📚 Formula Summary Handbook for IB Biology | IB 生物公式汇总手册
Mastering essential formulas is a cornerstone of success in IB Biology. From microscopy and cell counting to genetics and ecology, quantitative analysis helps you interpret experimental data and understand biological systems. This handbook compiles the key equations, statistical tests, and population models you will encounter in the course, with clear explanations and applied examples.
掌握核心公式是IB生物取得好成绩的基石。从显微镜使用、细胞计数到遗传学和生态学，定量分析帮助你解读实验数据、理解生物系统。本手册汇集了课程中涉及的关键方程、统计检验和种群模型，配合清晰的解释和应用实例。
1. Microscope Magnification | 显微镜放大倍数
Magnification is calculated as the ratio of image size to actual size. Always ensure that both measurements are in the same unit before dividing.
放大倍数等于图像大小与实际大小的比值。计算前务必确保两个测量值单位一致。
Magnification = Image size ÷ Actual size
放大倍数 = 图像大小 ÷ 实际大小
Rearrange the formula to find actual size: Actual size = Image size ÷ Magnification. This is essential when drawing scale bars or estimating cell dimensions from micrographs.
变形公式可求实际大小：实际大小 = 图像大小 ÷ 放大倍数。在绘制比例尺或从显微照片估算细胞尺寸时，这一运算至关重要。
Common unit conversions: 1 mm = 1000 µm; 1 µm = 1000 nm. For example, if a cell image measures 50 mm at ×400, the actual length is (50 × 1000 ÷ 400) = 125 µm.
常用单位换算：1 mm = 1000 µm；1 µm = 1000 nm。例如，某细胞图像在 ×400 时测得 50 mm，实际长度为 (50 × 1000 ÷ 400) = 125 µm。
2. Cell Counting with a Haemocytometer | 血球计数板细胞计数
A haemocytometer is used to estimate cell concentration. The central grid typically consists of 25 large squares, each of volume 0.1 mm³ (or 10⁻⁴ mL). Count cells in a known number of squares, then apply the dilution factor.
血球计数板用于估计细胞浓度。中央网格通常包含 25 个大方格，每个大方格体积为 0.1 mm³ (即 10⁻⁴ mL)。在已知方格数内计数细胞，再乘以稀释因子。
Cells per mL = (Average count per large square) × Dilution factor × 10⁴
每毫升细胞数 = (每个大方格平均计数) × 稀释因子 × 10⁴
The factor 10⁴ arises because each large square holds 10⁻⁴ mL. If you count four corner squares and obtain totals of 48, 52, 55 and 45, the average is 50. With a 1:1 dilution (dye:cell suspension), dilution factor = 2, so concentration = 50 × 2 × 10⁴ = 1.0 × 10⁶ cells/mL.
系数 10⁴ 源自每个大方格体积为 10⁻⁴ mL。若计数四个角方格，分别得到 48, 52, 55 和 45，平均值为 50。采用 1:1 稀释（染液:细胞悬液），稀释因子 = 2，则浓度 = 50 × 2 × 10⁴ = 1.0 × 10⁶ 个/mL。
3. Serial Dilution – C₁V₁ = C₂V₂ | 连续稀释 – C₁V₁ = C₂V₂
Serial dilutions are prepared by transferring a known volume of a stock solution and adding fresh solvent. The relationship C₁V₁ = C₂V₂ holds for any single dilution step.
连续稀释通过移取已知体积的储液并加入新溶剂实现。关系式 C₁V₁ = C₂V₂ 适用于每一次单步稀释。
C₁V₁ = C₂V₂
初始浓度 × 初始体积 = 稀释后浓度 × 稀释后体积
C₁ and C₂ are the concentrations before and after dilution; V₁ and V₂ are the corresponding volumes. The dilution factor for a single step is V₂ ÷ V₁. In a ten‑fold serial dilution, each step uses 1 part stock plus 9 parts diluent, giving a dilution factor of 10.
C₁ 和 C₂ 分别为稀释前后的浓度；V₁ 与 V₂ 为相应体积。单步稀释因子 = V₂ ÷ V₁。做 10 倍梯度稀释时，每步取 1 份储液加 9 份稀释液，稀释因子即为 10。
4. Enzyme Activity Rate | 酶活性速率
The rate of an enzyme‑catalysed reaction is commonly determined from the progress curve of product formation or substrate disappearance. For a linear initial‑rate region, rate = change in amount ÷ time.
酶促反应速率通常根据产物生成或底物消失的进程曲线确定。在线性初速率区间内，速率 = 变化量 ÷ 时间。
Rate = Δ[Product] ÷ Δt or Rate = 1 ÷ Time for endpoint assays
速率 = Δ[产物] ÷ Δt 或 终点法中速率 = 1 ÷ 时间
In classic iodine‑starch disappearance assays, the time taken for the blue‑black colour to vanish is recorded. The relative rate is expressed as 1/time (s⁻¹), which is proportional to enzyme activity under constant conditions.
在经典的碘‑淀粉消失实验中，记录蓝黑色褪去所需的时间。相对速率表示为 1/时间 (s⁻¹)，在条件恒定时与酶活性成正比。
5. Photosynthesis and Respiration Rates | 光合作用与呼吸速率
Photosynthesis rate can be measured by counting oxygen bubbles per minute, using a photosynthometer, or monitoring carbon dioxide uptake with a pH indicator. Respiration rate is often tracked by CO₂ production or O₂ consumption in a respirometer.
光合作用速率可通过每分钟计数气泡数、使用光合测定仪或利用 pH 指示剂监测二氧化碳吸收量来测定。呼吸速率通常借助呼吸计追踪 CO₂ 产生或 O₂ 消耗。
Rate = Change in measured variable (bubbles, Δabsorbance, volume) ÷ Time
速率 = 测定变量变化量 (气泡数、吸光度差、体积) ÷ 时间
When using a manometric respirometer, the movement of coloured liquid indicates volume of O₂ consumed. Remember to subtract control readings to account for temperature and pressure changes.
使用压力呼吸计时，有色液柱的移动反映 O₂ 消耗体积。记得减去对照读数，以校正温度和压力变化的影响。
6. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient indicates the type of metabolic fuel being respired. It is the ratio of carbon dioxide produced to oxygen consumed.
呼吸商反映被呼吸分解的代谢燃料类型，它是产生的二氧化碳与消耗的氧气之比。
RQ = CO₂ produced ÷ O₂ consumed
呼吸商 = 产生的 CO₂ ÷ 消耗的 O₂
Typical RQ values: carbohydrate ≈ 1.0, lipid ≈ 0.7, protein ≈ 0.9. A RQ >1 suggests anaerobic respiration, as extra CO₂ is released without O₂ use. A RQ <0.7 can occur in organisms undergoing gluconeogenesis from lipids.
典型 RQ 值：碳水化合物 ≈ 1.0，脂质 ≈ 0.7，蛋白质 ≈ 0.9。RQ > 1 提示存在无氧呼吸，因为释放了额外的 CO₂ 却没有消耗 O₂；RQ < 0.7 可能出现在利用脂质进行糖异生的生物中。
7. Chi‑Squared Test (χ²) for Genetics | 遗传学卡方检验 (χ²)
The chi‑squared test compares observed and expected frequencies to determine whether deviations are due to chance. It is widely used for Mendelian ratios.
卡方检验比较观察频数与期望频数，判断偏差是否由偶然造成。该检验广泛用于孟德尔比率分析。
χ² = Σ (O − E)² ÷ E
卡方值 = Σ (观察值 − 期望值)² ÷ 期望值
Degrees of freedom = number of categories − 1. Compare calculated χ² to the critical value at p = 0.05. If χ² < critical value, the null hypothesis (no significant difference) is accepted.
自由度 = 类别数 − 1。将计算所得 χ² 与 p = 0.05 时的临界值比较。若 χ² < 临界值，则接受无效假设（无显著性差异）。
8. Hardy‑Weinberg Equilibrium | 哈代‑温伯格平衡
In a non‑evolving population, allele and genotype frequencies remain constant. The principle provides a null model for detecting evolutionary change.
在未进化的群体中，等位基因频率和基因型频率保持不变。该定律提供了检测进化改变的无效模型。
p + q = 1
p² + 2pq + q² = 1
p = frequency of the dominant allele, q = frequency of the recessive allele. p², 2pq, and q² represent the frequencies of homozygous dominant, heterozygous, and homozygous recessive genotypes respectively. Use this to estimate carrier frequencies from disease incidence.
p = 显性等位基因频率，q = 隐性等位基因频率。p²、2pq 和 q² 分别代表纯合显性、杂合与纯合隐性的基因型频率。可用于从疾病发病率估算携带者频率。
9. Population Growth Equations | 种群增长方程
Exponential growth occurs under unlimited resources and is modelled by dN/dt = rN. Logistic growth incorporates a carrying capacity (K).
指数增长在资源无限时发生，模型为 dN/dt = rN。逻辑斯谛增长则纳入了环境容纳量 (K)。
Exponential: dN/dt = rN
指数增长：dN/dt = rN
Logistic: dN/dt = rN × (K − N) ÷ K
逻辑斯谛增长：dN/dt = rN × (K − N) ÷ K
Here N = population size, r = intrinsic rate of increase, K = carrying capacity. At N = K/2, the logistic growth rate is maximal. Exponential growth can also be written as Nₜ = N₀eʳᵗ.
其中 N = 种群大小，r = 内禀增长率，K = 容纳量。当 N = K/2 时，逻辑斯谛增长速率最快。指数增长也可写为 Nₜ = N₀eʳᵗ。
10. Lincoln Index (Mark‑Release‑Recapture) | 林肯指数 (标志重捕法)
The Lincoln index estimates the size of a motile animal population from a capture‑mark‑recapture study. It assumes random mixing and no migration, births, or deaths between samples.
林肯指数通过捕获‑标志‑重捕研究估算活动动物种群的大小。它假设两次采样间个体随机混合、无迁移、无出生或死亡。
N = (M × C) ÷ R
种群大小 = (标志数 × 重捕总数) ÷ 重捕中标志数
M = number of individuals marked and released in the first sample, C = total number caught in the second sample, R = number of marked individuals in the second sample. A large R improves accuracy.
M = 第一次捕捉并标志释放的个体数，C = 第二次捕捉总个体数，R = 第二次捕捉中已标志个体数。R 较大时准确度更高。
11. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s index quantifies biodiversity by considering both species richness and evenness. The reciprocal form used in IB Biology is:
辛普森指数通过兼顾物种丰富度与均匀度来量化生物多样性。IB生物课程中使用的指数倒数形式为：
D = N(N − 1) ÷ Σ n(n − 1)
辛普森多样性指数 = 总个体数 (总个体数 − 1) ÷ Σ 某物种个体数 (该物种个体数 − 1)
N = total number of organisms of all species, n = number of individuals of a particular species. Higher D values indicate greater diversity. This index ranges from 1 (no diversity) to a theoretical maximum equal to the number of species.
N = 所有物种的个体总数，n = 某一特定物种的个体数。D 值越高，多样性越大。该指数范围从 1 (无多样性) 到理论上等于物种数的最大值。
12. Basic Statistics for Biological Data | 生物数据基础统计
Descriptive statistics and the t‑test are routinely used to analyse experimental data in IB Biology internal assessments.
描述性统计和 t 检验常被用于IB生物内部评估的实验数据分析。
Mean (x̄) = Σx ÷ n — The arithmetic average.
平均值 (x̄) = Σx ÷ n — 算术平均数。
Standard deviation (SD) = √[ Σ(x − x̄)² ÷ (n − 1) ] — Measures spread around the mean.
标准差 (SD) = √[ Σ(x − x̄)² ÷ (n − 1) ] — 衡量数据围绕平均值的离散程度。
Standard error (SE) = SD ÷ √n — Indicates the precision of the sample mean as an estimate of the population mean.
标准误差 (SE) = SD ÷ √n — 反映样本均值作为总体均值估计值的精确度。
Unpaired t‑test:
非配对 t 检验：
t = (x̄₁ − x̄₂) ÷ √[ (SD₁² ÷ n₁) + (SD₂² ÷ n₂) ]
Degrees of freedom ≈ n₁ + n₂ − 2. Compare calculated t with critical value. If t > critical value, the difference between means is statistically significant at the chosen p‑level (usually 0.05).
自由度 ≈ n₁ + n₂ − 2。将计算所得 t 值与临界值比较。若 t > 临界值，两均值间的差异在所选显著性水平 (通常 p = 0.05) 下具有统计学意义。
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