📚 Pre-U OCR Geography: Formula & Theorem Quick Reference Handbook | Pre-U OCR 地理：公式定理速查手册
This concise handbook compiles essential formulae, definitions and theorems required for the Pre-U OCR Geography examination. It spans physical geography, human geography and practical skills, serving as a rapid revision tool to reinforce quantitative literacy and conceptual precision.
本简明手册汇集了 Pre-U OCR 地理考试所必需的核心公式、定义和定理，涵盖自然地理、人文地理和实地技能，可作为快速复习工具，助力提升量化素养和概念准确性。
1. Population & Demographic Formulae | 人口与人口统计公式
Crude Birth Rate (CBR) – computed as the number of live births per 1000 people in a given year.
粗出生率（CBR） – 某年内每千人的活产婴儿数。
CBR = (B ÷ P) × 1000
Crude Death Rate (CDR) – total deaths per 1000 population per year.
粗死亡率（CDR） – 每年每千人的死亡总数。
CDR = (D ÷ P) × 1000
Rate of Natural Increase (RNI) – the difference between CBR and CDR, usually expressed as a percentage.
自然增长率（RNI） – 粗出生率与粗死亡率之差，通常用百分比表示。
RNI (%) = (CBR − CDR) ÷ 10
Doubling time – the period required for a population to double in size, estimated by the Rule of 70.
人口倍增时间 – 人口数量翻一番所需的时间，常用 70 法则估算。
Doubling time (years) = 70 ÷ RNI (%)
Total Fertility Rate (TFR) – the average number of children a woman would bear during her reproductive years.
总和生育率（TFR） – 一名妇女在育龄期间平均生育的子女数。
TFR = Σ (ASFRₐ × 5) / 1000
where ASFRₐ is the age-specific fertility rate per 1000 women in each 5‑year age group.
其中 ASFRₐ 为每个 5 岁年龄组每千名妇女的年龄别生育率。
Population density – crude measure of inhabitants per unit area.
人口密度 – 单位面积上的居民数量。
Density = Total population ÷ Land area (km²)
2. Urbanisation & Segregation Indices | 城市化与隔离指数
Urbanisation level – the proportion of total population living in urban areas.
城市化水平 – 总人口中居住在城市地区的比例。
U (%) = (Urban population ÷ Total population) × 100
Index of Dissimilarity (ID) – measures evenness of distribution between two groups across spatial units (0 = perfect integration, 100 = complete segregation).
相异指数（ID） – 衡量两个群体在各空间单元中分布的均匀程度（0 = 完全融合，100 = 完全隔离）。
ID = ½ Σ |(gᵢ / G) − (oᵢ / O)| × 100
where gᵢ and oᵢ are the numbers of each group in sub‑area i, G and O are the total populations of the two groups.
其中 gᵢ 与 oᵢ 为子区域 i 中两群体的人数，G 与 O 为两群体的总人数。
Location Quotient (LQ) – indicates the concentration of a particular activity or group in an area relative to the national average.
区位商（LQ） – 表示某活动或群体在某一区域的集中程度，相对于全国平均的比值。
LQ = (eᵢ / eₜ) ÷ (Eᵢ / Eₜ)
where eᵢ is local employment in sector, eₜ is total local employment, Eᵢ national employment in sector, Eₜ total national employment.
eᵢ 为本地某行业就业人数，eₜ 为本地总就业人数，Eᵢ 为全国该行业就业人数，Eₜ 为全国总就业人数。
Gini coefficient (spatial income) – derived from the Lorenz curve, measures inequality. A value of 0 represents perfect equality, 1 maximal inequality.
基尼系数（空间收入） – 由洛伦兹曲线导出，衡量不平等程度。0 为完全平等，1 为最大不平等。
G = 1 − Σ (Pₖ − Pₖ₋₁)(Qₖ + Qₖ₋₁)
3. Hydrology & Fluvial Geomorphology | 水文与河流地貌
Drainage density – total stream length per unit area, reflecting landscape dissection.
河网密度 – 单位面积内的河道总长度，反映地表切割程度。
Dd = Lₜₒₜₐₗ ÷ A (km/km²)
Stream order (Strahler) – a hierarchical classification; when two streams of the same order join, the downstream segment increases by one order.
斯特拉勒水系分级 – 层级分类法：两条同级河流汇合后，下游河段级别增加一级。
Manning’s equation – estimates mean velocity of flow in an open channel.
曼宁公式 – 估算明渠水流平均流速。
V = (1/n) R⁰·⁶⁶⁷ S⁰·⁵
where R is hydraulic radius (A / wetted perimeter), S is channel slope (dimensionless), n is Manning’s roughness coefficient.
其中 R 为水力半径（A / 湿周），S 为河道比降（无量纲），n 为曼宁糙率系数。
Discharge – volume of water passing a cross‑section per unit time.
流量 – 单位时间内通过断面的水体体积。
Q = A × V (m³ s⁻¹)
Runoff coefficient – fraction of rainfall that becomes direct runoff.
径流系数 – 降雨转化为直接径流的比例。
C = Runoff depth ÷ Rainfall depth
Hydraulic radius – efficiency of channel cross‑section.
水力半径 – 表征河道断面输水效率。
R = A ÷ P (m)
P is the wetted perimeter.
P 为湿周。
4. Meteorology & Climatology | 气象与气候学
Relative humidity – ratio of actual water vapour pressure to saturation vapour pressure at a given temperature.
相对湿度 – 在一定温度下，实际水汽压与饱和水汽压之比。
RH (%) = (eₐ / eₛ) × 100
Dew point temperature (approximation) – temperature at which air becomes saturated when cooled at constant pressure.
露点温度（近似） – 在恒压下空气冷却至饱和时的温度。
Td ≈ T − ((100 − RH) ÷ 5)
Net radiation budget – balance between incoming and outgoing radiation.
净辐射收支 – 输入与输出辐射的差额。
Q* = K↓ − K↑ + L↓ − L↑
where K↓ is incoming shortwave, K↑ reflected shortwave, L↓ incoming longwave, L↑ outgoing longwave (all in W m⁻²).
K↓ 为入射短波辐射，K↑ 为反射短波，L↓ 为入射长波，L↑ 为射出长波（单位均为 W m⁻²）。
Wind chill index (simplified) – felt air temperature on exposed skin.
风寒指数（简化） – 裸露皮肤感受到的气温。
WCT = 13.12 + 0.6215T − 11.37V⁰·¹⁶ + 0.3965T V⁰·¹⁶
T is air temperature (°C), V is wind speed at 10 m height (km h⁻¹).
T 为气温（°C），V 为 10 米高度处的风速（km h⁻¹）。
Potential evapotranspiration – estimated by the simple Hargreaves formula where only temperature data are available.
潜在蒸散发 – 当仅有气温数据时可用简化哈格里夫斯公式估算。
PET = 0.0023 × Rₐ × (Tₘₑₐₙ + 17.8) × √(Tₘₐₓ − Tₘᵢₙ)
5. Environmental Change & Biogeography | 环境变化与生物地理
Carbon stock flux – net change in carbon stored in a pool over a given period.
碳储量通量 – 某时段内碳库中碳储量的净变化。
ΔC = Cₜ₂ − Cₜ₁
Carbon sequestration rate – amount of CO₂ absorbed per unit area per year.
碳汇速率 – 单位面积每年吸收的 CO₂ 量。
Sequestration = (Cₐₙₙᵤₐₗ ÷ Area) × 3.67
(converting carbon to CO₂ using the molar mass ratio 44/12 ≈ 3.67).
（利用摩尔质量比 44/12 ≈ 3.67 将碳换算为 CO₂）。
Simpson’s Diversity Index (D) – measures the probability that two individuals randomly selected belong to the same species; low D indicates high diversity.
辛普森多样性指数（D） – 随机抽取两个个体属于同一物种的概率；D 值越低，多样性越高。
D = Σ (nᵢ(nᵢ − 1)) / (N(N − 1))
where nᵢ is the number of individuals of species i, N is total individuals.
nᵢ 为物种 i 的个体数，N 为总个体数。
Normalised Difference Vegetation Index (NDVI) – derived from satellite imagery to assess vegetation greenness.
归一化植被指数（NDVI） – 利用卫星影像评估植被绿度。
NDVI = (NIR − RED) ÷ (NIR + RED)
6. Economic Geography & Spatial Interaction | 经济地理与空间交互
Gravity model – predicts the interaction flow between two places as a function of their size and distance.
引力模型 – 根据两地规模与距离预测它们之间的交互流量。
Iᵢⱼ = (Pᵢ × Pⱼ) ÷ dᵢⱼ²
where P represents population or mass, d is distance; sometimes a friction exponent β replaces the square.
P 为人口或规模，d 为距离；有时用摩擦指数 β 代替平方。
Distance decay – the declining intensity of interaction with increasing separation.
距离衰减 – 随距离增加而相互作用强度下降的规律。
F(d) = k / dᵅ
Threshold and range – threshold is the minimum demand needed to support a service; range is the maximum distance consumers travel.
门槛与范围 – 门槛是维持某项服务所需的最低需求门槛；范围是消费者愿意出行的最远距离。
Break point = dᵢⱼ / (1 + √(Pᵢ/Pⱼ))
(Reilly’s law of retail gravitation break point).
（赖利零售引力断点公式）。
Location quotient for employment – same as section 2 but applied to economic sectors to identify basic (export-oriented) employment.
就业区位商 – 与第 2 节相同，但用于经济部门以识别基础（外向型）就业。
7. Topographic Surveying & Cartography | 地形测量与制图
Gradient (slope) – ratio of vertical rise to horizontal distance, often expressed as a ratio or percentage.
坡度（梯度） – 垂直高差与水平距离之比，常以比值或百分比表示。
Gradient = Δh ÷ d
Vertical exaggeration – used in cross‑sections to emphasise relief.
垂直夸大 – 在剖面图中用于突出地形起伏。
VE = Horizontal scale ÷ Vertical scale
Scale conversion – relates map distance to ground distance.
比例尺换算 – 图面距离与实际地面距离的关系。
Ground distance = Map distance × Scale denominator
Area from a map – using grid squares or planimeter.
图上面积 – 利用方格法或求积仪测量。
Area (km²) = (Counted squares × grid cell area) ÷ (Scale denominator)²
8. Statistical & Research Methods | 统计与研究方法
Mean, median, mode – measures of central tendency essential for describing data sets collected in geographical investigations.
均值、中位数、众数 – 描述地理调查数据集集中趋势的核心指标。
Mean (x̄) = Σxᵢ ÷ n
Standard deviation – quantifies the spread around the mean; the square root of variance.
标准差 – 衡量数据围绕均值的离散程度，是方差的平方根。
σ = √(Σ(xᵢ − x̄)² ÷ n)
Spearman’s rank correlation coefficient (rₛ) – measures the strength and direction of association between two ranked variables.
斯皮尔曼秩相关系数（rₛ） – 衡量两个排序变量之间关联的强度和方向。
rₛ = 1 − (6 Σd²) / (n(n² − 1))
where d is the difference between ranks for each pair, n is the number of pairs.
d 为每对数据的位次差，n 为数据对数。
Chi‑squared test (χ²) – tests the association between categorical variables in a contingency table.
卡方检验（χ²） – 用于检验列联表中分类变量间的相关性。
χ² = Σ ((O − E)² ÷ E)
O is observed frequency, E is expected frequency. Degrees of freedom = (rows − 1) × (columns − 1).
O 为观察频数，E 为期望频数。自由度 = (行数 − 1) × (列数 − 1)。
Confidence interval (95%) – for a sample mean, roughly ± 1.96 standard errors.
置信区间（95%） – 样本均值的范围约 ± 1.96 个标准误。
95% CI = x̄ ± 1.96 × (σ / √n)
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