📚 A-Level CAIE Geography: Formula and Key Calculations Quick Reference | A-Level CAIE 地理：公式与关键计算速查手册
This quick-reference handbook brings together the essential formulas, indices and statistical tests that appear throughout the CAIE A-Level Geography course (9696). Whether you are analysing river discharge, calculating population change or applying Spearman’s rank correlation, memorising these equations and understanding their components will save time in the exam and strengthen your data-response answers.
本速查手册汇集了 CAIE A-Level 地理课程 (9696) 中贯穿始终的关键公式、指数和统计检验。无论你是在分析河流流量、计算人口变化还是应用斯皮尔曼等级相关，熟记这些方程并理解其组成部分都能在考试中为你节省时间，并强化数据应答题的答案。
1. River Discharge and Velocity | 河流流量与流速
Discharge (Q) is the volume of water flowing past a given cross-section per unit of time. It is fundamental to understanding flood risk, channel capacity and the energy available for erosion and transportation.
流量（Q）是单位时间内流过某一断面的水体体积。它是理解洪水风险、河道容量以及可用于侵蚀和搬运的能量的基础。
Q = A × V
Where Q = discharge (m³ s⁻¹ or cumecs), A = cross-sectional area of the channel (m²), and V = mean velocity of the flow (m s⁻¹). The area is calculated as width × mean depth for a roughly rectangular channel.
其中 Q = 流量（m³ s⁻¹ 或 cumecs），A = 河道横截面积（m²），V = 平均流速（m s⁻¹）。对于近似矩形的河道，面积可按宽度 × 平均水深计算。
Velocity can be measured directly with a flowmeter or float, but it is often estimated using the Manning equation in more advanced physical geography questions. Always check that the units are consistent: if width and depth are given in metres, area will be in m² and discharge in m³ s⁻¹.
流速可直接用流速仪或浮标法测量，但在更高阶的自然地理题目中常通过曼宁公式进行估算。务必检查单位是否统一：若给定宽度与水深以米为单位，则面积单位就是 m²，流量即为 m³ s⁻¹。
2. Hydraulic Radius and Manning’s Equation | 水力半径与曼宁公式
The hydraulic radius (R) describes the efficiency of a channel cross-section to convey water. A larger hydraulic radius means less friction per unit of water, so flow is more efficient.
水力半径（R）描述河道截面的输水效率。水力半径越大，单位水体的摩擦越小，因此水流更高效。
R = A / WP
Where A = cross-sectional area (m²) and WP = wetted perimeter (m) — the length of the channel bed and banks in contact with the water. For a wide, shallow trapezoidal channel, R is approximately equal to mean depth.
其中 A = 横截面积（m²），WP = 湿周（m）——与水接触的河床及河岸线段长度。对于宽浅的梯形河道，R 约等于平均水深。
Manning’s equation links velocity to hydraulic radius, channel slope and roughness:
曼宁公式将流速与水力半径、河道坡降和粗糙度联系起来：
V = (R²/³ × S¹/²) / n
Here S is the water surface slope (gradient, dimensionless or m m⁻¹), and n is Manning’s roughness coefficient (dimensionless). Roughness increases with boulders, vegetation and sinuous alignments. This equation is used to estimate flood-stage discharge when velocity cannot be measured directly.
式中 S 为水面比降（无量纲或 m m⁻¹），n 为曼宁粗糙系数（无量纲）。粗糙度随大石块、植被及蜿蜒平面形态而增大。当无法直接测量流速时，可用此方程估算洪水期流量。
3. Channel Geometry: Gradient and Sinuosity | 河道几何：坡降与弯曲度
Gradient measures the steepness of the channel or valley and strongly influences flow velocity and the stream’s capacity to erode and transport sediment.
坡降衡量河道或谷底的陡峭程度，对流速及水流侵蚀、搬运泥沙的能力有显著影响。
Gradient = vertical interval / horizontal distance
For example, a drop of 50 m over a horizontal distance of 2000 m gives a gradient of 50 ÷ 2000 = 0.025, often expressed as 1 in 40 or 2.5 %.
例如，水平距离 2000 m 上落差 50 m，坡降为 50 ÷ 2000 = 0.025，常表示为 1:40 或 2.5 %。
Sinuosity describes how much a channel meanders compared with the straight-line valley distance. A sinuosity greater than 1.5 defines a meandering channel.
弯曲度描述河道相对于谷底直线距离的蜿蜒程度。弯曲度大于 1.5 即定义为曲流河道。
Sinuosity = channel length / straight‑line valley length
Both measurements can be obtained from topographic maps. High sinuosity reduces gradient per unit channel length and encourages deposition on point bars.
两项长度均可从地形图上量取。高弯曲度降低了单位河道长度的比降，促使凸岸点坝沉积。
4. Drainage Basin Analysis: Density and Stream Order | 流域分析：河网密度与河流等级
Drainage density is a numerical expression of how well or poorly a basin is drained by surface streams. It is sensitive to geology, slope, vegetation cover and climate.
河网密度是地表水系对流域排水程度的数值表达，对岩性、坡度、植被覆盖和气候十分敏感。
Drainage density = total length of streams (km) / basin area (km²)
High drainage density (e.g. on impermeable clays) indicates rapid runoff and a flashy hydrograph; low density (e.g. on permeable chalk) suggests slower, delayed responses.
高河网密度（如在不透水黏土上）表明坡面汇流迅速，形成暴涨暴落的流量过程线；低密度（如在透水白垩上）则意味着较缓慢、延迟的响应。
Stream order (Strahler method) is used to classify streams by hierarchy: when two first-order streams join, they form a second-order stream; two second-order streams form a third order, and so on. Bifurcation ratio can then be calculated.
河流等级（斯特拉勒法）用于按层次对河道分级：两条一级河流汇合形成二级河流；两条二级河流汇合形成三级，依此类推。进而可计算分叉比。
5. Population Change Indicators | 人口变动指标
The crude birth rate (CBR) and crude death rate (CDR) are expressed per 1000 of the total population to allow comparison between countries of different sizes.
粗出生率（CBR）与粗死亡率（CDR）以每千人口表示，便于不同规模国家间进行比较。
CBR = (number of live births / total population) × 1000
CDR = (number of deaths / total population) × 1000
Natural increase (or natural change) is simply the difference between CBR and CDR, and it is often converted to a percentage:
自然增长（或自然变动）即为 CBR 与 CDR 的差值，常转换为百分比：
Rate of natural increase (%) = (CBR – CDR) / 10
The total population growth rate also includes net migration. Remember that net migration = in-migrants – out-migrants, and the overall growth rate = (natural increase + net migration) ÷ mid-year population × 100.
人口总增长率还包括净迁移。注意净迁移 = 迁入人口 – 迁出人口，总增长率 =（自然增长 + 净迁移）÷ 年中总人口 × 100。
6. Fertility, Mortality and Dependency Ratios | 生育率、死亡率与抚养比
The total fertility rate (TFR) is the average number of children a woman would bear if she experienced the prevailing age-specific fertility rates throughout her childbearing years (usually ages 15–49). It is more precise than CBR.
总和生育率（TFR）是指一代妇女按当前各年龄组生育率度过整个育龄期（通常 15–49 岁）平均生育的子女数，比 CBR 更精确。
The dependency ratio is a key measure of the economic pressure on the productive population.
抚养比是衡量生产性人口经济压力的关键指标。
Dependency ratio = [(population 0–14 + population 65+) / population 15–64] × 100
A high dependency ratio suggests a greater burden on the working-age group. The youth dependency ratio and old-age dependency ratio can also be calculated separately to distinguish between young and ageing structures.
高抚养比意味着劳动年龄群体负担更重。少儿抚养比与老年抚养比可分别计算，以区分年轻型与老龄型人口结构。
Infant mortality rate (IMR) = (deaths of infants under 1 year / live births in same year) × 1000. It is a sensitive indicator of healthcare, nutrition and sanitation.
婴儿死亡率（IMR）=（年内未满1周岁婴儿死亡数 / 同年活产数）× 1000。它是衡量医疗、营养和环境卫生水平的敏感指标。
7. Population Density and Urbanisation | 人口密度与城市化率
Arithmetic (crude) population density is the simplest measure of how crowded a given area is.
算术（粗）人口密度是衡量某区域拥挤程度最简单的指标。
Population density = total population / land area (km²)
Physiological density refines this by relating population to arable (cultivable) land, revealing pressure on food-producing resources.
生理密度则将人口与可耕地面积相联系，能揭示粮食生产资源的压力。
Urbanisation can be expressed as the percentage of a country’s population living in urban areas. The rate of urbanisation is the average annual change in this percentage.
城市化可用一个国家城市人口占总人口的百分比表示。城市化速度即为该百分比年均变幅。
Urbanisation rate (% per year) = (Uₜ₂ – Uₜ₁) / (t₂ – t₁)
where U represents the urban percentage at times t₁ and t₂. This helps compare the pace of rural-urban shift across countries.
式中 U 表示 t₁ 和 t₂ 时刻的城市人口百分比。这有助于比较各国城乡迁移的快慢。
8. Statistical Tests: Spearman’s Rank Correlation | 统计检验：斯皮尔曼等级相关系数
Spearman’s rank correlation (rₛ) is used to measure the strength and direction of the association between two sets of ranked data. It is widely applied in geography fieldwork, for example, to test whether river discharge increases with distance downstream or whether pedestrian density falls with distance from the CBD.
斯皮尔曼等级相关系数（rₛ）用于衡量两组排序数据之间关联的强度和方向。它广泛应用于地理实地考察中，例如检验河流流量是否随下游距离增大，或行人密度是否随离中心商务区距离增加而下降。
rₛ = 1 − [6 Σ d²] / [n (n² − 1)]
Where d = difference in ranks for each pair of values, and n = number of paired observations. The result ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation). A value close to 0 suggests no relationship.
其中 d = 每对数值的等级差，n = 配对观测值个数。结果范围从 −1（完全负相关）到 +1（完全正相关）。接近 0 则表明无关系。
When n is between 4 and 30, compare rₛ with critical values in Spearman’s rank table. If the calculated rₛ exceeds the critical value at the 0.05 significance level, the null hypothesis can be rejected and the relationship is statistically significant.
当 n 在 4 到 30 之间时，将 rₛ 与斯皮尔曼等级临界值表对比。若计算值 rₛ 超过 0.05 显著性水平的临界值，即可拒绝零假设，说明关系在统计上显著。
9. Chi-Squared Test | 卡方检验
The chi-squared (χ²) test is used to examine whether an observed frequency distribution differs significantly from an expected distribution. In geography, it can be applied to land-use mapping, questionnaire responses and settlement spacing.
卡方（χ²）检验用于检验观测频数分布是否与期望分布存在显著差异。在地理学中，可应用于土地利用制图、问卷调查和聚落间距分析。
χ² = Σ [ (O − E)² / E ]
O = observed frequency, E = expected frequency. Expected frequencies are typically calculated on the assumption of an even distribution or from a theoretical model. Degrees of freedom (df) = (number of categories – 1) when testing one variable.
O = 观测频数，E = 期望频数。期望值通常基于均匀分布假设或理论模型计算。检验单个变量时，自由度（df）= 类别数 − 1。
Compare the calculated χ² with the critical value at a chosen significance level (commonly p = 0.05). If χ² exceeds the critical value, the null hypothesis is rejected, indicating a significant difference between observed and expected patterns.
将计算所得 χ² 值与所选显著性水平（常用 p = 0.05）的临界值比较。若 χ² 超过临界值，则拒绝零假设，表明观测模式与期望模式存在显著差异。
10. Climate Calculations: Mean, Range and Variability | 气候计算：平均值、较差与变率
Describing a climate station’s data starts with basic measures of central tendency and spread.
描述气候站数据始于基本的集中趋势与离散程度量数。
Mean annual temperature = Σ monthly temperatures / 12
Annual temperature range = warmest monthly mean − coldest monthly mean
Precipitation totals are usually summed for the year, but the inter-annual variability can be expressed by the coefficient of variation or a simple variability index:
降水量通常按年求和，但年际变率可用变异系数或简易变率指数表示：
Precipitation variability index (%) = (standard deviation of annual totals / mean annual total) × 100
A high index indicates unreliable rainfall regime, a key concept when discussing semi-arid environments and food security. Additionally, the percentage contribution of each month to the annual total highlights the seasonal distribution.
指数高表明降水状况不稳定，这在讨论半干旱环境和粮食安全时是一个关键概念。此外，各月降水量占年总量的百分比可突出季节分配特征。
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