📚 Formula & Theorem Quick-Reference Handbook | Year 13 剑桥生物公式定理速查手册
This handbook brings together the essential formulae, equations and quantitative relationships that every Cambridge Year 13 Biology student needs to apply with confidence. Whether you are calculating magnification, analysing population genetics or interpreting respirometer data, having these expressions at your fingertips speeds up revision and deepens understanding.
本手册汇集了每一位剑桥 Year 13 生物学学生必须自信运用的核心公式、方程与数量关系。无论你是在计算放大率、分析群体遗传学，还是解读呼吸计数据，随手查阅这些表达式都能加速复习并加深理解。
1. Microscopy & Magnification | 显微术与放大率
Magnification (M) describes how much larger an image appears compared to the real object. It can be calculated from the image size (I) and the actual specimen size (A).
放大率 (M) 描述图像比真实物体大多少倍，可由图像大小 (I) 与标本实际大小 (A) 求得。
The standard formula is:
标准公式为：
M = I ÷ A
actual size: A = I ÷ M, or image size: I = A × M. Always use the same unit (usually mm or µm) for I and A.
该式可变形为实际大小：A = I ÷ M，或图像大小：I = A × M。I 和 A 必须使用相同单位（通常为 mm 或 µm）。
When a scale bar is provided, measure the bar length, divide by the represented distance, and then use the resulting scale factor to convert measurements in the image into actual sizes.
若给出比例尺，可测量比例尺长度，除以其所代表的实际距离，再用所得比例因子将图像测量值换算为实际大小。
2. Surface Area to Volume Ratio | 表面积与体积比
The surface area to volume ratio (SA:V) dictates how efficiently a cell or organism can exchange substances with its environment. Smaller objects have a larger SA:V ratio.
表面积与体积之比 (SA:V) 决定了细胞或生物体与环境交换物质的效率。体积越小，SA:V 比值越大。
For simple geometric shapes the ratio can be expressed mathematically:
对于简单几何形状，该比值可用数学式表达：
For a cube: SA : V = 6 / l
For a sphere: SA : V = 3 / r
Here l is the side length of the cube and r is the radius of the sphere. As l or r increases, the SA:V ratio decreases, limiting diffusion efficiency in large multicellular organisms.
此处 l 为立方体边长，r 为球体半径。当 l 或 r 增大时，SA:V 比值降低，限制大型多细胞生物中的扩散效率。
3. Hardy–Weinberg Principle | 哈迪–温伯格定律
The Hardy–Weinberg principle provides mathematical predictions about allele and genotype frequencies in a non-evolving population.
哈迪–温伯格定律对一个非进化群体中的等位基因频率和基因型频率做出了数学预测。
If a gene has two alleles with frequencies p and q, then:
若某基因有两个等位基因，其频率为 p 和 q，则：
p + q = 1
And the genotype frequencies in a large, randomly mating population with no mutation, selection, gene flow or genetic drift will be:
在一个随机交配、无突变、无选择、无基因流动、无遗传漂变的大群体中，基因型频率将为：
p² + 2pq + q² = 1
frequency of homozygous recessive. Use observed homozygous recessive frequency to find q = √q², then derive p = 1 − q and calculate the other genotype frequencies.
p² = 纯合显性频率，2pq = 杂合子频率，q² = 纯合隐性频率。利用观察到的纯合隐性频率求出 q = √q²，再计算 p = 1 − q 并推导其他基因型频率。
4. Exponential Population Growth | 种群指数增长
When resources are unlimited, a population can grow exponentially. The number of individuals N at time t is given by:
当资源不受限制时，种群可呈指数增长。t 时刻的个体数 N 由下式给出：
N(t) = N0 ert
N0 = initial population size, e = base of natural logarithms (~18), r = intrinsic rate of natural increase, t = time. The doubling time can be found as tdouble = ln(2) / r.
N0 = 初始种群大小，e = 自然对数底 (~2.718)，r = 内禀增长率，t = 时间。倍增时间可由 tdouble = ln(2) / r 求得。
5. Lincoln Index (Capture–Mark–Recapture) | 林肯指数（标记-重捕法）
The Lincoln Index estimates the size of a motile animal population. It is based on the proportion of marked individuals in a second capture.
林肯指数用于估算运动动物种群的大小，依据是第二次捕获样本中标记个体的比例。
N = (M × C) ÷ R
ht in the second sample, R = number of marked individuals recaptured in the second sample.
N = 估计总种群数量，M = 首次捕获、标记并释放的个体数，C = 第二次捕获的总个体数，R = 第二次捕获中带标记的个体数。
Assumptions include: marks are not lost, marked individuals mix randomly, no immigration/emigration, and handling does not affect survival.
假设包括：标记不脱落、标记个体随机混入、无迁入/迁出、捕捉不影响存活。
6. Simpson’s Index of Diversity | 辛普森多样性指数
Simpson’s Index (D) quantifies biodiversity, taking into account both species richness and evenness. A common form used by Cambridge is:
辛普森指数 (D) 在考虑物种丰富度与均匀度的基础上量化生物多样性。剑桥常用的一个形式为：
D = 1 − [ Σ n(n−1) ] / [ N(N−1) ]
iduals of all species. The value of D ranges from 0 (no diversity) to nearly 1 (high diversity).
n = 某一物种的个体数，N = 所有物种的总个体数。D 值从 0（无多样性）到接近 1（高多样性）不等。
7. Gross & Net Primary Productivity | 总初级生产力与净初级生产力
Primary productivity measures the rate at which producers convert light energy into chemical energy. The key relationship is:
初级生产力衡量生产者将光能转化为化学能的速率，其关键关系为：
NPP = GPP − R
oductivity (energy available to primary consumers). NPP can be expressed as kJ m⁻² year⁻¹.
GPP = 总初级生产力（光合作用固定的总能量），R = 植物呼吸热损耗，NPP = 净初级生产力（可供初级消费者利用的能量）。NPP 单位可为 kJ m⁻² year⁻¹。
8. Respiratory Quotient (RQ) | 呼吸商 (RQ)
The respiratory quotient is the ratio of the volume of carbon dioxide produced to the volume of oxygen consumed during aerobic respiration.
呼吸商是有氧呼吸过程中产生的二氧化碳体积与消耗的氧气体积之比。
RQ = CO₂ produced ÷ O₂ consumed
1.0, lipids about 0.7 and proteins about 0.9. RQ values allow inference of the predominant respiratory substrate.
碳水化合物呼吸的 RQ ≈ 1.0，脂质约为 0.7，蛋白质约为 0.9。RQ 值可用于推断主要的呼吸底物。
9. Efficiency of Energy Transfer | 能量传递效率
The efficiency of energy transfer between trophic levels can be calculated as:
营养级之间的能量传递效率可计算为：
Efficiency (%) = (Energy in higher level ÷ Energy in lower level) × 100
Typically, only about 10% of energy is passed on; the rest is lost through respiration, faeces and uneaten parts.
通常仅有约 10% 的能量被传递，其余通过呼吸、粪便和未食用部分散失。
10. Chi-Squared (χ²) Test | 卡方 (χ²) 检验
The chi-squared test compares observed frequencies with expected frequencies to determine if a difference is statistically significant.
卡方检验比较观察频数与期望频数，以判断差异是否具有统计学显著性。
χ² = Σ [ (O − E)² ÷ E ]
. Σ means sum over all categories. Compare the calculated χ² against a critical value from a table, using degrees of freedom (number of categories − 1).
O = 观察值，E = 期望值，Σ 意为对所有类别求和。将计算出的 χ² 值与查表所得的临界值比较，自由度为类别数减 1。
11. Student’s t-Test | 学生 t 检验
Student’s t-test is used to compare the means of two sets of data. For two independent samples, assuming unequal variance (Welch’s approximation):
学生 t 检验用于比较两组数据的平均值。对于两个独立样本，假定方差不齐时（Welch 近似）：
t = (x̄₁ − x̄₂) ÷ √ s₁²/n₁ + s₂²/n₂
x̄₁, x̄ culated t is compared with the critical t-value at a chosen probability level and appropriate degrees of freedom.
x̄₁、x̄₂ = 样本均值，s₁²、s₂² = 样本方差，n₁、n₂ = 样本大小。将计算出的 t
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