ² 值需与卡方分布表中给定自由度（n – 1）下的临界值进行比较。若 χ² 大于临界值，则拒绝原假设。
8. Standard Deviation & Standard Error | 标准差与标准误差
Standard deviation quantifies the spread of data around the mean. A small SD indicates data points are clustered closely, while a large SD signals wide dispersion.
标准差量化了数据围绕均值的分布情况。很小的 SD 表示数据点紧密聚集，很大的 SD 则表明数据分散较广。
SD = √[ Σ (x – x̄)² / (n – 1) ]
Standard error of the mean shows how accurately the sample mean estimates the population mean. It decreases as sample size increases.
均值的标准误差显示了样本均值对总体均值的估计精度。样本量越大，标准误差越小。
SE = SD / √n
ping error bars often suggest no significant difference, although a statistical test provides a more rigorous conclusion.
在图表中绘制均值时，添加表示 ±1 SE 的误差棒。误差棒重叠通常提示无显著差异，但统计检验可以给出更严谨的结论。
9. Hardy-Weinberg Equilibrium | 哈迪–温伯格平衡
The Hardy-Weinberg principle predicts allele frequencies in a non-evolving population. It provides a useful null model for detecting evolutionary change.
哈迪–温伯格定律可预测非进化种群中的等位基因频率，它为检测进化变化提供了有用的零模型。
p + q = 1 and p² + 2pq + q² = 1
Here p represents the frequency of the dominant allele and q the frequency of the recessive allele. p² denotes the proportion of homozygous dominant individuals, 2pq the heterozygotes, and q² the homozygous recessive individuals. Use this to calculate carrier frequencies for genetic disorders.
此处 p 代表显性等位基因的频率，q 代表隐性等位基因的频率。p² 表示纯合显性个体的比例，2pq 为杂合子比例，q² 为纯合隐性个体比例。可用它计算遗传病的携带者频率。
10. Simpson's Diversity Index | 辛普森多样性指数
Simpson's Diversity Index measures the biodiversity of a habitat, taking into account both species richness and evenness. A higher value indicates greater diversity.
辛普森多样性指数衡量一个栖息地的生物多样性，同时考虑了物种丰富度和均匀度。数值越高，多样性越大。
D = 1 – Σ (n / N)²
n is the number of individuals of a particular species, and N is the total number of organisms of all species. The summation is carried out across all species present. To compare different communities, calculate D for each and note that values range from 0 (no diversity) to just below 1.
n 是某一物种的个体数，N 是所有物种的总个体数。对所有物种完成求和。要比较不同的群落，可计算每个群落的 D 值，并注意 D 值范围从 0（无多样性）到接近 1。
11. Water Potential and Solute Potential | 水势与溶质势
Water potential (Ψ) determines the direction of water movement across plant cell membranes. It is the sum of solute potential (Ψs) and pressure potential (Ψp), with pure water having a value of zero under standard conditions.
水势（Ψ）决定了水分跨越植物细胞膜的移动方向。它是溶质势（Ψs）与压力势（Ψp）之和，纯水在标准条件下的水势为零。
Ψ = Ψs + Ψp
The solute potential of a solution can be calculated using the van 't Hoff relationship. The negative sign indicates that solutes reduce water potential.
溶液的溶质势可以用范特霍夫关系式计算。负号表明溶质降低了水势。
Ψs = – i C R T
Here i is the ionisation constant (e.g. 1 for sucrose, 2 for NaCl), C is the molar concentration of the solute (mol L⁻¹), R is the pressure constant (0.0831 L bar mol⁻¹ K⁻¹) and T is temperature in Kelvin. This equation helps explain why a plant cell becomes flaccid in a concentrated salt solution.
其中 i 是电离常数（例如蔗糖为 1，氯化钠为 2），C 是溶质的摩尔浓度（mol L⁻¹），R 是压力常数（0.0831 L bar mol⁻¹ K⁻¹），T 是开尔文温度。该方程有助于解释为什么植物细胞在浓盐溶液中会变得质壁分离。
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