📚 Year 13 CIE Geography: Formula & Theorem Quick Reference Handbook | Year 13 CIE 地理：公式定理速查手册
This quick-reference handbook collects the essential quantitative tools, indices and theoretical models that Year 13 CIE Geography students need to apply confidently in Paper 3 and Paper 4. Each section presents a concise formula or theorem, explains its purpose and provides a worked example where appropriate. Use it alongside your case studies to strengthen data-response and evaluation skills.
本速查手册汇总Year 13 CIE地理学生在试卷3和试卷4中必须自信运用的核心定量工具、指数和理论模型。每个小节都以简洁的公式或定理展开，说明其用途并在适当处配以计算示例。配合案例研究使用，可有效提升数据应答与评估能力。
1. Crude Birth Rate, Crude Death Rate & Natural Increase | 粗出生率、粗死亡率与自然增长率
The Crude Birth Rate (CBR) is the number of live births per 1 000 population in a given year. It is calculated as: CBR = (B ÷ P) × 1 000, where B = total live births and P = mid-year total population.
粗出生率 (CBR) 指某年每千人口中的活产数。计算公式为：CBR = (B ÷ P) × 1 000，其中B为活产总数，P为年中总人口。
Similarly, the Crude Death Rate (CDR) = (D ÷ P) × 1 000. The Rate of Natural Increase (RNI) is then obtained as (CBR − CDR) ÷ 10, expressed as a percentage. If CBR is 28 ‰ and CDR is 8 ‰, RNI = (28 − 8) ÷ 10 = 2.0% per annum.
类似地，粗死亡率 (CDR) = (D ÷ P) × 1 000。自然增长率 (RNI) 则为 (CBR − CDR) ÷ 10，以百分比表示。若CBR为28‰、CDR为8‰，则RNI = (28 − 8) ÷ 10 = 2.0% 每年。
Doubling time, assuming exponential growth, can be approximated by the ‘Rule of 70’: Doubling time (years) ≈ 70 ÷ RNI (%). A 2% growth rate gives a doubling time of about 35 years.
在指数增长假设下，可用“70法则”估算人口倍增时间：倍增时间（年）≈ 70 ÷ RNI (%)。2%的增长率对应约35年的倍增时间。
2. Dependency Ratio & Ageing Index | 抚养比与老龄化指数
The total dependency ratio measures the pressure on the productive population. It is defined as: [(P(0–14) + P(65+)) ÷ P(15–64)] × 100. A result of 60 means 60 dependents for every 100 working-age people.
总抚养比衡量生产性人口的负担，公式为：[(P(0–14) + P(65+)) ÷ P(15–64)] × 100。结果为60表示每100名劳动年龄人口需抚养60名非劳动年龄人口。
The youth dependency ratio uses only P(0–14) in the numerator, and the old-age dependency ratio uses P(65+). Both help identify different pressures on health, education and pension systems.
少儿抚养比仅以 P(0–14) 为分子，老年抚养比则以 P(65+) 为分子。两者有助于识别健康、教育和养老金体系面临的不同压力。
The Ageing Index is another useful metric: (P(65+) ÷ P(0–14)) × 100. Values above 100 indicate more elderly than children, a key sign of an ageing society such as Japan or Italy.
老龄化指数是另一实用指标：(P(65+) ÷ P(0–14)) × 100。数值超过100表示老年人口多于儿童，是日本、意大利等老龄化社会的关键标志。
3. Population Density & Urbanisation Level | 人口密度与城市化水平
Arithmetic population density = Total population ÷ Total land area (persons per km²). This crude measure hides internal variations. Agricultural density = Rural population ÷ Arable land area, useful for studying subsistence pressure.
算术人口密度 = 总人口 ÷ 土地总面积（人/km²）。这一粗略指标掩盖了内部差异。农业密度 = 农村人口 ÷ 耕地面积，用于研究生计压力。
Urbanisation level is the proportion of a country’s population living in urban areas: (Urban population ÷ Total population) × 100. The rate of urbanisation refers to the annual percentage increase in this proportion, not to be confused with urban population growth rate.
城市化水平指一国城镇人口占总人口的比例：(城镇人口 ÷ 总人口) × 100。城市化速度则指该比例的年均百分点增长，不可与城镇人口增长率混淆。
4. Rank-Size Rule | 位序–规模法则
The rank-size rule describes the relationship between city size and its rank in the urban hierarchy. If the largest city has population P1, the population of the rth ranked city is approximately P1 ÷ r. In a perfect rank-size distribution, the 2nd city has half the population of the 1st, the 3rd has one-third, and so on.
位序–规模法则描述了城市规模与其在城市体系中位序的关系。若最大城市人口为 P1，则第 r 位城市的人口约为 P1 ÷ r。在完美位序–规模分布中，第二大城市人口为第一大城市的一半，第三大城市为三分之一，以此类推。
A generalised form is Pr = P1 × r−q, where q is a constant often close to 1. Deviations from the rule (q ≠ 1) indicate primacy or a fragmented urban system. CIE candidates should be able to plot log city size against log rank and interpret the slope.
一般形式为 Pr = P1 × r−q，其中 q 为常数，通常接近1。偏离该法则（q ≠ 1）表明存在首位分布或破碎化的城市体系。CIE考生应能绘制对数城市规模与对数位序图，并解释其斜率。
5. Primacy Index | 首位度指数
The two-city primacy index = P1 ÷ P2. A value greater than 2 often signals a primate city where the largest city dominates the urban landscape economically and politically, as in Bangkok or London.
二城市首位度指数 = P1 ÷ P2。若数值大于2，通常表明存在首位城市，最大城市在经济和政治上主导城市体系，如曼谷、伦敦。
The four-city primacy index = P1 ÷ (P2 + P3 + P4). This broader index reduces the distortion caused by a single small second city. It is often used alongside the Gini coefficient to measure urban concentration.
四城市首位度指数 = P1 ÷ (P2 + P3 + P4)。该指标覆盖更广，可减少因单一小规模第二城市带来的扭曲，常与基尼系数结合使用以衡量城市集中度。
6. Gravity Model & Reilly’s Law of Retail Gravitation | 重力模型与赖利零售引力定律
The gravity model states that the interaction (I) between two places is proportional to the product of their populations and inversely proportional to the distance between them: Iij = k (Pi × Pj) ÷ dij2, where k is a constant. It explains migration and trade flows.
重力模型指出两地间的相互作用 (I) 与它们的人口乘积成正比，与距离的平方成反比：Iij = k (Pi × Pj) ÷ dij2，其中 k 为常数。该模型用于解释人口迁移和贸易流。
Reilly’s law adapts this to retail trade: two competing centres A and B attract customers from an intermediate settlement in proportion to their size and inversely to the square of distance. It helps define retail hinterlands.
赖利定律将其应用于零售贸易：两个竞争中心A、B从中间居民点吸引的顾客量与各自规模成正比，与距离平方成反比，可用于划定零售腹地。
7. Breaking Point (Converse’s) Formula | 断裂点（康弗斯）公式
The breaking point between two competing urban centres is the location where customers are equally likely to visit either centre. Converse’s formula gives the distance from centre A: dA = DAB ÷ (1 + √(PB ÷ PA)), where DAB is the total distance between A and B.
两个竞争城市中心之间的断裂点是消费者前往任一中心概率相等的位置。康弗斯公式给出距中心A的距离：dA = DAB ÷ (1 + √(PB ÷ PA))，其中 DAB 为A与B间的总距离。
Example: A has 160 000 residents, B has 40 000, and they are 60 km apart. dA = 60 ÷ (1 + √(40 000 ÷ 160 000)) = 60 ÷ (1 + 0.5) = 40 km from A. The breaking point lies closer to the smaller centre.
示例：A拥有160 000人口，B拥有40 000人口，相距60 km。dA = 60 ÷ (1 + √(40 000 ÷ 160 000)) = 60 ÷ (1 + 0.5) = 40 km，断裂点更靠近较小的中心。
8. Network Connectivity: Beta, Alpha & Gamma Indices | 网络连通性：Beta、Alpha与Gamma指数
Transport network analysis employs graph theory. The Beta Index (β) measures average connections per node: β = e ÷ v, where e = number of edges (links) and v = number of vertices (nodes). Higher β means greater network complexity.
交通网络分析运用图论。Beta指数 (β) 衡量每个节点的平均连接数：β = e ÷ v，其中 e 为边（连线）数，v 为顶点（节点）数。β值越高代表网络越复杂。
The Alpha Index (α) evaluates the number of circuits: α = (e − v + 1) ÷ (2v − 5) × 100%. It ranges from 0% (no redundancy) to 100% (maximum connectivity). The Gamma Index (γ) is the ratio of observed to maximum possible edges: γ = e ÷ (3(v − 2)) × 100%. These indices help compare subway, road or rail networks.
Alpha指数 (α) 评价回路数量：α = (e − v + 1) ÷ (2v − 5) × 100%。范围从0%（无冗余）至100%（最大连通度）。Gamma指数 (γ) 为实际边数与最大可能边数之比：γ = e ÷ (3(v − 2)) × 100%。这些指数可用于比较地铁、公路或铁路网络。
9. Location Quotient (LQ) | 区位商
Location Quotient assesses whether a region has a higher share of employment in a specific industry compared to the national average. LQ = (ei ÷ e) ÷ (Ei ÷ E), where ei = local employment in industry i, e = total local employment, Ei = national employment in industry i, E = total national employment.
区位商用于判断某地区特定产业就业比例是否高于全国平均水平。LQ = (ei ÷ e) ÷ (Ei ÷ E)，其中 ei 为地方i产业就业人数，e 为地方就业总人数，Ei 为全国i产业就业人数，E 为全国就业总人数。
An LQ > 1 indicates a regional economic specialisation; LQ < 1 suggests underrepresentation. It is widely used in economic geography to map clusters and analyse comparative advantage.
LQ > 1 表明该地区经济具有专业化特征；LQ < 1 则表示该产业代表性不足。这一指数广泛用于经济地理的产业集聚识别和比较优势分析。
10. Simpson’s Diversity Index | 辛普森多样性指数
Simpson’s Index measures the probability that two randomly selected individuals belong to different groups, often applied to ethnic diversity or ecosystem species richness. D = 1 − Σ (ni ÷ N)2, where ni is the number of individuals in group i and N is the total sample size.
辛普森指数衡量随机选取的两个个体属不同群体的概率，常用于民族多样性或生态系统物种丰富度研究。D = 1 − Σ (ni ÷ N)2，其中 ni 为第i组个体数，N 为样本总数。
Values near 1 indicate high diversity; values near 0 indicate low diversity. The reciprocal form 1 ÷ Σ (ni/N)2 gives the effective number of equally common groups. CIE questions may provide tables requiring manual calculation.
数值趋近1表示高度多样；趋近0表示低多样。其倒数形式 1 ÷ Σ (ni/N)2 可求出同等常见组的有效数量。CIE考题常提供表格要求手工计算。
11. Drainage Basin Morphometry: Drainage Density & Stream Gradient | 流域形态测量：河网密度与河流梯度
Drainage density (Dd) indicates how well a basin is drained: Dd = total stream length (km) ÷ drainage basin area (km²). High values (>2 km/km²) often correlate with impermeable surfaces, steep slopes and rapid runoff, increasing flood risk.
河网密度 (Dd) 表示流域排水程度：Dd = 河道总长度 (km) ÷ 流域面积 (km²)。高值（>2 km/km²）常与透水差的地表、陡坡和快速径流相关，增加洪水风险。
Stream gradient is the steepness of a river channel segment: Gradient = vertical drop (m) ÷ horizontal distance (m), often expressed as a ratio or percentage. It influences flow velocity, erosion and sediment transport capacity.
河流梯度是河段陡峭程度的量度：梯度 = 垂直落差 (m) ÷ 水平距离 (m)，通常以比率或百分比表示，影响流速、侵蚀和输沙能力。
The bifurcation ratio (Rb) from Horton-Strahler ordering is another theorem-based measure: Rb = number of streams of one order ÷ number of streams of the next higher order. In natural basins, Rb typically ranges 3–5.
基于霍顿-斯特拉勒分级的分叉比 (Rb) 是另一项定理式指标：Rb = 某级河流数目 ÷ 下一更高级河流数目。天然流域中Rb 通常保持在3–5之间。
12. Lorenz Curve & Gini Coefficient | 洛伦兹曲线与基尼系数
The Lorenz curve depicts the cumulative share of income (or another attribute) against the cumulative share of population. A 45° line represents perfect equality. The Gini coefficient quantifies inequality: G = A ÷ (A + B), where A is the area between the line of equality and the Lorenz curve, and B is the area below the Lorenz curve.
洛伦兹曲线描绘收入（或其他属性）累积份额与人口累积份额的关系。45°线代表完全平等。基尼系数定量衡量不平等程度：G = A ÷ (A + B)，A为平等线与洛伦兹曲线之间的面积，B为洛伦兹曲线下方的面积。
Gini values range from 0 (perfect equality) to 1 (maximum inequality). A practical approximation using decile data is: G ≈ 1 − Σ (xi − xi−1)(yi + yi−1), where x is cumulative population share and y is cumulative income share. Students should interpret Gini figures in connection with development levels and social policies.
基尼系数值域为0（绝对平等）到1（绝对不平等）。利用十分位数据的实用近似公式为：G ≈ 1 − Σ (xi − xi−1)(yi + yi−1)，其中x为累积人口份额，y为累积收入份额。学生应结合发展水平和社会政策解读基尼系数。
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