📚 A-Level OCR Geography: Quick Reference Handbook for Formulas and Theorems | A-Level OCR 地理：公式定理速查手册
In A-Level OCR Geography, quantitative skills and the application of relevant formulas play a critical role in both physical and human geography, as well as in the geographical investigation (NEA). This handbook compiles the essential equations, statistical tests, and conceptual models you must be confident in using during your exams and fieldwork write‑ups. Each entry is presented with a clear definition, the formula itself in accessible notation, and a short explanation of its geographic significance. Whether you are measuring river discharge, analysing population change, or testing for relationships in your data, this revision guide will help you work accurately and efficiently.
在 A-Level OCR 地理课程中，定量技能和相关公式的应用在自然地理、人文地理以及地理探究（NEA）中都起着至关重要的作用。本手册汇编了你在考试和实地考察报告中必须熟练掌握的核心公式、统计检验和概念模型。每一个知识点都配有清晰的定义、用易读符号呈现的公式，以及对其地理意义的简要说明。无论你是在测量河流流量、分析人口变化，还是在检验数据之间的关系，这本复习指南都能帮助你准确、高效地完成任务。
1. River Discharge and Hydraulic Radius | 河流流量与水力半径
River discharge is the volume of water flowing through a cross‑section of a channel per unit time. It is a foundational measure in hydrology for comparing river regimes and flood risk.
河流流量是指单位时间内流过河道某一横截面的水体体积。它是水文学中比较河流情势和洪水风险的基础指标。
Q = A × V
Where Q = discharge (m³ s⁻¹), A = cross‑sectional area (m²), V = mean velocity (m s⁻¹).
其中 Q = 流量（立方米/秒），A = 横截面积（平方米），V = 平均流速（米/秒）。
Hydraulic radius helps assess channel efficiency. It is calculated as cross‑sectional area divided by wetted perimeter.
水力半径用来衡量河道的输水效率，其计算公式为横截面积除以湿周。
R = A / P
Where R = hydraulic radius (m), P = wetted perimeter (m). A larger hydraulic radius generally indicates greater efficiency and higher velocity.
其中 R = 水力半径（米），P = 湿周（米）。水力半径越大，通常意味着输水效率越高、流速越快。
2. Bradshaw Model | 布拉德肖模型
The Bradshaw Model is a conceptual framework that predicts how key river characteristics change systematically from source to mouth. It can be summarised using proportional arrows rather than a single formula, but the underlying relationships can be expressed in simplified mathematical terms.
布拉德肖模型是一个概念框架，用于预测河流从源头到河口关键特征的系统变化。它通常用比例箭头而非单一公式来概括，但背后的关系可以用简化的数学语言来表达。
Discharge increases downstream: Q increases as more tributaries join, so A and V both tend to rise. Velocity generally increases downstream despite local variations caused by friction and channel roughness. The relationship can be remembered as: as distance from source (d) increases, Q, V, channel depth, channel width, and hydraulic radius all tend to increase, while gradient and bedload size decrease.
流量向下游递增：随着更多支流汇入，Q 增加，因此 A 和 V 都趋于上升。流速总体上向下游增大，尽管受摩擦和河床糙度影响会有局部变化。这套关系可以记为：随着距源头距离（d）的增加，Q、V、河道深度、河道宽度和水力半径都趋于增大，而坡度和推移质粒径趋于减小。
In an exam context, you may be required to interpret scattergraphs of these variables against distance downstream and comment on the degree of scatter or anomalies that relate to human intervention or geological controls.
在考试中，你可能需要解读这些变量随下游距离变化的散点图，并对与人类活动或地质控制因素相关的离散程度或异常值进行评述。
3. Population Change Formulas | 人口变化公式
Crude Birth Rate (CBR) is the number of live births per 1000 population per year.
粗出生率是每年每千人口中的活产婴儿数。
CBR = (B / P) × 1000
Where B = number of live births, P = mid‑year population. Similarly, Crude Death Rate (CDR) = (D / P) × 1000, with D representing number of deaths.
其中 B = 活产数，P = 年中人口数。类似地，粗死亡率 CDR = (D / P) × 1000，D 代表死亡人数。
Rate of Natural Increase (RNI) is the difference between CBR and CDR, often expressed as a percentage.
自然增长率（RNI）是粗出生率与粗死亡率的差值，通常以百分数表示。
RNI (%) = (CBR − CDR) / 10
This conversion is needed because CBR and CDR are per 1000, whereas RNI is conventionally quoted as a percentage. Population growth rate also accounts for net migration: Growth Rate = (Natural Increase + Net Migration) / Population × 100.
需要进行这种换算，因为 CBR 和 CDR 以每千人为单位，而 RNI 习惯上以百分数表示。人口增长率还需计入净迁移：增长率 = (自然增长 + 净迁移) / 人口 × 100。
4. Population Density and Dependency Ratio | 人口密度与抚养比
Population density measures how crowded an area is. It is a simple but powerful indicator in urban and resource geography.
人口密度衡量一个地区的拥挤程度，是城市与资源地理中简单而有用的指标。
Population Density = Total Population / Land Area (km²)
The dependency ratio expresses the proportion of the economically dependent population (0–14 and 65+) relative to the working‑age population (15–64).
抚养比表示非劳动适龄人口（0–14 岁和 65 岁以上）相对于劳动适龄人口（15–64 岁）的比例。
Dependency Ratio = [(Population 0–14 + Population 65+) / Population 15–64] × 100
A high dependency ratio indicates greater pressure on the productive workforce. OCR questions often ask you to compare dependency ratios between countries at different stages of the Demographic Transition Model.
高抚养比表明劳动力人群承受的压力更大。OCR 考题常要求比较处于人口转变模型不同阶段的国家之间的抚养比。
5. Urbanisation and Location Quotient | 城市化率与区位商
Urbanisation rate can be measured using the percentage of a population living in areas classified as urban.
城市化率可以用居住在城市分类区域的人口所占百分比来衡量。
Urban Population (%) = (Urban Population / Total Population) × 100
The Location Quotient (LQ) allows geographers to identify spatial concentrations of a particular industry or activity relative to a wider region.
区位商（LQ）帮助地理学者识别某一特定产业或活动相对于更大全域的空间集中程度。
LQ = (e / t) / (E / T)
Where e = local employment in the sector, t = total local employment, E = regional (or national) employment in that sector, T = total regional (or national) employment. An LQ > 1 indicates a higher concentration than the wider area.
其中 e = 当地该部门就业人数，t = 当地总就业人数，E = 更大区域（或全国）该部门就业人数，T = 更大区域总就业人数。LQ > 1 表示比更大区域更为集中。
6. Measures of Central Tendency and Dispersion | 集中趋势与离散程度测量
In geographical investigation, summarising data using averages and spread is essential before applying more advanced statistics.
在地理探究中，使用平均数和分散度概括数据是应用更高级统计方法前的基础步骤。
Mean (x̄) = Σ x / n
Median = middle value in ordered list
The interquartile range (IQR) reduces the influence of outliers.
四分位距（IQR）可减少离群值的影响。
IQR = Q₃ − Q₁
Where Q₁ is the 25th percentile and Q₃ is the 75th percentile. The median and IQR are often preferred when data are skewed, as with settlement sizes or flood magnitudes.
其中 Q₁ 是第 25 百分位数，Q₃ 是第 75 百分位数。当数据偏斜时（如聚落规模或洪水强度），往往更宜采用中位数和四分位距。
7. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s Rank is used to test the strength and direction of a relationship between two sets of ranked data, such as distance downstream and bedload size, or GNI per capita and access to clean water.
斯皮尔曼等级相关系数用于检验两组排序数据之间关系的强度和方向，例如下游距离与推移质粒径，或人均国民总收入与清洁水获取率。
rₛ = 1 − [ (6 Σ d²) / (n(n² − 1)) ]
Where d = difference in ranks for each paired observation, n = number of pairs. The value lies between −1 and +1. You must then test for statistical significance using a critical values table for Spearman’s Rank. OCR expects you to state a null hypothesis and determine whether the correlation is significant at the 95% or 99% confidence level.
其中 d = 每对观测值的秩次差，n = 配对数。数值介于 −1 到 +1 之间。随后必须查阅斯皮尔曼等级相关系数临界值表进行统计显著性检验。OCR 大纲要求你提出零假设，并判断相关在 95% 或 99% 置信水平上是否显著。
8. Chi‑Squared Test (χ²) | 卡方检验
Chi‑squared is a technique that compares observed frequencies with expected frequencies to determine whether there is a significant difference between them. It is used for categorical data, for instance, to compare the distribution of different land uses in a transect or to test regional variations in soil type.
卡方检验是一种将观察频数与期望频数进行比较、以确定两者是否存在显著差异的方法。它用于分类数据，例如比较样带内不同土地利用的分布，或检验土壤类型的区域差异。
χ² = Σ [ (O − E)² / E ]
Where O = observed frequency, E = expected frequency. The computed value is compared with a critical value from the χ² distribution table using (number of categories − 1) degrees of freedom. You must remember that expected frequencies should be at least 5 in most cases for the test to be valid.
其中 O = 观察频数，E = 期望频数。将计算值与卡方分布表的临界值进行比较，自由度为（分类数 − 1）。必须记住，在大多数情况下，期望频数应至少为 5，检验才有效。
9. Mann‑Whitney U Test | 曼‑惠特尼 U 检验
The Mann‑Whitney U test is a non‑parametric test used to determine whether there is a significant difference between two independent sets of data that are not normally distributed. It is widely applied in geography fieldwork, for example, when comparing infiltration rates at two contrasting sites.
曼‑惠特尼 U 检验是一种非参数检验，用于判断两列非正态分布的独立数据之间是否存在显著差异。它在地理实地考察中被广泛应用，例如比较两个不同地点的人渗率。
U₁ = n₁n₂ + ½ n₁(n₁ + 1) − R₁
U₂ = n₁n₂ + ½ n₂(n₂ + 1) − R₂
The smaller U value is used for comparison with the critical value. R₁ and R₂ are the sums of ranks for each group. As with other significance tests, you set up a null hypothesis and reject it only if the calculated U is equal to or less than the critical value.
取较小的 U 值与临界值进行比较。R₁ 和 R₂ 为各组秩次的总和。与其他显著性检验一样，你需要建立零假设，并且仅当计算出的 U 值等于或小于临界值时，才拒绝零假设。
10. Standard Deviation and Variance | 标准差与方差
Standard deviation quantifies the spread of data around the mean. A small standard deviation indicates data points are clustered closely near the mean, while a large one shows wide dispersion. This is particularly useful in geographical inquiry when comparing variations in pebble size along a beach or variations in annual precipitation.
标准差衡量数据围绕平均值的分散程度。标准差小表示数据点紧密聚集在均值附近，标准差大则表示离散程度高。在地理探究中，这在比较海滩砾石大小变化或年降水量变化时尤为有用。
σ = √ [ Σ (x − x̄)² / n ]
For a sample, the formula uses n − 1 in the denominator. Variance is simply σ². OCR may ask you to comment on what the standard deviation reveals about the reliability of the mean or to compare standard deviations across different sample sites.
对于样本数据，公式的分母应使用 n − 1。方差就是 σ²。OCR 可能会要求你评述标准差对均值可靠性的揭示作用，或比较不同采样点的标准差。
11. Soil Moisture Budget | 土壤水分收支
A soil moisture budget is a seasonal accounting system that compares precipitation (P) with potential evapotranspiration (PET) to determine soil water status. It is not a single formula but a sequence of calculations that you need to interpret on graphs.
土壤水分收支是按季节划分的一种核算体系，通过比较降水量（P）与潜在蒸散量（PET）来判断土壤水分状况。它不是一个单一公式，而是一系列需要结合图表解读的计算过程。
When P > PET, there is a soil moisture surplus and recharge. When PET > P and the soil moisture store is exhausted, a deficit occurs. The calculation of actual evapotranspiration depends on available soil moisture. You may be presented with a graph of the annual regime and asked to label periods of surplus, deficiency, recharge, and utilisation.
当 P > PET 时，出现土壤水分盈余和补充；当 PET > P 且土壤水分储存消耗殆尽时，则出现亏缺。实际蒸散量的计算取决于可用的土壤水分。你可能会遇到年度水分收支图，并被要求标出盈余、亏缺、补充和利用的时段。
Fieldwork applications often utilise basic water balance: ΔS = P − (Q + ET + I), where ΔS is the change in storage, Q is runoff, ET is evapotranspiration, and I is interception. This is useful in drainage basin studies.
野外调查中常应用基础水量平衡方程：ΔS = P − (Q + ET + I)，其中 ΔS 为储水变化量，Q 为径流，ET 为蒸散，I 为截流。这在流域研究中非常有用。
12. Key Practical Checks in Quantitative Work | 定量分析中的关键实用检查
When applying any formula in OCR Geography, always check your units. Discharge, for example, must be expressed in cubic metres per second, and cross‑sectional area in square metres. In rates of natural increase, ensure you correctly convert between per‑mille and percentage forms.
在 OCR 地理中应用任何公式时，务必检查单位。比如，流量必须以立方米/秒表示，横截面积须为平方米。在自然增长率中，要确保千分比与百分比的正确转换。
Always include a clear statement of the null hypothesis where a statistical test is used. Show all working and state the degrees of freedom or p‑value where relevant. In fieldwork reports, it is good practice to discuss the limitations of your chosen statistical test and the possible influence of outliers or sample size.
凡使用统计检验，都要写明清晰的零假设。展示全部计算过程，并在适用时给出自由度或 p 值。在实地考察报告中，最佳做法是讨论所选统计检验的局限以及离群值或样本量可能产生的影响。
Published by TutorHao | Geography Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导