📚 Year 11 OCR Geography: Formula and Theorem Quick Reference Handbook | Year 11 OCR 地理：公式定理速查手册
Welcome to your essential quick-reference tool for the Year 11 OCR Geography course. This handbook brings together the key equations, quantitative relationships and core geographical models that support analysis of population change, urban systems, river dynamics, climate, development and spatial interaction. Each entry is presented as a clear formula or a concise statement of a geographical theorem, followed by a bilingual explanation designed to strengthen your exam skills.
欢迎使用 Year 11 OCR 地理课程的必备速查工具。本手册汇集了支持人口变化、城市系统、河流动态、气候、发展及空间相互作用分析的关键方程、定量关系与核心地理模型。每个条目都以清晰的公式或地理定理的简明陈述呈现，并配有双语解释，旨在强化你的考试技能。
1. Population Density and Vital Rates | 人口密度与基本人口率
Population density describes the concentration of people within a given area and is the most fundamental measure of human geography.
人口密度描述给定区域内人口的集中程度，是人文地理最基本的度量。
Population Density = Total Population / Land Area (persons per km²)
This simple ratio allows comparison between urban cores, rural peripheries and megacities. OCR questions often ask you to calculate density from a data table and explain the reasons for high or low values.
这个简单的比率有助于比较城市核心区、乡村边缘区与巨型城市。OCR 考题常要求根据数据表计算人口密度，并解释数值高低的原因。
The crude birth rate (CBR) and crude death rate (CDR) are expressed per 1000 people to make them comparable across differently sized populations.
粗出生率 (CBR) 和粗死亡率 (CDR) 以每千人为单位表示，以便在不同规模的人口中进行比较。
CBR = (Number of live births / Total population) × 1000
CDR = (Number of deaths / Total population) × 1000
Natural increase is the difference between these two rates. When combined with net migration, it gives the overall population growth rate, a key dynamic in the demographic transition.
自然增长是这两个率值之差。再结合净迁移，就得到总人口增长率——这是人口转变中的关键动态。
Natural Increase Rate = CBR - CDR
Population Growth Rate = (Natural increase + Net migration) / Total population × 1000
In fieldwork and exam contexts, always check whether rates are given per 100 or per 1000 and convert accordingly.
在实地调查和考试情境中，务必核对率值是以每百人还是每千人给出的，并相应转换。
2. Dependency Ratios | 抚养比率
The dependency ratio indicates the economic pressure on the working-age population. It is a vital statistic for analysing age structure and planning public services.
抚养比率显示劳动年龄人口承受的经济压力，是分析年龄结构与规划公共服务的重要统计量。
Total Dependency Ratio = [(Population 0–14 + Population 65+) / Population 15–64] × 100
A high dependency ratio, common in Stage 2 of the DTM and in ageing Stage 5 societies, means fewer workers support more dependents. You may be asked to split this into youth dependency and old-age dependency ratios.
高抚养比率常见于 DTM 第二阶段和老龄化的第五阶段社会，意味着较少的劳动人口赡养更多的被抚养者。考题可能要求你将这个指标拆分为少儿抚养比和老年抚养比。
Youth Dependency Ratio = (Population 0–14 / Population 15–64) × 100
Old-age Dependency Ratio = (Population 65+ / Population 15–64) × 100
Rising old-age dependency is closely linked to the need for increased healthcare and pension expenditure in HICs.
老年抚养比的上升与高收入国家医疗与养老金支出需求增加密切相关。
3. Population Balancing Equation and the Demographic Transition Model | 人口平衡方程与人口转变模型
The demographic transition model (DTM) is a foundational theorem linking birth and death rates to economic development. Its quantitative backbone is the population balancing equation.
人口转变模型 (DTM) 是将出生率、死亡率与经济发展相联系的基础定理，其量化支柱是人口平衡方程。
Pₜ = P₀ + B - D + I - E
where Pₜ is the population at time t, P₀ the initial population, B births, D deaths, I immigration and E emigration. This equation underpins all census updates and population projections.
式中 Pₜ 为 t 时刻人口，P₀ 为初始人口，B 为出生数，D 为死亡数，I 为迁入数，E 为迁出数。该方程是所有人口普查更新和人口预测的基础。
The DTM itself describes predictable shifts: Stage 1 high fluctuating, Stage 2 early expanding, Stage 3 late expanding, Stage 4 low fluctuating, and Stage 5 decline. You should be able to apply these stages to case studies and interpret population pyramids that reflect each pattern.
DTM 本身描述了可预测的转变：第一阶段高位振荡、第二阶段早期扩张、第三阶段后期扩张、第四阶段低位振荡和第五阶段衰退。你应能将这些阶段应用于案例研究，并解释反映各模式的人口金字塔。
4. Urbanisation Rate and Connectivity Indices | 城市化率与连通性指数
Urbanisation rate measures the shift of population from rural to urban areas, a core indicator in GCSE Human Geography.
城市化率衡量人口从农村向城市转移的程度，是 GCSE 人文地理的核心指标。
Urbanisation Rate (%) = (Urban population / Total population) × 100
Rapid urbanisation in LIDCs is often driven by rural push factors and urban pull factors; the formula helps quantify the resulting spatial concentration.
低收入与中低收入国家的快速城市化常由农村推力因素和城市拉力因素驱动；这一公式有助于量化由此产生的空间集聚。
For transport networks, the beta index evaluates connectivity by comparing the number of edges (links) to vertices (nodes).
在交通网络中，贝塔指数通过比较边（连线）与顶点（节点）的数量来评估连通性。
Beta Index β = e / v
vertices. A value greater than 1 suggests a well-connected network; β < 1 indicates a simple tree-like structure. This is often applied to urban transport or river drainage networks.
其中 e 为边的数量，v 为顶点数量。指数大于 1 表明网络连通良好；β < 1 则表明简单的树状结构。这一指数常用于城市交通或河流排水网络的分析。
The gravity model is a spatial interaction theorem used to predict movement between two places.
引力模型是一种空间相互作用定理，用于预测两地之间的流动。
Interaction I = (P₁ × P₂) / d²
P₁ and P₂ are the populations of the two centres, and d is the distance between them. The model illustrates why larger, closer cities generate more commuting, trade and migration flows.
P₁ 与 P₂ 为两个中心的人口，d 为两地间的距离。该模型说明为何规模较大且相距较近的城市会产生更多的通勤、贸易和迁移流。
5. River Discharge and Hydrological Equations | 河流流量与水文方程
River discharge, measured in cubic metres per second (cumecs), is a fundamental variable in fluvial geomorphology and flood risk management.
河流流量以立方米每秒（cumecs）为单位，是河流地貌学与洪水风险管理中的基本变量。
Discharge Q = A × v
where A is the cross-sectional area of the channel (m²) and v is the mean flow velocity (m/s). Fieldwork often requires students to calculate Q from channel width, depth and velocity measurements made with a flowmeter.
其中 A 是河道横截面积（m²），v 是平均流速（m/s）。实地调查常要求学生根据用流速仪测得的河宽、水深和流速来计算 Q。
The water balance equation summarises the pathways of precipitation in a drainage basin.
水平衡方程概括了流域内降水的各种路径。
Runoff = Precipitation - Evapotranspiration ± Change in storage
This relationship helps explain why some storms generate flash floods while others do not; it depends on antecedent soil moisture and urban surface sealing.
这一关系有助于解释为何有些暴雨会引发山洪暴发而另一些则不会；这取决于前期土壤湿度与城市地面的不透水覆盖。
6. Storm Hydrograph and the Bradshaw Model | 暴雨洪水曲线与布拉德肖模型
A storm hydrograph shows river discharge over time after a rainfall event. The lag time—the interval between peak rainfall and peak discharge—is a critical quantifier of basin response.
暴雨洪水曲线显示降雨事件后河流流量随时间的变化。滞时——即降雨峰值与流量峰值之间的时间间隔——是衡量流域响应的关键量化指标。
Lag Time = Time of peak discharge - Time of peak rainfall
Short lag times, typical of urbanised catchments with impermeable surfaces, produce flashy hydrographs with high flood risk. OCR exams frequently test interpretation of hydrograph shapes and the calculation of lag time from graph axes.
短滞时常见于具有不透水表面的城市化流域，产生陡峭的洪水过程线，其洪水风险较高。OCR 考试常测试对过程线形态的解读以及根据图轴计算滞时的能力。
The Bradshaw Model is a geographical theorem that describes systematic downstream changes in channel characteristics. Instead of a single formula, it is best summarised as a set of expected trends:
布拉德肖模型是一种地理定理，描述河道特征沿下游方向有规律的变化。它并非单一公式，而最好通过一组预期趋势来总结：
| Channel variable | Downstream change |
|---|---|
| Discharge | Increases |
| Channel width & depth | Increase |
| Mean velocity | Increases |
| Channel bed roughness | Decreases |
| Sediment particle size | Decreases (becomes finer) |
| Slope gradient | Decreases |
The model helps explain why lower-course environments are more susceptible to flooding but also support larger rivers that transport huge sediment loads.
该模型有助于解释为何下游河段环境更易受洪水影响，但也承载着运输巨大泥沙量的大型河流。
7. Hjulström Curve – Erosion, Transport and Deposition Thresholds | 尤尔斯特罗姆曲线——侵蚀、搬运与沉积阈值
The Hjulström curve is a foundational theorem in fluvial geomorphology, established by Filip Hjulström. It links flow velocity to particle size and shows the critical velocities needed for erosion, transportation and deposition of sediment.
尤尔斯特罗姆曲线是河流地貌学的基本定理，由 Filip Hjulström 建立。它将水流速度与颗粒粒径联系起来，展示泥沙侵蚀、搬运与沉积所需的临界速度。
Coarser sand and gravel require the highest velocities to be eroded, while fine silt and clay need surprisingly high velocities due to cohesion. Once particles are in suspension, transport can continue at much lower speeds. The curve is a fundamental tool for interpreting river bank stability and the formation of alluvial landforms.
较粗的砂砾石需要最高的速度才能被侵蚀，而细粉砂和粘土因粘聚力而需要惊人的高速。一旦颗粒处于悬浮状态，它们可以在低得多的速度下继续搬运。该曲线是解释河岸稳定性及冲积地貌形成的基本工具。
Students should be able to sketch the Hjulström curve, label its axes (mean velocity in cm/s against particle diameter in mm) and annotate the erosion envelope, transport zone and settling line.
学生应能绘制尤尔斯特罗姆曲线的草图，标注坐标轴（平均流速 cm/s 与颗粒直径 mm），并注明侵蚀包络线、搬运区与沉降线。
8. Climatic Calculations | 气候计算
Climate graphs and statistical summaries demand a few reliable arithmetic formulae. The annual temperature range is a simple measure of continentality and seasonal variation.
气候图与统计汇总需要几个可靠的算术公式。年温差是衡量大陆度和季节变化的简单指标。
ual Temperature Range = Highest monthly mean temperature - Lowest monthly mean temperature
l climate, whereas London exhibits a much smaller range (≈13°C) due to maritime influence.
例如，莫斯科的年温差非常大（约 27°C），彰显其大陆性气候，而伦敦由于受海洋影响，年温差则小得多（约 13°C）。
Total annual precipitation is derived by summing the twelve monthly totals. You may also need to calculate mean daily, monthly or annual temperatures from raw data.
年总降水量由十二个月的降水量总和得出。可能还需要根据原始数据计算日平均、月平均或年平均气温。
an Monthly Temperature = Sum of daily mean temperatures in month / Number of days in month
he In addition, the calculation of the number of dry months (precipitation < 2 × temperature) is used for Köppen climate classification, a system frequently referenced in OCR Geography.
此外，干旱月份数（降水量 < 2 × 气温）的计算用于柯本气候分类，该分类在 OCR 地理中经常被引用。
9. Development Indicators – GDP, HDI and Others | 发展指标——GDP、HDI 及其他
Gross Domestic Product per capita is the most widely used economic indicator of development, although it masks internal inequality.
人均国内生产总值是最广泛使用的经济发展指标，尽管它掩盖了内部的不平等。
GDP per capita = Total GDP / Total population
For exam analysis, you should be comfortable converting between raw GDP figures and per capita values, and discussing the difference between nominal GDP and purchasing power parity (PPP).
在考试分析中，你应能熟练地在 GDP 总量与人均数值之间进行转换，并讨论名义 GDP 与购买力平价 (PPP) 的区别。
The Human Development Index (HDI) is a composite theorem-like indicator that combines health, education and income into a single score between 0 and 1.
人类发展指数 (HDI) 是一种综合性的定理式指标，将健康、教育和收入合并为一个介于 0 到 1 之间的分值。
HDI = (Ihealth × I_education × Iincome)1/3
where Ihealth dex, I_education is the education index (mean years of schooling and expected years) and Iincome is the GNI per capita index. Even without computing the exact indices, you need to know that HDI provides a more rounded measure than income alone.
其中 Ihealth 为预期寿命指数，I_education 为教育指数（平均受教育年限与预期受教育年限），Iincome 为人均国民总收入指数。即便不精确计算各项指数，你也需要知道 HDI 比单纯的收入指标提供了更全面的衡量。
Other useful measures include the fertility rate, literacy rate and the percentage of population with access to clean water. These can be calculated using similar percentage formulas and often appear in data-response questions.
其他有用的指标包括生育率、识字率以及获得清洁饮用水的人口百分比。这些都可以使用类似的百分比公式计算，并经常出现在数据响应类问题中。
10. Spatial Interaction Theorems and Distance Decay | 空间相互作用定理与距离衰减法则
Spatial interaction in geography is governed by the distance decay principle: the intensity of any flow, connection or interaction generally decreases as distance increases.
地理学中的空间相互作用受距离衰减法则支配：任何流、连接或相互作用的强度通常随距离增加而减弱。
Interaction ∝ 1 / distance²
This theorem underpins the gravity model and helps explain phenomena such as the decline in shopping frequency as travel time to a retail centre grows.
这一定理是引力模型的基础，有助于解释诸如随着前往零售中心的出行时间增加
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