OCR Year 13 Geography: Formula & Theorem Quick Reference Guide | OCR 13年级地理：公式定理速查手册

📚 OCR Year 13 Geography: Formula & Theorem Quick Reference Guide | OCR 13年级地理：公式定理速查手册

This revision resource compiles the essential quantitative formulas and theoretical expressions that frequently appear in OCR A Level Geography (H481) examinations. Mastery of these tools is critical for data response questions, fieldwork analysis, and the synoptic Geographical Debates paper. The guide covers hydrological measurements, demographic indices, statistical tests, inequality measures, and spatial interaction models, providing clear definitions and applications for each.

本速查手册汇总了OCR A Level地理（H481）考试中频繁出现的关键定量公式与理论表达式。掌握这些工具对于数据响应题、实地考察分析以及综合性的地理议题试卷至关重要。指南涵盖水文测量、人口指数、统计检验、不平等指标和空间相互作用模型，为每一项提供清晰的定义和应用说明。

---

1. River Discharge & Hydraulic Radius | 河流流量与水力半径

Discharge is the volume of water passing a cross‑section of a channel per unit time. It is the product of the channel’s cross‑sectional area and the mean flow velocity. An accurate discharge value underpins flood hydrograph construction and water resource assessments.

流量是单位时间内通过河道某一横断面的水体体积，它是河道横截面积与平均流速的乘积。准确的流量值是洪水过程线绘制和水资源评估的基础。

Q = A × v

Here, Q is discharge (m³ s⁻¹), A is the wetted cross‑sectional area (m²), and v is the mean velocity (m s⁻¹). Velocity is often measured with a flow meter at multiple points across the channel and then averaged.

其中Q为流量（m³ s⁻¹），A为湿横截面积（m²），v为平均流速（m s⁻¹）。流速通常使用流速仪在河道不同点位测量后取平均值。

The hydraulic radius (R) expresses the efficiency of a channel’s shape in conveying water. It is the ratio of cross‑sectional area to wetted perimeter.

水力半径（R）反映了河道断面形状输送水流的效率，它是横截面积与湿周长的比值。

R = A / P

A larger hydraulic radius indicates less frictional resistance from the channel bed and banks, therefore a higher velocity for the same gradient. This concept is used in Manning’s equation for open channel flow.

水力半径越大，表明来自河床与河岸的摩擦阻力越小，因此在同一比降下流速越高。这一概念在曼宁明渠流方程中得到应用。

---

2. Demographic Indicators | 人口统计指标

Demographic calculations allow geographers to compare population structures and dynamics across countries. The crude birth rate (CBR) and crude death rate (CDR) are annual numbers of births and deaths per 1000 population.

人口统计学计算使地理学者能够比较不同国家的人口结构与动态。粗出生率（CBR）和粗死亡率（CDR）是每年每千人口中的出生和死亡人数。

CBR = (Total births / Total population) × 1000
CDR = (Total deaths / Total population) × 1000

Natural increase (NI) is the difference between births and deaths, and the rate of natural increase (RNI) is usually expressed as a percentage.

自然增长（NI）是出生与死亡人数之差，自然增长率（RNI）通常以百分比表示。

RNI (%) = [(CBR − CDR) / 10]

The dependency ratio measures the pressure on the productive population (aged 15–64) from the young (0–14) and old (65+) cohorts. A high ratio suggests a greater economic burden on the working‑age group.

抚养比衡量的是少儿（0–14岁）和老年（65岁以上）人口对劳动适龄人口（15–64岁）的压力。该比值较高意味着劳动年龄群体承受更大的经济负担。

Dependency Ratio = [(Pop₀₋₁₄ + Pop₆₅₊) / Pop₁₅₋₆₄] × 100

These indicators are key to interpreting the Demographic Transition Model and evaluating population policies.

这些指标是解读人口转变模型和评估人口政策的关键。

---

3. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数

Spearman’s rank tests the strength and direction of a monotonic relationship between two sets of ordinal or continuous data. It is frequently used in geographical investigations, for example comparing settlement size and the number of services.

斯皮尔曼秩相关检验用于测量两组顺序数据或连续数据之间单调关系的强度和方向。它在地理调查中经常使用，例如比较聚落规模与服务数量之间的关系。

rₛ = 1 − (6∑D²) / [n(n² − 1)]

In this formula, D is the difference between the ranks of each paired observation, and n is the number of data pairs. The coefficient rₛ ranges from +1 (perfect positive correlation) to −1 (perfect negative correlation), with 0 indicating no association.

公式中，D是每对观测值秩次的差值，n是数据对的数量。系数rₛ的取值范围从+1（完全正相关）到−1（完全负相关），0表示无关联。

Before computing rₛ, you must rank both datasets separately, then find the differences. The significance of the result is tested against critical values in Spearman’s rank table, considering the degrees of freedom (n − 2).

在计算rₛ之前，必须对两组数据分别进行排序，然后计算差值。结果的显著性需对照斯皮尔曼秩相关表中的临界值进行检验，自由度取（n − 2）。

---

4. Chi‑squared Test | 卡方检验

The chi‑squared (χ²) test determines whether there is a significant difference between observed frequencies and expected frequencies in one or more categories. It is widely applied to questionnaires, land‑use surveys, and distributions across space.

卡方（χ²）检验旨在判断一个或多个类别中观测频数与期望频数之间是否存在显著差异，广泛应用于问卷调查、土地利用调查和空间分布分析。

χ² = ∑ [(O − E)² / E]

O represents the observed frequency in each category, and E is the expected frequency under the null hypothesis. For a goodness‑of‑fit test, the expected value is often calculated by assuming an equal distribution across categories. For a test of association in a contingency table, expected values are derived from row and column totals.

O代表每一类别的观测频数，E代表原假设下的期望频数。对于拟合优度检验，期望值通常假设各类别均匀分布来计算。对于列联表中的关联检验，期望值则根据行列的合计数导出。

Once χ² is calculated, the degrees of freedom (df = number of categories − 1 for simplicity) are used to find the critical value. If χ² exceeds the critical value, the null hypothesis is rejected.

计算χ²后，使用自由度（简单拟合优度检验为类别数 − 1）查找临界值。若χ²大于临界值，则拒绝原假设。

---

5. Mann‑Whitney U Test | 曼‑惠特尼U检验

The Mann‑Whitney U test is a non‑parametric alternative to the t‑test for comparing two independent groups. It assesses whether one sample tends to have larger values than the other, using the ranks of all observations.

曼‑惠特尼U检验是t检验的一种非参数替代方法，用于比较两个独立组。它利用所有观测值的秩，判断是否一个样本的数值倾向于大于另一个样本。

U₁ = n₁n₂ + ½ n₁(n₁ + 1) − R₁
U₂ = n₁n₂ + ½ n₂(n₂ + 1) − R₂

Here, n₁ and n₂ are the sample sizes of the two groups, and R₁ and R₂ are the sum of ranks for each group. The smaller of U₁ and U₂ is compared with the critical value from the Mann‑Whitney U table. A significant result indicates a difference in the median values of the two populations.

其中n₁和n₂为两组样本量，R₁和R₂为每组的秩和。取U₁与U₂中的较小值，与曼‑惠特尼U表中的临界值进行比较。结果显著则表明两个总体的中位数存在差异。

Geographers use this test when data do not meet the normality assumption required for a t‑test, for instance in comparing environmental quality scores between two urban wards.

当数据不满足t检验的正态性假设时，地理学者会使用该检验，例如比较两个城市选区的环境质量评分。

---

6. Student’s t‑test | 学生t检验

The independent samples t‑test compares the means of two unrelated groups to determine if they are statistically different. It is suitable for continuous data that are approximately normally distributed.

独立样本t检验比较两个不相关组均值的差异是否具有统计学意义，适用于近似正态分布的连续数据。

t = (x̄₁ − x̄₂) / √[sₚ²(1/n₁ + 1/n₂)]

where the pooled variance sₚ² is given by

其中合并方差sₚ²由下式给出：

sₚ² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)

x̄₁ and x̄₂ are the sample means, s₁² and s₂² are the sample variances, and n₁ and n₂ are the sample sizes. Degrees of freedom equal (n₁ + n₂ − 2).

x̄₁与x̄₂为样本均值，s₁²与s₂²为样本方差，n₁与n₂为样本量，自由度为（n₁ + n₂ − 2）。

Once t is computed, compare it with the critical t‑value at the chosen significance level. If |t| exceeds the critical value, reject the null hypothesis of equal means.

计算出t值后，将其与选定显著性水平下的临界t值比较。若|t|大于临界值，则拒绝均值相等的原假设。

---

7. Location Quotient | 区位商

The location quotient (LQ) measures the concentration of a particular industry or economic activity in a local area relative to a larger reference region, typically the nation. It identifies regional specialisation.

区位商（LQ）衡量某一特定行业或经济活动在局部地区相对于更大参照区域（通常为国家）的集中程度，用以识别地区专门化。

LQ = (e₁ / E₁) ÷ (e₂ / E₂)

e₁ is the local employment in the industry, E₁ is total local employment, e₂ is national employment in that industry, and E₂ is total national employment. An LQ greater than 1 indicates a higher concentration than the national average, suggesting the region exports the good or service.

e₁为局部地区该产业就业人数，E₁为局部总就业人数，e₂为全国该产业就业人数，E₂为全国总就业人数。LQ大于1表示集中度高于全国平均水平，暗示该地区出口该产品或服务。

LQ is a foundational tool in economic geography for mapping functional regions and analysing agglomeration economies.

区位商是经济地理学中绘制功能区域和分析集聚经济的基础工具。

---

8. Gini Coefficient | 基尼系数

The Gini coefficient quantifies the degree of inequality in a distribution, such as income or land ownership, on a scale from 0 (perfect equality) to 1 (perfect inequality). It is derived from the Lorenz curve.

基尼系数量化某一分布（如收入或土地所有权）中的不平等程度，取值范围从0（完全平等）到1（完全不平等）。它源自洛伦兹曲线。

G = A / (A + B)

In this geometric definition, A is the area between the line of perfect equality and the Lorenz curve, while A+B is the total area under the diagonal. For discrete data, the coefficient can be approximated by numerical methods.

在这种几何定义中，A是完全平等线与洛伦兹曲线之间的面积，A+B是对角线下的总面积。对于离散数据，该系数可通过数值方法近似计算。

While A‑level exams rarely require manual computation from raw data, you must be able to interpret Gini values and relate them to spatial patterns of development or segregation.

虽然A Level考试很少要求根据原始数据进行手工计算，但必须能够解读基尼系数值，并将其与发展或隔离的空间格局联系起来。

---

9. Simpson’s Diversity Index | 辛普森多样性指数

Simpson’s Diversity Index (D) is used in ecosystem studies to measure biodiversity, accounting for both species richness and evenness. A higher D value indicates greater diversity.

辛普森多样性指数（D）用于生态系统研究中衡量生物多样性，同时考虑了物种丰富度和均匀度。D值越高表示多样性越大。

D = 1 − [∑ n(n − 1)] / [N(N − 1)]

where n is the number of individuals of a particular species and N is the total number of individuals of all species. The fraction ∑ n(n−1) / N(N−1) represents the probability that two randomly selected individuals belong to the same species.

其中n为某一特定物种的个体数，N为所有物种的个体总数。分数∑ n(n−1) / N(N−1)表示随机抽取的两个个体属于同一物种的概率。

This index is applied in sand dune succession studies, woodland sampling, and comparative ecosystem health assessments within OCR geographical investigations.

在OCR地理调查中，该指数常用于沙丘演替研究、林地采样以及生态系统健康状况比较评估。

---

10. Gravity Model | 引力模型

The gravity model is a spatial interaction theorem predicting the movement of people, goods, or information between two places, based on their size and the distance separating them. It draws an analogy with Newton’s law of gravitation.

引力模型是一条空间相互作用定理，它根据两地规模和相隔距离，预测人口、货物或信息之间的流动。该模型与牛顿万有引力定律有类比关系。

I = k × (P₁ × P₂) / D²

I is the interaction or flow between places 1 and 2, P₁ and P₂ are population sizes (or other mass variables like GDP), D is the distance between them, and k is a constant that accounts for other facilitating or hindering factors.

I为地点1与2之间的相互作用或流量，P₁和P₂为人口规模（或GDP等其他质量变量），D为两地之间距离，k为常数，用于解释其他促进或阻碍因素。

The model helps explain trade patterns, migration flows, and retail catchment areas. An increase in population size or a reduction in distance increases predicted interaction, which can be tested empirically.

该模型有助于解释贸易格局、迁移流和零售商圈。人口规模增大或距离缩短会增加预测的相互作用量，这可通过实证加以检验。

---

Published by TutorHao | Geography Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）