📚 Year 12 OCR Biology: Quick Reference Guide to Formulas and Theorems | Year 12 OCR 生物：公式定理速查手册
This handbook brings together all the essential mathematical formulas and biological theorems you will encounter in Year 12 of the OCR A Level Biology course. Use it to refresh your memory before assessments, to complete practice questions accurately, and to build confidence in applying quantitative skills across topics such as cell structure, exchange surfaces, transport in animals, and biodiversity.
本手册汇集了你在 Year 12 OCR A Level 生物学课程中会遇到的所有关键数学公式和生物学定理。你可以用它来在评估前快速回顾，准确地完成练习题，并在细胞结构、交换表面、动物运输和生物多样性等主题中建立应用定量技能的自信。
1. Magnification Formula | 放大倍率公式
Magnification describes how much larger an image appears compared with the real object. The formula is fundamental for microscopy work and is regularly tested in practical assessments.
放大倍率描述了图像相比于实物放大了多少。该公式是显微镜工作的基础，并在实践评估中经常被考查。
Magnification = Image size ÷ Actual size
放大倍率 = 图像大小 ÷ 实际大小
All measurements must be converted to the same unit before calculation. Typical units are micrometres (µm) or millimetres (mm). Remember that 1 mm = 1000 µm.
在计算前，必须将所有测量值转换为相同的单位。常用单位是微米 (µm) 或毫米 (mm)。记住 1 mm = 1000 µm。
| Symbol | Meaning | 符号 | 含义 |
| I | Image size (measured from micrograph or drawing) | 图像大小（从显微照片或绘图测得） | |
| A | Actual size of the specimen | 样本的实际大小 | |
| M | Magnification (e.g. ×400) | 放大倍率（例如 ×400） |
When a question gives magnification and one of the size values, simply rearrange the equation. For instance, Actual size = Image size ÷ Magnification.
当题目给出放大倍率和其中一个尺寸数值时，只需整理公式。例如，实际大小 = 图像大小 ÷ 放大倍率。
2. Eyepiece Graticule Calibration | 目镜测微尺校准
An eyepiece graticule must be calibrated for each objective lens using a stage micrometer. This allows you to convert graticule units into real lengths.
目镜测微尺必须使用镜台测微尺针对每个物镜进行校准。这使你可以将测微尺单位转换为真实长度。
One graticule unit = (Number of micrometre divisions × Length of one stage division) ÷ Number of graticule divisions
一个测微尺单位 = （镜台刻度数 × 每个镜台刻度的长度）÷ 目镜测微尺的刻度数
Usually, 1 stage micrometre division = 10 µm (since the smallest divisions are 0.01 mm). Count how many graticule divisions align with a known number of stage divisions, then calculate the length of one graticule unit.
通常，1 个镜台测微尺刻度 = 10 µm（因为最小刻度为 0.01 mm）。数出多少目镜刻度与已知数量的镜台刻度对齐，然后计算一个目镜测微尺单位的长度。
Once calibrated, the eyepiece graticule can be used to measure actual sizes of cells. Multiply the number of graticule units by the calibrated value.
校准后，目镜测微尺可用于测量细胞的实际大小。将测微尺单位数乘以校准值即可。
3. Surface Area to Volume Ratio | 表面积与体积比
Small organisms have a large surface area to volume ratio (SA:V), which aids efficient diffusion across their body surface. As organisms increase in size, SA:V decreases, creating a need for specialised exchange surfaces and transport systems.
小型生物拥有较大的表面积与体积比 (SA:V)，这有助于物质在其体表进行高效扩散。随着生物体增大，SA:V 会减小，从而需要特化的交换表面和运输系统。
SA:V = Surface Area ÷ Volume
SA:V = 表面积 ÷ 体积
For a cube, surface area = 6 × side², volume = side³. Because volume increases as the cube of linear dimensions while surface area increases as the square, the ratio falls as the organism gets larger.
对于正方体，表面积 = 6 × 边长²，体积 = 边长³。由于体积随线性尺度的立方增长而表面积随平方增长，因此该比值随着生物体变大而下降。
The concept explains why large multicellular organisms rely on lungs, gills, or internal transport systems, while single-celled organisms can exchange substances directly across their plasma membrane.
这一概念解释了为什么大的多细胞生物依赖肺、鳃或内部运输系统，而单细胞生物可以直接通过质膜进行物质交换。
4. Fick’s Law of Diffusion | 菲克扩散定律
Fick’s Law quantifies the rate of diffusion across an exchange surface. It is often cited when comparing the efficiency of gas exchange in different organisms or organs.
菲克定律量化了物质跨越交换表面的扩散速率。在比较不同生物或器官的气体交换效率时，常引用该定律。
Rate of diffusion ∝ (Surface area × Concentration difference) ÷ Diffusion distance
扩散速率 ∝ （表面积 × 浓度差）÷ 扩散距离
- Large surface area – provided by alveoli, villi or root hairs.
- Steep concentration gradient – maintained by ventilation or blood flow.
- Short diffusion distance – thin epithelium, flattened cells.
- 大的表面积 – 由肺泡、绒毛或根毛提供。
- 陡的浓度梯度 – 通过通气或血流维持。
- 短的扩散距离 – 薄的上皮、扁平细胞。
To maximise diffusion rate, organisms have evolved structures that increase surface area, maintain gradients, and minimise the thickness of the exchange surface.
为使扩散速率最大化，生物体进化出了增大表面积、维持浓度梯度并尽量减小交换表面厚度的结构。
5. Water Potential Equation | 水势方程
Water potential (Ψ) determines the direction of water movement by osmosis. It is measured in pressure units, usually kilopascals (kPa). Pure water has a water potential of 0 kPa; solutions have negative water potentials.
水势 (Ψ) 决定了渗透作用中水分子的运动方向。它以压力单位计量，通常为千帕 (kPa)。纯水的水势为 0 kPa；溶液具有负的水势。
Ψ = Ψs + Ψp
水势 = 溶质势 + 压力势
| Component | Description | 组成部分 | 描述 |
| Ψs (solute potential) | Always negative or zero; becomes more negative as solute concentration increases. | 溶质势 | 始终为负或零；随溶质浓度升高而变得更负。 |
| Ψp (pressure potential) | Can be zero or positive; represents physical pressure on the solution (e.g. turgor pressure in plant cells). | 压力势 | 可为零或正值；表示施加于溶液上的物理压力（如植物细胞的膨压）。 |
Water always moves from a region of higher (less negative) water potential to a region of lower (more negative) water potential.
水分总是从水势较高（负值较小）的区域向水势较低（负值较大）的区域移动。
6. Cardiac Output | 心输出量
Cardiac output is the volume of blood pumped by one ventricle per minute. It links heart rate and stroke volume, both of which can be measured or calculated from data.
心输出量是指一个心室每分钟泵出的血液体积。它将心率和每搏输出量联系起来，两者都可以从数据中测量或计算得出。
Cardiac Output (CO) = Stroke Volume (SV) × Heart Rate (HR)
心输出量 (CO) = 每搏输出量 (SV) × 心率 (HR)
Cardiac output is expressed in litres per minute (L min-1). A typical resting value is around 5 L min-1 but can rise significantly during exercise.
心输出量以升每分钟 (L min-1) 表示。典型的静息值约为 5 L min-1，但在运动时可显著升高。
Stroke volume is affected by venous return, myocardial contractility, and preload. You should be able to rearrange the formula if two of the three values are given.
每搏输出量受静脉回流量、心肌收缩力和前负荷的影响。如果已知三个变量中的两个，你应该能够整理公式求出第三个。
7. Pulmonary Ventilation | 肺通气量
Pulmonary ventilation quantifies the volume of air moved into and out of the lungs per minute. It is essential when studying gas exchange in mammals.
肺通气量量化了每分钟进出肺部的空气体积。在研究哺乳动物的气体交换时，这一点至关重要。
Pulmonary Ventilation = Tidal Volume × Breathing Rate
肺通气量 = 潮气量 × 呼吸频率
Tidal volume is the volume of air moved in or out during one normal breath (about 0.5 L at rest). Breathing rate is the number of breaths per minute. Ventilation is given in L min-1.
潮气量是一次正常呼吸中吸入或呼出的空气体积（静息时约 0.5 L）。呼吸频率为每分钟的呼吸次数。肺通气量以 L min-1 表示。
During exercise both tidal volume and breathing rate increase, leading to a large rise in pulmonary ventilation to meet the higher oxygen demand.
运动时潮气量和呼吸频率均会增加，导致肺通气量大幅上升以满足增加的氧气需求。
8. Body Mass Index (BMI) | 身体质量指数
BMI is a simple index used to classify individuals as underweight, normal, overweight or obese. It appears in the context of health and disease.
BMI 是一个简单的指数，用于将个体分为体重过轻、正常、超重或肥胖。它出现在健康与疾病的相关内容中。
BMI = Mass (kg) ÷ (Height (m))²
BMI = 体重 (kg) ÷ （身高 (m)）²
For example, a person weighing 70 kg with a height of 1.75 m has a BMI of 70 ÷ (1.75²) = 22.9 kg m-2. BMI values between 18.5 and 24.9 are generally considered healthy.
例如，一个重 70 kg、身高 1.75 m 的人，其 BMI = 70 ÷ (1.75²) = 22.9 kg m-2。BMI 值在 18.5 至 24.9 之间通常被认为是健康的。
While BMI is useful for population studies, it does not distinguish between muscle and fat mass and therefore has limitations for athletes.
尽管 BMI 在人群研究中很有用，但它无法区分肌肉和脂肪质量，因此对于运动员有一定局限性。
9. Simpson’s Index of Diversity | 辛普森多样性指数
Simpson’s Index (D) measures the biodiversity of a habitat, taking into account both species richness and evenness. A higher value of D indicates greater diversity.
辛普森指数 (D) 衡量一个栖息地的生物多样性，兼顾物种丰富度和均匀度。D 值越高表明多样性越高。
D = 1 − Σ (n ÷ N)²
D = 1 − Σ (n ÷ N)²
| Symbol | Meaning | 符号 | 含义 |
| n | Number of individuals of a particular species | n | 某一特定物种的个体数 |
| N | Total number of organisms of all species | N | 所有物种的总个体数 |
| Σ | Sum of | Σ | 求和 |
Work out (n ÷ N) for each species, square it, sum all those values, then subtract from 1. The index ranges from 0 (very low diversity) to 1 (very high diversity).
为每个物种计算 (n ÷ N)，然后平方，将所有值求和，最后从 1 中减去该总和。该指数范围从 0（极低多样性）到 1（极高多样性）。
10. Chi-squared (χ²) Test | 卡方检验
The chi-squared test is a statistical test used to determine whether there is a significant difference between observed and expected frequencies. It is frequently applied to genetics or ecological data.
卡方检验是一种统计检验，用于确定观察频数与期望频数之间是否存在显著差异。它常用于遗传学或生态学数据。
χ² = Σ (O − E)² ÷ E
χ² = Σ (O − E)² ÷ E
O = observed frequency, E = expected frequency. You calculate the difference for each category, square it, divide by E, and sum all the results. The degrees of freedom (df) = number of categories − 1.
O = 观察频数，E = 期望频数。你需计算每个类别的差值，平方，除以 E，然后将所有结果求和。自由度 (df) = 类别数 − 1。
Compare your calculated χ² value with a critical value from a chi-squared distribution table. If the calculated value exceeds the critical value at p = 0.05, you reject the null hypothesis and conclude that the difference is statistically significant.
将你计算出的 χ² 值与卡方分布表中的临界值进行比较。若计算值大于 p = 0.05 时的临界值，则拒绝零假设，认为差异在统计上显著。
11. Standard Deviation | 标准偏差
Standard deviation measures the spread of data around the mean. A small standard deviation indicates that data points are close to the mean; a large one indicates wide spread. It is essential for interpreting error bars on graphs and for assessing the reliability of results.
标准偏差衡量数据围绕平均值的离散程度。小的标准偏差表明数据点接近平均值；大的标准偏差表明分布较广。它对于解读图表上的误差线以及评估结果的可靠性至关重要。
s = √[ Σ (x − x̄)² ÷ (n &
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