Pre-U Edexcel Geography: Formula & Theorem Quick Reference Guide | Pre-U Edexcel 地理：公式定理速查手册

📚 Pre-U Edexcel Geography: Formula & Theorem Quick Reference Guide | Pre-U Edexcel 地理：公式定理速查手册

This quick reference guide compiles the essential quantitative formulas, models and theorems required for the Pre‑U Edexcel Geography course. Whether you are analysing population change, measuring development gaps or applying spatial interaction models, having these tools at your fingertips will sharpen your data‑response and essay answers. All formulas are presented in their simplest symbolic form, using only standard Unicode characters; no specialist typesetting is needed.

本速查手册汇集了 Pre‑U Edexcel 地理课程中必备的定量公式、模型与定理。无论是分析人口变化、衡量发展差距，还是运用空间相互作用模型，掌握这些工具都能让你的数据应答和论述答题更加犀利。所有公式均以最简单的符号形式呈现，仅使用标准 Unicode 字符，无需特殊排版。

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1. Population Structure and Change | 人口结构与变动

The most fundamental demographic formulas measure births, deaths and natural change. Always express rates per 1000 population unless otherwise specified.

最基本的人口统计公式用于衡量出生、死亡和自然变动。除非另有说明，率均以每千人为单位表达。

Crude Birth Rate (CBR) = (Number of live births ÷ Total population) × 1000

CBR = (B ÷ P) × 1000

Crude Death Rate (CDR) = (Number of deaths ÷ Total population) × 1000

CDR = (D ÷ P) × 1000

Natural Increase Rate (NIR) = CBR − CDR

NIR = (B − D) ÷ P × 1000

When examining population growth over time, the doubling time can be estimated by the ‘Rule of 70’:

在考察人口随时间增长时，倍增时间可用“70法则”估算：

Doubling time (years) ≈ 70 ÷ Annual growth rate (%)

T₂ ≈ 70 ÷ r

These simple rates underpin the Demographic Transition Model and are often the starting point for any population‑resource analysis.

这些简单的比率支撑着人口转变模型，通常是任何人口‑资源分析的起点。

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2. Fertility and Mortality Measures | 生育率与死亡率度量

Crude rates can be misleading because they ignore age structure. More refined measures are essential for comparing populations with different age profiles.

粗率可能具有误导性，因为它们忽略了年龄结构。在比较不同年龄构成的人口时，更精细的度量必不可少。

General Fertility Rate (GFR) = (Live births ÷ Women aged 15−49) × 1000

GFR = (B ÷ W₁₅₋₄₉) × 1000

Total Fertility Rate (TFR) = Sum of Age‑Specific Fertility Rates (ASFRs) over reproductive ages

TFR = Σ ASFRₓ (where x = 15 to 49)

Infant Mortality Rate (IMR) = (Deaths under 1 year ÷ Live births) × 1000

IMR = (D₀₋₁ ÷ B) × 1000

Life Expectancy at Birth (e₀) is derived from a life table and represents the average number of years a newborn would live if current age‑specific mortality rates remain constant.

出生时预期寿命 (e₀) 导自生命表，表示在当前的年龄别死亡率保持不变的条件下，新生儿预期存活的平均年数。

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3. Migration and Net Migration | 迁移与净迁移

Population change in a region results from both natural change and net migration. The basic balance equation is:

一个地区的人口变化由自然变动和净迁移共同决定。基本平衡方程为：

Population Change = (Births − Deaths) + (Immigration − Emigration)

ΔP = (B − D) + (I − E)

Net Migration Rate = [(Immigration − Emigration) ÷ Total population] × 1000

NMR = (I − E) ÷ P × 1000

Push–pull models and Lee’s migration theory both use these quantities qualitatively, but the net migration rate quantifies the impact. For internal migration, the in‑migration rate and out‑migration rate are calculated similarly.

推拉模型和李氏迁移理论都定性使用这些量，但净迁移率量化了影响。对于国内迁移，迁入率和迁出率以类似方式计算。

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4. Urbanisation and Density | 城市化与密度指标

Urbanisation metrics are central to the study of settlement and development. Two essential formulas are:

城市化指标是聚落与发展研究的核心。两个基本公式为：

Percentage Urban = (Urban population ÷ Total population) × 100

U% = (P_urban ÷ P_total) × 100

Population Density = Total population ÷ Land area (km²)

D = P ÷ A

For more nuanced analysis, physiological density (population per unit of arable land) and agricultural density (farmers per unit of arable land) can reveal pressure on productive land.

为了更细致的分析，生理密度（每单位可耕地人口）和农业密度（每单位可耕地农民数）能揭示生产性土地的压力。

Physiological Density = P ÷ Arable land area

D_phys = P ÷ A_arable

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5. Economic and Development Indicators | 经济与发展指标

Pre‑U candidates must handle a range of composite indices. The most used is the Human Development Index (HDI), essentially a geometric mean of three normalised indices:

Pre‑U 考生必须处理一系列综合指数。最常用的是人类发展指数 (HDI)，实质上是三个归一化指数的几何平均数：

HDI = (I_health × I_education × I_income)^(1/3)

HDI = ∛(I_h × I_e × I_y)

Each dimension index is calculated using the formula: Index = (Actual value − Minimum) ÷ (Maximum − Minimum). For income, the natural logarithm of GNI per capita is used.

每个维度指数使用以下公式计算：指数 = (实际值 − 最小值) ÷ (最大值 − 最小值)。对于收入，使用人均国民总收入的自然对数。

Gender Inequality Index (GII) and Multidimensional Poverty Index (MPI) follow similar normalisation principles, though their structures are more complex. You are expected to interpret, not recalculate, these indices in the exam.

性别不平等指数 (GII) 和多维贫困指数 (MPI) 遵循类似的归一化原则，但结构更为复杂。考试要求你会解读而非重新计算这些指数。

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6. Location Quotient | 区位商

The location quotient (LQ) is a simple ratio used to measure the concentration of an industry or occupation in a region relative to a larger reference area. It is invaluable for economic geography and regional planning questions.

区位商 (LQ) 是一个简单比率，用于衡量某产业或职业在一个地区相对于较大参照区域的集中程度。它对经济地理和区域规划问题极有价值。

LQ = (Regional employment in sector ÷ Total regional employment) ÷ (National employment in sector ÷ Total national employment)

LQ = (eᵢ ÷ e_total) ÷ (Eᵢ ÷ E_total)

An LQ greater than 1 indicates a higher concentration than the national average, suggesting the sector may be part of the region’s basic (export) economy. An LQ less than 1 suggests a net importing sector.

LQ 大于 1 表示集中度高于全国平均水平，暗示该产业可能是区域基础（出口）经济的一部分。LQ 小于 1 则暗示该产业为净进口型。

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7. Spatial Interaction: The Gravity Model | 空间相互作用：重力模型

The gravity model predicts the interaction between two places based on their size and distance. Its general form is:

重力模型根据两地的大小和距离预测它们之间的相互作用。其一般形式为：

Interaction Iᵢⱼ = k × (Pᵢ × Pⱼ) ÷ Dᵢⱼᵝ

Iᵢⱼ = k (Pᵢ Pⱼ) / Dᵢⱼᵝ

where P represents population (or mass, e.g., employment, retail floorspace), D is the distance between them, k is a constant, and β is the distance‑decay exponent (typically between 1 and 2). When using retail floorspace, P is replaced by W (attractiveness).

其中 P 表示人口（或质量，如就业、零售营业面积），D 为两者之间的距离，k 是常数，β 为距离衰减指数（通常在 1 到 2 之间）。使用零售面积时，P 用 W（吸引力）替换。

In a retail context, Reilly’s Law of Retail Gravitation gives the break‑point between two centres:

在零售情境中，赖利零售引力定律给出了两个中心之间的断裂点：

Dₐ = Dₐ₆ ÷ (1 + √(P₆/Pₐ))

Dₐ = Dₐ₆ / [1 + √(P₆/Pₐ)]

where Dₐ is the distance from centre A to the break point, Dₐ₆ is the distance between A and B, and Pₐ, P₆ are the populations of the two centres.

其中 Dₐ 是从中心 A 到断裂点的距离，Dₐ₆ 是 A 与 B 之间的距离，Pₐ、P₆ 是两个中心的人口。

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8. Distance Decay and Diffusion | 距离衰减与扩散

Distance decay describes how interaction declines as separation increases. A simple exponential decay is often used:

距离衰减描述了相互作用如何随着分离距离增加而下降。通常使用简单的指数衰减：

Interaction I = I₀ × e^(−λd) or I = 1 ÷ dⁿ

I = I₀ e^(−λd) or I = d⁻ⁿ

The concept underpins Hägerstrand’s spatial diffusion model, where the rate of adoption of an innovation depends on the ‘neighbourhood effect’ and distance from the source. The probability of contact is often modelled as:

这一概念支撑着哈格斯特朗的空间扩散模型，其中创新采纳率取决于“近邻效应”和与源地的距离。接触概率常建模为：

P = a × d⁻ᵇ

P = a d⁻ᵇ

These formulas, though simple, are powerful in explaining the wave‑like spread of phenomena such as a pandemic, a new farming technique or a retail chain.

这些公式虽然简单，但在解释大流行病、新农业技术或零售连锁店等波浪式传播现象时十分有力。

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9. Central Place Theory and the Range of a Good | 中心地理论与商品范围

Christaller’s Central Place Theory defines the range of a good or service as the maximum distance a consumer is willing to travel. It can be expressed using a simple distance function:

克里斯泰勒的中心地理论将商品或服务的范围定义为消费者愿意出行的最大距离。可用简单的距离函数表达：

Range = D_max such that Demand ≥ Threshold

R = D_max, where D ≤ (price indifference) / (travel cost per km)

The threshold is the minimum demand needed to support a service. The central place hierarchy (K=3, K=4, K=7) specifies the number of lower‑order settlements served by a higher‑order centre, but it is a spatial theorem, not a numeric formula in the exam.

门槛 是支撑一项服务所需的最低需求量。中心地等级体系 (K=3, K=4, K=7) 规定了一个高级中心所服务的低级居民点的数量，但它在考试中是一个空间定理，而非数值公式。

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10. Rank-Size Rule | 等级规模法则

The rank‑size rule describes the statistical regularity between the population of a city and its rank in the urban hierarchy. The classic form is:

等级规模法则描述了城市人口与其在城市等级体系中位序之间的统计规律。经典形式为：

Population of city of rank r = Population of largest city ÷ r

P_ r = P₁ ÷ r

For greater accuracy, many geographers use P_ r = P₁ ÷ r^q where q is the Zipf coefficient (often close to 1). A value of q > 1 indicates a more primate distribution; q < 1 suggests a more even size distribution.

为了更精确，许多地理学家使用P_ r = P₁ ÷ r^q，其中 q 是齐普夫系数（通常接近 1）。q > 1 表示更集中的首位分布；q < 1 则表明城市规模分布更均匀。

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11. Environmental and Sustainability Metrics | 环境与可持续性度量

Ecological footprint and carbon footprint are increasingly examined. A simplified carbon footprint formula is:

生态足迹和碳足迹正日益成为考点。一个简化的碳足迹公式为：

Carbon Footprint = Σ (Activity data × Emission factor)

CF = Σ (AD × EF)

For water stress, the Falkenmark Indicator is a commonly used formula:

对于水资源压力，法尔肯马克指标是一个常用公式：

Water per capita = Total renewable water resources ÷ Population

Water per capita (m³/year) = Q ÷ P

A value below 1700 m³ per person per year indicates water stress; below 1000 m³ indicates scarcity. These thresholds appear frequently in data‑response questions.

每年人均低于 1700 m³ 表示水资源压力；低于 1000 m³ 表示水资源稀缺。这些阈值经常出现在数据应答题中。

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12. Spearman’s Rank Correlation | 斯皮尔曼等级相关系数

When testing relationships between ranked data (e.g., GDP rank and HDI rank), Spearman’s rank correlation coefficient is the required tool. The formula is:

在检验排序数据（如 GDP 排名与 HDI 排名）之间的关系时，斯皮尔曼等级相关系数是必备工具。其公式为：

r_s = 1 − [6 Σ d² ÷ n(n² − 1)]

r_s = 1 − (6 Σ d²) / [n(n² − 1)]

where d is the difference between the ranks of each pair, and n is the number of pairs. A value of +1 indicates perfect positive correlation, −1 perfect negative correlation. You should always test the significance using a critical values table.

其中 d 为每对数据的等级差，n 为数据对的数量。值为 +1 表示完全正相关，−1 为完全负相关。你应始终使用临界值表来检验显著性。

While not a ‘theorem’, this statistical technique is embedded in geographical enquiry skills and frequently appears in Pre‑U examination tasks.

虽然这并非“定理”，但此统计技术是地理探究技能的组成部分，并频繁出现在 Pre‑U 考试任务中。

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