📚 Year 13 CAIE Geography: Formula & Theorem Quick Reference Handbook | A Level地理公式定理速查手册
This handbook compiles essential quantitative formulae, indices and geographical theorems commonly required for Year 13 CAIE Geography (9696) data‑response, fieldwork and examination questions. Each entry is presented with a plain‑English explanation followed by an equivalent Chinese explanation, together with the corresponding mathematical expression using standard Unicode notation. You can use this as a rapid revision aid before Paper 2 and Paper 4 assessments.
本手册汇编了 Year 13 CAIE 地理（9696）数据题、实地调查题和考试中常见的核心定量公式、指数和地理定律。每个条目先给出英文说明，再配以对应的中文解释，并用标准 Unicode 符号呈现数学表达式。可将其作为 Paper 2 与 Paper 4 考前快速复习工具。
1. Demographic Rates and Theorems | 人口统计率与定律
Crude Birth Rate (CBR) is the number of live births in a year per 1,000 people in the total mid‑year population.
CBR = (Total live births / Total population) × 1000
粗出生率（CBR）指一年内每1000名年中总人口对应的活产婴儿数。
Crude Death Rate (CDR) applies the same logic to total deaths.
CDR = (Total deaths / Total population) × 1000
粗死亡率（CDR）同理，以总死亡人数计算。
The Natural Increase Rate (NIR) captures the surplus of births over deaths, often expressed as a percentage.
NIR = (CBR – CDR) / 10 (% per annum)
自然增长率（NIR）表示出生多于死亡的部分，通常以百分比表示。
Net Migration Rate accounts for the difference between immigrants and emigrants.
Net migration rate = (Immigrants – Emigrants) / Total population × 1000
净迁移率计算迁入与迁出人口的差额。
Population doubling time is estimated using the ‘Rule of 70’.
Doubling time (years) = 70 / Annual growth rate (%)
人口翻倍时间可用“70法则”估算。
Ravenstein’s laws of migration summarise key empirical regularities: most migrants move short distances, migration occurs in stages, and each major flow generates a counter‑flow. These laws underpin many migration‑analysis questions.
拉文斯坦迁移定律总结了关键的实证规律：大多数迁移者移动短距离、迁移呈阶梯状、每一主要迁移流都引发反向流。这些定律是许多迁移分析题的基石。
2. Population Density Measures | 人口密度度量
Arithmetic density expresses population pressure on total land.
Arithmetic density = Total population / Total land area (persons/km²)
算术人口密度反映人口对全部国土的压力。
Physiological density relates population to cultivable land, revealing subsistence pressure.
Physiological density = Total population / Arable land area
生理密度将人口与耕地面积联系，揭示生存压力。
Agricultural density compares the number of farmers to arable land, an indicator of agricultural efficiency.
Agricultural density = Number of farmers / Arable land area
农业密度比较农民数量与耕地面积，反映农业生产效率。
3. Urbanisation Indices | 城市化指数
Level of urbanisation indicates the proportion of people living in urban areas.
Urbanisation level = (Urban population / Total population) × 100
城市化水平表示居住在城镇区域的人口比例。
Rate of urbanisation measures the speed of urban growth, typically using an exponential growth formula.
Annual urban growth rate = [ (Pₜ / P₀)1/n – 1 ] × 100
城市化速度通常用指数增长公式计算年均增长率。
Urban primacy index highlights the dominance of the largest city over the second‑largest.
Primacy index = Population of largest city / Population of second‑largest city
城市首位度指数突出最大城市相对于第二大城市的优势程度。
4. Hydrological Formulae | 水文学公式
River discharge is the volume of water passing a cross‑section per unit time.
Discharge (Q) = Cross‑sectional area (A) × Mean velocity (V) (m³/s)
河流流量指单位时间通过横截面的水体体积。
Drainage density quantifies the total stream length per unit area, reflecting basin efficiency.
Drainage density = Total stream length / Drainage basin area (km/km²)
河网密度量化单位面积上的河道总长，反映流域排水效率。
Stream frequency counts the number of channel segments per unit area.
Stream frequency = Number of stream segments / Drainage basin area
河流频数统计单位面积上的河段数目。
5. Agricultural Productivity Calculations | 农业生产力计算
Crop yield is the most direct measure of land productivity.
Crop yield = Total output (kg or tonnes) / Area under crop (ha)
农作物产量是土地生产力的最直接量度。
Livestock density reflects grazing pressure.
Livestock density = Number of livestock units / Area of pasture or grassland
牲畜密度反映草场承载压力。
The location quotient (LQ) reveals regional specialisation in a particular crop or activity.
LQ = (Share of crop A in region / Share of crop A in nation)
区位商揭示某地区在特定作物或产业上的专业化程度。
6. Climatic Indices and Variability | 气候指数与变率
Mean daily temperature is frequently approximated by averaging the maximum and minimum.
Tₘₑₐₙ = (Tₘₐₓ + Tₘᵢₙ) / 2
日平均气温通常取最高气温与最低气温的平均值。
Diurnal temperature range indicates daily thermal variation.
Diurnal range = Tₘₐₓ – Tₘᵢₙ
气温日较差反映一日内的温度变化幅度。
Precipitation variability is captured by the coefficient of variation, essential for assessing rainfall reliability.
ability coefficient = (Standard deviation of annual rainfall / Mean annual rainfall) × 100
降水变率用变异系数衡量，是评价降雨可靠性的关键指标。
7. Economic and Development Indicators | 经济与发展指标
GDP per capita is a common proxy for average living standards.
GDP per capita = Total GDP / Total population
人均国内生产总值是平均生活水平的常用代表。
Dependency ratios measure the burden on the productive population.
Youth dependency ratio = (Population aged 0–14 / Population aged 15–64) × 100
Old‑age dependency ratio = (Population aged 65+ / Population aged 15–64) × 100
抚养比衡量劳动年龄人口所承受的抚养负担。
The Gini coefficient is derived from the Lorenz curve and measures income inequality.
Gini = 1 – Σ (Xₖ – Xₖ₋₁)(Yₖ + Yₖ₋₁)
基尼系数由洛伦兹曲线导出，用于量度收入差距程度。
8. Statistical Methods for Fieldwork | 田野调查的统计方法
Spearman’s rank correlation tests the strength and direction of association between two ranked variables.
rₛ = 1 – (6 Σ d²) / (n (n² – 1))
斯皮尔曼等级相关系数检验两个排序变量之间的关联强度与方向。
The mean, or arithmetic average, summarises central tendency.
Mean (x̄) = Σx / n
算术平均数总括数据的集中趋势。
Sample standard deviation quantifies dispersion around the mean.
s = √[ Σ(x – x̄)² / (n – 1) ]
样本标准差衡量数据相对于均值的离散程度。
9. Gradient and Rate of Change Formulae | 坡度与变化率公式
Slope gradient is expressed as a ratio of vertical rise to horizontal distance, commonly used on topographic maps and beach profiles.
Gradient = Vertical interval / Horizontal distance
坡度用垂直高差与水平距离的比值表达，常见于地形图与海滩剖面计算。
Rate of coastal erosion or recession is a linear measure of shoreline retreat.
Erosion rate = Change in shoreline position (m) / Time period (years)
海岸侵蚀速率是岸线后退的线性量度。
Rate of urban sprawl can be estimated from built‑up area change.
Sprawl rate = (New built‑up area – Old built‑up area) / Time period
城市蔓延速率可用建成区面积变化来衡量。
10. Gravity Model and Distance Decay | 重力模型与距离衰减
The gravity model predicts interaction between two places based on their population sizes and the distance separating them, illustrating the distance‑decay principle.
Interaction I = (P₁ × P₂) / D²
重力模型基于两地人口规模与距离预测相互作用强度，体现了距离衰减原理。
Distance decay is a spatial theorem stating that interaction decreases as distance increases; often graphed as a negative exponential curve. It underpins retail location, migration and service provision models.
距离衰减是一条空间定律，指相互作用随距离增加而减弱，常表现为负指数曲线。它是零售区位、迁移与服务供应模型的基础。
11. Atmospheric Moisture and Environmental Indices | 大气湿度与环境指数
Relative humidity links the actual moisture content of the air to its maximum capacity at a given temperature.
RH = (eₐ / eₛ) × 100
相对湿度将空气实际水汽含量与同温度下的最大容量联系起来。
Ecological footprint per capita provides a measure of human demand on natural resources.
Per capita footprint = Total ecological footprint / Total population
人均生态足迹衡量人类对自然资源的需求压力。
Percentage forest cover is a simple environmental quality indicator.
Forest cover (%) = (Forest area / Total land area) × 100
森林覆盖率是一个简单的环境质量指标。
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