📚 AS OCR Geography: Formula & Theorem Quick Reference Guide | AS OCR 地理：公式定理速查手册
This hands-on reference compiles the essential equations, statistical tests, and empirical theorems that underpin the AS OCR Geography specification. From river discharge and water balance to population metrics, inequality indices, and spatial interaction models, every formula is presented with a clear definition, worked notation, and applied context. Use this guide to strengthen data-response skills, sharpen fieldwork analysis, and speed up revision of the quantitative methods required for both Physical and Human Geography papers.
本速查手册汇集了 AS OCR 地理课程所依托的关键公式、统计检验与经验定理。从河流流量、流域水平衡到人口指标、不平等指数和空间相互作用模型，每个公式都配有清晰的定义、标注符号与应用情境。借助本手册，你可以强化数据响应技能、提升野外作业分析水平，并快速梳理自然地理与人文地理试卷中必备的定量方法。
1. Drainage Basin Water Balance | 流域水平衡
The fundamental water balance equation for a drainage basin expresses the conservation of mass: P = Q + E + ΔS, where P is precipitation, Q is runoff (streamflow), E is evapotranspiration, and ΔS is the change in storage (soil moisture, groundwater, surface reservoirs) over a given time period. All components are typically expressed in millimetres per unit time.
流域水平衡的基本方程体现了质量守恒：P = Q + E + ΔS，其中 P 代表降水量，Q 为径流量（河川流量），E 代表蒸发散，ΔS 为给定时段内储存量（土壤水分、地下水、地表水库）的变化量。所有分量通常以单位时间的毫米数表示。
When ΔS is positive, storage is increasing and less precipitation immediately becomes runoff; a negative ΔS indicates the basin is releasing stored water, often sustaining baseflow during dry periods. In AS exam data exercises, you may be asked to calculate one missing term when the others are provided.
当 ΔS 为正值时，储水量上升，降水即时转化为径流的比例减少；负的 ΔS 表示流域正在释放储存水，常在于旱期维持基流。在 AS 考试的数据练习题中，你可能需要根据已知项计算缺失的一项。
P = Q + E + ΔS
2. River Discharge and Mean Velocity | 河流流量与平均流速
Discharge (Q) is the volume of water passing a cross-section per second. It is calculated by Q = A × v, where A is the cross-sectional area of the channel (in m²) and v is the mean velocity (in m/s). Discharge is usually given in cubic metres per second (cumecs).
流量 (Q) 是每秒通过河道横断面的水体体积，计算公式为 Q = A × v，其中 A 为河道横断面积（平方米），v 为平均流速（米/秒）。流量单位通常为立方米每秒（cumecs）。
Field measurement often uses the float method: timing a floating object over a known distance gives surface velocity. Multiply by a correction factor (commonly 0.8 for turbulent flow) to estimate mean velocity. The area is derived from width and a series of depth measurements across the channel.
野外常采用浮标法测速：记录浮标流过已知距离的时间，得到水面流速；再乘以校正系数（湍流常用 0.8）估算平均流速。横断面积由河宽和一系列水深测量数据求得。
Q = A × v v = d / t
3. Storm Hydrograph Parameters | 洪水过程线参数
A storm hydrograph records discharge over time in response to a rainfall event. Lag time is the interval between peak rainfall intensity and peak discharge. The rising limb shows how quickly discharge increases, while the recession (falling) limb reflects the rate at which water stored in the basin drains away.
洪水过程线记录了一次降雨事件中流量随时间的变化。滞时是降雨强度峰值与洪峰流量之间的时间差。涨水段显示流量上升的快慢，退水段则反映流域内储存水排出的速率。
The rainfall-runoff relationship can be quantified by the runoff coefficient: C = Qtotal / Ptotal, where Qtotal is total storm runoff depth and Ptotal is total rainfall depth (in mm). A high coefficient (>0.5) suggests rapid runoff generation, often linked to impermeable surfaces or saturated soils.
降雨-径流关系可通过径流系数量化：C = Q 总 / P 总，其中 Q 总 为暴雨总径流深度（毫米），P 总 为总降雨深度（毫米）。系数偏高（>0.5）表明径流生成迅速，往往与不透水地表或饱和土壤有关。
C = Qtotal / Ptotal
4. Population Dynamics – Vital Rates | 人口动态——关键率
Crude Birth Rate (CBR) and Crude Death Rate (CDR) are expressed per 1000 population per year. CBR = (B / P) × 1000 and CDR = (D / P) × 1000, where B is live births, D is deaths, and P is mid-year population. The Rate of Natural Increase (RNI) is then CBR − CDR, usually stated as a percentage: RNI(%) = (CBR − CDR) / 10.
粗出生率 (CBR) 和粗死亡率 (CDR) 以每年每千人为单位。CBR =（B / P）× 1000，CDR =（D / P）× 1000，其中 B 为活产数，D 为死亡数，P 为年中人口。自然增长率 (RNI) 为 CBR − CDR，通常以百分比表示：RNI(%)= (CBR − CDR) / 10。
Population doubling time provides an intuitive measure of growth momentum. Under exponential growth, Td ≈ 70 / RNI(%), where RNI is expressed as a percentage. This ‘rule of 70’ is widely used to compare demographic trajectories of countries at different stages of the Demographic Transition Model.
人口倍增时间直观地衡量增长势头。在指数增长假设下，T 倍增 ≈ 70 / RNI(%)，其中 RNI 以百分比表示。这一”70 法则”广泛用于比较处于人口转变模型不同阶段的国家轨迹。
CBR = (B / P) × 1000 CDR = (D / P) × 1000 RNI(%) = (CBR − CDR)/10
5. Migration and Net Migration Rate | 迁移与净迁移率
Net migration is the balance between immigration (I) and emigration (E): NM = I − E. The Net Migration Rate (NMR) per 1000 population is NMR = (I − E) / P × 1000. A positive NMR indicates net in-migration; a negative value signals net out-migration.
净迁移是入境移民 (I) 与出境移民 (E) 的差额：NM = I − E。每千人的净迁移率 (NMR) 为 NMR = (I − E) / P × 1000。NMR 为正表示净迁入，为负则表示净迁出。
Total population change over a period combines natural change and net migration: ΔP = (B − D) + (I − E). In exam scenarios, you may be given vital rates and migration totals to compute population growth or decline, testing your ability to synthesise demographic data.
某一时期的总人口变动结合了自然变化和净迁移：ΔP = (B − D) + (I − E)。考试情境中，可能会给出生命率和迁移总量要求你计算人口增长或减少，这是对你整合人口数据能力的检验。
NM = I − E NMR = (I − E) / P × 1000
6. Age-Sex Structure and Dependency Ratios | 年龄-性别结构与赡养比
The Total Dependency Ratio (TDR) measures the proportion of economically dependent individuals relative to the working-age population. TDR = [(P0−14 + P65+) / P15−64] × 100. The young-age and old-age dependency ratios can be calculated separately to reveal the specific pressures on social services.
总赡养比 (TDR) 衡量经济依赖人口与劳动年龄人口的比值。TDR = [(P ₀₋₁₄ + P ₆₅₊) / P ₁₅₋₆₄] × 100。少儿赡养比与老年赡养比可单独计算，以揭示社会服务面临的特定压力。
A rising old-age dependency ratio signals increased demand for pensions and healthcare, characteristic of Stage 5 of the DTM. Conversely, a high young-age dependency ratio indicates investment needs in education and child health. Both ratios are central to evaluating the ‘demographic dividend’ window.
老年赡养比上升预示养老金和医疗需求增加，是 DTM 第五阶段的典型特征。反之，少儿赡养比高则意味着教育和儿童健康方面投资需求大。两者对评估”人口红利”窗口期至关重要。
TDR = (P0–14 + P65+) / P15–64 × 100
7. Urbanisation and Population Density | 城市化与人口密度
Urbanisation level is the percentage of a country’s population living in areas designated as urban: U(%) = (Purban / Ptotal) × 100. The rate of urbanisation refers to the annual change in that percentage, not the absolute growth of the urban population.
城市化水平是指居住在被划定为城区的人口占总人口的百分比：U(%) = (P 城区 / P 总) × 100。城市化率则是指该百分比每年的变化量，而非城区人口的绝对增量。
Arithmetic population density is the simplest spatial measure: D = P / A, where P is total population and A is land area (km²). Physiological density refines this by using arable land area, shedding light on the potential pressure on food-producing resources.
算术人口密度是最简单的空间测度：D = P / A，P 是总人口，A 是土地面积（平方公里）。生理密度则改用可耕地面积，更能揭示对粮食生产资源的潜在压力。
Urbanisation (%) = Purban / Ptotal × 100 Density = P / A
8. Measures of Central Tendency and Spread | 集中趋势与离散度指标
Mean (x̄) = Σx / n, median is the middle value in ordered data, and mode is the most frequent value. In fieldwork, the median is often preferred when data contain outliers, because it is less skewed by extreme values.
平均数 (x̄) = Σx / n，中位数是有序数据中的中间值，众数是出现频次最高的值。野外调查中，若数据含有异常值，通常更倾向于采用中位数，因为它受极端值的偏影响较小。
Range = max − min. Interquartile range (IQR) = Q3 − Q1, encompassing the middle 50% of data. Standard deviation measures the average distance of data points from the mean. For a sample: s = √[ Σ(x − x̄)² / (n − 1) ].
极差 = 最大值 − 最小值。四分位距 (IQR) = Q₃ − Q₁，覆盖数据的中间 50％。标准差衡量各数据点与平均值的平均距离。样本标准差：s = √[ Σ(x − x̄)² / (n − 1) ]。
x̄ = Σx / n IQR = Q3 − Q1 s = √[ Σ(x − x̄)² / (n − 1) ]
9. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rank (rₛ) assesses the strength and direction of a monotonic relationship between two sets of ranked data. The formula is rₛ = 1 − [ 6 Σd² / (n(n² − 1)) ], where d is the difference in ranks for each pair and n is the number of pairs. Values range from −1 (perfect negative) to +1 (perfect positive).
斯皮尔曼等级相关系数 (rₛ) 用于衡量两组定序数据之间单调关系的强度与方向。公式为 rₛ = 1 − [ 6 Σd² / (n(n² − 1)) ]，其中 d 为每对数据的等级差，n 为样本对数。取值从 −1（完全负相关）到 +1（完全正相关）。
In AS skills questions, you typically rank each variable, compute d², sum them, and substitute into the formula. Use a critical values table to determine statistical significance at the 95% or 99% confidence level. Always state a null hypothesis before testing.
在 AS 技能题中，你通常要对每个变量排序、计算 d²、求和并代入公式。查阅临界值表可判断 95% 或 99% 置信水平下的统计显著性。检验前务必先写明零假设。
rₛ = 1 − [ 6 Σd² / (n(n² − 1)) ]
10. Chi-Squared Test | 卡方检验
Chi-squared (χ²) is used to test whether the frequency distribution of categorical data differs significantly from expected frequencies. The formula is χ² = Σ [ (O − E)² / E ], where O is the observed frequency and E is the expected frequency under the null hypothesis. Expected values are calculated by (row total × column total) ÷ grand total.
卡方 (χ²) 用于检验分类数据的频数分布与期望频数是否存在显著差异。公式为 χ² = Σ [ (O − E)² / E ]，其中 O 为观测频数，E 为基于零假设的期望频数。期望值计算方式为（行合计 × 列合计）÷ 总计。
Degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). Compare the calculated χ² value against the critical value at a chosen significance level (usually 0.05). If χ² exceeds the critical value, reject the null hypothesis.
自由度 (df) =（行数 − 1）×（列数 − 1）。将算得的 χ² 值与所选显著性水平（通常为 0.05）的临界值比较。若 χ² 大于临界值，则拒绝零假设。
χ² = Σ [ (O − E)² / E ] df = (r − 1) × (c − 1)
11. Lorenz Curve and Gini Coefficient | 洛伦兹曲线与基尼系数
The Lorenz curve plots cumulative percentage of total income (or other resource) against the cumulative percentage of population. Perfect equality follows the 45° line. The Gini coefficient quantifies the degree of inequality: G = Area A / (Area A + Area B), where A is the area between the line of equality and the Lorenz curve, and B is the area under the Lorenz curve.
洛伦兹曲线描绘总收入的累计百分比与人口累计百分比的关系。完全平等表现为 45° 线。基尼系数将不平等程度量化为 G = 面积 A /（面积 A + 面积 B），其中 A 为完全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。
An alternative numerical approximation often used with grouped data is: Gini ≈ 1 − Σ (Xk − Xk−1)(Yk + Yk−1), where X is the cumulative proportion of population and Y the cumulative proportion of income at group k. Values range from 0 (perfect equality) to 1 (perfect inequality).
对于分组数据，常使用数值近似公式：基尼系数 ≈ 1 − Σ (Xₖ − Xₖ₋₁)(Yₖ + Yₖ₋₁)，其中 X 是累计人口比例，Y 是第 k 组的累计收入比例。取值在 0（完全平等）到 1（完全不平等）之间。
Gini ≈ 1 − Σ (Xk − Xk−1)(Yk + Yk−1)
12. Gravity Model and Location Quotient | 重力模型与区位商
The gravity model predicts the interaction flow (Tij) between two places i and j as Tij = k (Mi Mj) / Dijb, where M represents the mass (population, employment, or retail floorspace), D is distance, and k and b are constants derived from calibration. Larger masses increase interaction; greater distance reduces it exponentially.
重力模型预测两地 i 与 j 之间的相互作用流 (Tij) 为 Tij = k (Mi Mj) / Dijb，其中 M 代表质量（人口、就业或零售面积），D 为距离，k 和 b 是由校准得出的常数。质量越大相互作用越强；距离越远则呈指数级衰减。
The Location Quotient (LQ) measures the concentration of a specific industry or function in a local area compared to a reference region. LQ = (ei / e) / (Ei / E), where ei is local employment in sector i, e is total local employment, Ei is reference-region employment in the same sector, and E is total reference-region employment. An LQ greater than 1.0 indicates specialisation.
区位商 (LQ) 衡量某一特定产业或功能在地方相对于参照区域的集中程度。LQ = (ei / e) / (Ei / E)，其中 ei 是地方 i 部门就业，e 是地方总就业，Ei 是参照区域同部门就业，E 是参照区域总就业。LQ 大于 1.0 表示专业化集聚。
Tij = k (Mi Mj) / Dijb LQ = (ei / e) / (Ei / E)
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