📚 Pre-U CIE Geography: Formula & Theorem Quick Reference Handbook | Pre-U CIE 地理：公式定理速查手册
This handbook compiles the essential formulas, equations and theoretical models required for the Pre-U CIE Geography course. It is designed as a rapid revision tool to help students quickly recall quantitative methods and key theorems for data response questions and examinations.
本手册汇编了 Pre-U CIE 地理课程中必备的公式、方程与理论模型。手册旨在作为快速复习工具，帮助学生迅速回忆定量方法和关键定理，以便应对数据分析题和考试。
1. Demographic Indicators | 人口统计指标
Crude Birth Rate (CBR) = (Number of live births in a year ÷ Total mid-year population) × 1000
粗出生率（CBR）=（一年内活产婴儿数 ÷ 年中总人口）× 1000
Crude Death Rate (CDR) = (Number of deaths in a year ÷ Total mid-year population) × 1000
粗死亡率（CDR）=（一年内死亡人数 ÷ 年中总人口）× 1000
Natural Increase Rate (%) = [(Number of births − Number of deaths) ÷ Total population] × 100, or (CBR − CDR) ÷ 10
自然增长率（%）= [（出生人数 − 死亡人数）÷ 总人口] × 100，或（CBR − CDR）÷ 10
Fertility Rate: General Fertility Rate (GFR) = (Number of live births ÷ Number of women aged 15–49) × 1000
一般生育率（GFR）=（活产婴儿数 ÷ 15–49 岁女性人数）× 1000
Total Fertility Rate (TFR) is the average number of children a woman would have, calculated by summing age-specific fertility rates and multiplying by 5 if in five-year groups.
总和生育率（TFR）指一名妇女一生平均生育子女数，采用各年龄组生育率求和（若按五岁分组则乘以5）来估算。
2. Population Density and Distribution | 人口密度与分布公式
Population Density = Total population ÷ Land area (persons per km²)
人口密度 = 总人口 ÷ 土地面积（人/平方公里）
Arithmetic Density is calculated using total land area, while Physiological Density uses arable land area: Physiological Density = Total population ÷ Arable land area.
算术密度采用土地总面积计算，而生理密度使用耕地面积：生理密度 = 总人口 ÷ 耕地面积。
Dependency Ratio = [(Population aged 0–14 + Population aged 65+) ÷ Population aged 15–64] × 100
抚养比 = [（0–14 岁人口 + 65 岁及以上人口）÷ 15–64 岁人口] × 100
3. Fertility and Mortality Formulas | 生育与死亡率公式
Infant Mortality Rate (IMR) = (Number of deaths of infants under 1 year ÷ Number of live births) × 1000
婴儿死亡率（IMR）=（一岁以下婴儿死亡数 ÷ 活产婴儿数）× 1000
Child Mortality Rate = (Number of deaths of children aged 1–4 ÷ Total population aged 1–4) × 1000
儿童死亡率 =（1–4 岁儿童死亡数 ÷ 1–4 岁总人口）× 1000
Life Expectancy at birth is derived from life tables; the calculation involves summing person-years lived beyond age x and dividing by survivors at age x. In Pre-U, you are expected to interpret rather than compute life expectancy directly.
出生时预期寿命由生命表推导；计算过程涉及累加 x 岁后的人口生存年数并除以 x 岁存活人数。在 Pre-U 阶段，要求学生能够解读预期寿命而非直接计算。
4. Migration and Urbanization Equations | 迁移与城市化公式
Net Migration = Number of immigrants − Number of emigrants
净迁移 = 迁入人数 − 迁出人数
Net Migration Rate = [(Immigrants − Emigrants) ÷ Total population] × 1000
净迁移率 = [（迁入人数 − 迁出人数）÷ 总人口] × 1000
Urbanization Level = (Urban population ÷ Total population) × 100
城市化水平 =（城镇人口 ÷ 总人口）× 100
Urban Growth Rate = [(Urban population at end of period − Urban population at start) ÷ Urban population at start] × 100, for a given time interval.
城镇人口增长率 = [（期末城镇人口 − 期初城镇人口）÷ 期初城镇人口] × 100，适用于给定时间段。
5. River Discharge and Gradient | 河流流量与坡度
Discharge (Q) = Cross-sectional area (A) × Mean velocity (V), where A is measured in m² and V in m/s, giving Q in m³/s.
流量（Q）= 横截面积（A）× 平均流速（V），其中 A 单位为平方米，V 单位为米/秒，Q 单位为立方米/秒。
Gradient of a river channel or slope = Vertical rise ÷ Horizontal distance, often expressed as a ratio (e.g., 1:50) or as a percentage: Gradient (%) = (Rise ÷ Run) × 100.
河床或山坡的坡度 = 垂直高差 ÷ 水平距离，常以比值表示（如 1:50）或百分比形式：坡度（%）=（高差 ÷ 平距）× 100。
Hydraulic Radius (R) = Cross-sectional area (A) ÷ Wetted perimeter (P), important for assessing channel efficiency.
水力半径（R）= 横截面积（A）÷ 湿周（P），对于评估河道效率很重要。
6. Spearman’s Rank Correlation | 斯皮尔曼等级相关系数
rₛ = 1 − [6 Σ d² ÷ (n(n² − 1))]
where d = difference in ranks of each pair, and n = number of pairs. The value ranges from −1 (perfect negative correlation) to +1 (perfect positive).
其中 d = 各对数据的秩次差，n = 数据对的个数。值域从 −1（完全负相关）到 +1（完全正相关）。
This test is used frequently in geography to assess the relationship between two variables such as distance from a river and sediment size, or GDP and life expectancy.
该检验在地理中常用于评估两个变量之间的关系，例如距河距离与沉积物粒径的关系，或 GDP 与预期寿命的关系。
Steps: rank each set of data separately, compute differences, square them, sum, and apply the formula. A significance table is used to test whether the correlation is statistically significant.
步骤：分别对两组数据排序赋秩，计算秩次差，平方后求和，再代入公式。通过与显著性临界值表比较，判断相关性是否统计显著。
7. The Gravity Model | 引力模型
Iᵢⱼ = k × (Pᵢ × Pⱼ) ÷ D²
where Iᵢⱼ is the interaction between places i and j, Pᵢ and Pⱼ are population sizes, D is the distance between them, and k is a constant. This model is used to predict flows of people, goods, or information.
其中 Iᵢⱼ 是 i 地和 j 地之间的相互作用，Pᵢ 和 Pⱼ 是人口规模，D 是两地距离，k 为常数。该模型用于预测人流、物流或信息流。
In geographical analysis, the exponent of distance may vary; a squared distance (power 2) is common, but it can be adjusted to reflect the friction of distance for different phenomena.
在地理分析中，距离的指数可能变化；平方距离（幂次 2）常见，但可根据不同现象的“距离摩擦”进行调整。
The gravity model is foundational for understanding spatial interaction and is linked to retail location, migration patterns, and trade flows.
引力模型是理解空间相互作用的基础，与零售业选址、迁移模式和贸易流相关联。
8. Rank-Size Rule | 位序–规模法则
Pₙ = P₁ ÷ n
where Pₙ is the population of the nth largest city, P₁ is the population of the largest city, and n is the rank. This rule describes a regular pattern in many developed-country urban systems.
其中 Pₙ 是第 n 大城市的人口，P₁ 是最大城市的人口，n 为位序。该法则描述了在许多发达国家的城镇体系中存在的规律模式。
When plotted on logarithmic scales, a country that follows the rank-size rule will show a straight-line relationship between rank and population size.
在双对数坐标上，遵循位序–规模法则的国家，其城市位序与人口规模会呈现直线关系。
A primate city occurs where the largest city is more than twice the size of the second city, indicating deviation from the rule; thus the rank-size rule helps identify primacy.
当最大城市的人口超过第二大城市的两倍时，即为“首位城市”，表明偏离此法则；因此位序–规模法则有助于识别首位分布。
9. Central Place Theory | 中心地理论
Christaller’s central place theory proposes a hierarchy of settlements with hexagonal market areas. The theory uses the concepts of range (maximum distance consumers will travel) and threshold (minimum population needed to support a service).
克里斯塔勒的中心地理论提出了六边形市场区的聚落等级体系。该理论使用“商品服务范围”（消费者愿意出行的最远距离）和“门槛人口”（维持服务所需的最低人口）两个概念。
The k-value (e.g., k=3, k=4, k=7) determines the arrangement of central places: a k=3 network follows a marketing principle, k=4 follows a transport principle, and k=7 an administrative principle.
K 值（如 k=3, k=4, k=7）决定了中心地的布局：k=3 网络遵循市场原则，k=4 遵循交通原则，k=7 遵循行政原则。
Formulas for calculating the number of next lower-order centres served by a higher-order centre depend on the chosen k-value, but the emphasis in Pre-U is on understanding spatial patterns rather than extensive computation.
高等级中心所服务的低等级中心数量的计算公式取决于所选的 k 值，但 Pre-U 的重点在于理解空间格局，而非大量计算。
10. Development Indicators | 发展指标
Human Development Index (HDI) combines three dimensions: health (life expectancy at birth), education (mean years of schooling and expected years), and standard of living (GNI per capita).
人类发展指数（HDI）综合了三个维度：健康（出生时预期寿命）、教育（平均受教育年限和预期受教育年限）和生活水平（人均国民总收入）。
Each dimension is normalised to an index between 0 and 1 using the formula: Dimension index = (actual value − minimum value) ÷ (maximum value − minimum value). The HDI is the geometric mean of the three dimension indices.
每个维度指标通过公式标准化为 0~1 之间的值：维度指数 =（实际值 − 最小值）÷（最大值 − 最小值）。HDI 是三个维度指数的几何平均数。
Gini coefficient summarises income inequality; while not formula-based in Pre-U exams, it is derived from the Lorenz curve. It ranges from 0 (perfect equality) to 1 (perfect inequality).
基尼系数描述收入不平等程度；尽管 Pre-U 考试不要求公式计算，它通过洛伦兹曲线求得。系数范围从 0（完全平等）到 1（完全不平等）。
11. Climate and Weather Formulas | 气候与天气公式
Annual Temperature Range = Mean temperature of warmest month − Mean temperature of coldest month.
年温差 = 最暖月均温 − 最冷月均温。
Total Annual Precipitation = Sum of monthly precipitation for all 12 months.
年降水总量 = 12 个月各月降水量之和。
Potential Evapotranspiration (PET) often estimated using Thornthwaite’s formula, but in Pre-U students are more likely to interpret a water-balance diagram: P − PET = surplus or deficit.
潜在蒸散量（PET）常用 Thornthwaite 公式估算，但 Pre-U 学生更可能解读水量平衡图：降水量 − PET = 盈余或亏缺。
Humidity calculations: Relative Humidity (%) = (Actual vapour pressure ÷ Saturation vapour pressure) × 100.
湿度计算：相对湿度（%）=（实际水汽压 ÷ 饱和水汽压）× 100。
12. Distance Decay Function | 距离衰减函数
The general distance decay model states that interaction declines as distance increases. A simple form: I = k × P₁ × P₂ × e⁻ᵇᵈ or inversely proportional to some power of distance.
距离衰减的一般模型表明，相互作用随距离增加而减弱。简化形式：I = k × P₁ × P₂ × e⁻ᵇᵈ，或与距离的某次幂成反比。
In spatial geography, this concept underpins theories such as the gravity model and helps explain the friction of distance affecting migration, shopping trips, and information flow.
在空间地理中，此概念是引力模型等理论的基础，有助于解释影响迁移、购物出行和信息流的距离摩擦。
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