📚 Formula and Theorem Quick Reference for AS WJEC Geography | AS WJEC 地理：公式定理速查手册
This quick-reference guide brings together the essential formulas, equations and theorems you will need throughout your AS WJEC Geography course. It covers river hydrology, population change, settlement patterns, urbanisation and the key statistical techniques that underpin fieldwork skills. Each entry is presented with a clear explanation and a ready-to-use formula, so you can check your work quickly and deepen your understanding of quantitative geography.
本速查手册汇总了 AS WJEC 地理课程中必备的公式、方程与定理，涵盖河流水文学、人口变迁、聚落模式、城市化过程以及支撑实地调查技能的关键统计方法。每一条目均配有清晰的解释和可直接使用的公式，方便你快速核对计算结果，加深对定量地理学的理解。
1. River Discharge and Velocity | 河流流量与流速
River discharge (Q) is the volume of water flowing past a cross-section of the channel per unit time. It is obtained by multiplying the cross-sectional area of the wetted channel by the mean flow velocity. Discharge is usually expressed in cubic metres per second (m³/s).
河流流量（Q）是指单位时间内通过河道某一横截面的水体体积，由湿周横截面积乘以平均流速得出，通常以立方米每秒（m³/s）为单位。
Q = A × V
Where A = cross-sectional area (m²) calculated from width and mean depth, and V = mean velocity (m/s) measured with a flow meter or by float method.
其中 A 为横截面积（m²），由河宽与平均水深计算得出；V 为平均流速（m/s），可用流速仪或浮标法测定。
2. Drainage Basin Gradient | 流域坡度
Gradient describes the steepness of a slope or channel reach. It is the ratio of the vertical difference in height to the horizontal distance between two points. Fieldwork often uses a clinometer or map contours to obtain these values; the gradient is usually given as a fraction or ratio.
坡度描述斜坡或河段的陡峭程度，为两点间高差与水平距离的比值。野外工作中常使用测斜仪或地图等高线获取数据，坡度通常以分数或比的形式表示。
Gradient = Vertical Rise (m) / Horizontal Distance (m)
For example, a vertical rise of 5 m over a horizontal distance of 100 m gives a gradient of 1:20 or 0.05.
例如高差 5 m 与水平距离 100 m 之比，可得坡度为 1:20 或 0.05。
3. Population Change and Growth Rate | 人口变化与增长率
Population change is driven by natural change (births minus deaths) and net migration (immigration minus emigration). The crude birth rate (CBR) and crude death rate (CDR) are expressed per 1 000 people per year, making comparisons between countries straightforward.
人口变迁由自然变动（出生减死亡）与净迁移（迁入减迁出）共同驱动。粗出生率（CBR）与粗死亡率（CDR）以每年每千人为单位表示，便于国家间比较。
Natural Increase Rate (%) = (CBR – CDR) / 10
Total Growth Rate (%) = [(Births – Deaths) + (Immigration – Emigration)] / Total Population × 100
Remember to convert births and deaths per 1 000 into actual numbers when using total population figures.
注意当使用总人口数据时，需将每千人的出生与死亡人数换算为实际数值。
4. Population Density and Dependency Ratio | 人口密度与抚养比
Population density measures how crowded a country or region is, while the dependency ratio indicates the economic burden on the working-age population. The ratio is particularly useful when analysing demographic change and planning for health, education and pensions.
人口密度衡量一个国家或区域的拥挤程度，而抚养比则反映劳动年龄人口所承受的经济负担。抚养比在分析人口结构变化和规划医疗、教育及养老金时尤为实用。
Population Density = Total Population / Land Area (km²)
Dependency Ratio = [(Population aged 0–14) + (Population aged 65+)] / (Population aged 15–64) × 100
A high dependency ratio suggests a greater pressure on the productive sector to support dependants.
较高的抚养比表明生产性部门承受着更大的扶养压力。
5. Population Doubling Time | 人口倍增时间
Doubling time is a quick estimate of the number of years it will take for a population to double in size, assuming a constant annual growth rate. The rule of 70 provides a simple approximation widely used in demography.
倍增时间用于快速估算在年增长率不变的条件下人口规模翻番所需的年数。“70 规则”给出了人口学中广泛使用的一个简单近似值。
Doubling Time (years) ≈ 70 / Annual Growth Rate (%)
If a country’s population is growing at 2 % per year, doubling time ≈ 70 ÷ 2 = 35 years. The same principle can be applied if you need to calculate halving time for a declining population.
若某国人口年增长率为 2%，则倍增时间约为 70 ÷ 2 = 35 年。同理也可用于计算人口下降时的减半时间。
6. Urbanisation Level | 城市化水平
Urbanisation level measures the proportion of a country’s population living in urban settlements. It is a vital indicator of development and can be tracked over time to reveal the pace of rural-to-urban migration.
城市化水平衡量一国人口中居住在城镇聚落的比例，是反映发展水平的重要指标，并可连续追踪以揭示乡村向城市迁移的速度。
Urbanisation Level (%) = (Urban Population / Total Population) × 100
When using census data, ensure the definition of ‘urban’ is consistent across different time periods or regions.
使用人口普查数据时，要确保“城镇”的定义在不同时期或区域保持一致。
7. Break-Point Formula (Settlement) | 断裂点公式（聚落）
Reilly’s law of retail gravitation leads to the break-point formula, which calculates the boundary of the market area between two competing settlements. It helps to explain where shoppers are likely to travel and is useful for centre hierarchy studies.
赖利的零售引力定律引出了断裂点公式，用以计算两个竞争聚落之间的市场区域边界。这一公式有助于解释消费者的出行意愿，常用于中心地等级研究。
dₐ = D / (1 + √(P_b / P_a))
Where dₐ = distance from settlement A to the break point, D = distance between settlement A and B, and Pₐ, P_b are the populations (or retail floorspace) of the two settlements.
其中 dₐ 为从聚落 A 到断裂点的距离，D 为 A、B 两聚落之间的距离，Pₐ、P_b 分别为两聚落的人口（或零售面积）。
8. Gravity Model of Spatial Interaction | 空间相互作用重力模型
The gravity model predicts the volume of interaction (such as migration, trade or commuting) between two places based on their mass (commonly population) and the distance separating them. It is an analogue of Newton’s law of universal gravitation.
重力模型根据两地规模（通常以人口表示）和间隔距离预测它们之间的相互作用量（如迁移、贸易或通勤），其原理类似牛顿万有引力定律。
I_ab = (P_a × P_b) / d²
I_ab represents the interaction between place a and b, P_a and P_b are the populations, and d is the distance. Sometimes the exponent is adjusted to reflect different types of friction of distance.
I_ab 表示地点 a 与 b 间的相互作用，P_a 和 P_b 为两地人口，d 为距离。有时会调整指数以反映不同类型的距离摩擦效应。
9. Measures of Central Tendency and Dispersion | 集中趋势与离散程度的度量
Mean, median and mode summarise the centre of a data set. Range, inter-quartile range and standard deviation describe the spread. Standard deviation is especially important when comparing the variability of two sets of geographical data.
均值、中位数和众数概括数据集的中心趋势；全距、四分位距和标准差则描述离散程度。比较两组地理数据的变异性时，标准差尤为重要。
Range = Maximum value – Minimum value
Inter-Quartile Range (IQR) = Q₃ – Q₁
Standard Deviation (sample) s = √( Σ (x – x̄)² / (n – 1) )
Where x̄ is the mean, n is the number of observations. Use the sample version (n – 1) when working with a sample, as is common in fieldwork.
其中 x̄ 为均值，n 为观测个数。地理实地调查通常使用样本，因此采用样本标准差计算公式（分母为 n – 1）。
10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rank tests the strength and direction of a relationship between two sets of ordinal data, such as settlement rank and number of services. It produces a coefficient between –1 and +1, where +1 indicates a perfect positive correlation and –1 a perfect negative correlation.
斯皮尔曼等级相关用于检验两组序数数据（如聚落等级与服务数量）之间关系的强度与方向。系数取值在 –1 到 +1 之间，+1 表示完全正相关，–1 表示完全负相关。
rₛ = 1 – (6 Σ d²) / (n (n² – 1))
Where d = difference in ranks for each pair, n = number of pairs. After calculating rₛ, compare it with a critical value for the chosen significance level to decide whether the correlation is statistically meaningful.
其中 d 为每对数据的等级差，n 为数据对的数量。计算出 rₛ 后，需与选定显著性水平的临界值比较，以判断相关是否具有统计意义。
11. Chi-Squared Test | 卡方检验
Chi-squared (χ²) determines whether there is a significant difference between observed and expected frequencies in categorical data. It is frequently applied to land-use surveys, questionnaire responses or sediment size distributions.
卡方（χ²）用于判断分类数据中观测频率与期望频率之间是否存在显著差异，常应用于土地利用调查、问卷调查或沉积物粒径分布分析。
χ² = Σ ( (O – E)² / E )
O = observed frequency, E = expected frequency. The degrees of freedom (df) = (number of rows – 1) × (number of columns – 1). The calculated χ² value is compared with a critical value from statistical tables.
O 为观测频率，E 为期望频率。自由度 df =（行数 – 1）×（列数 – 1）。计算出的 χ² 值需与统计表中的临界值比较。
12. Percentage Change and Rate of Increase | 百分比变化与增长率
Percentage change is a foundational arithmetic skill in geography, used to analyse trends in population, economic output, land cover and more. It expresses the magnitude of change relative to the original value.
百分比变化是地理学中的一项基础算术技能，用于分析人口、经济产出、土地覆盖等方面的趋势，反映变化幅度相对初始值的大小。
Percentage Change (%) = [(New value – Original value) / Original value] × 100
A positive result means growth, a negative result indicates decline. Always check whether the original value is zero, as the formula becomes meaningless in that case.
正值表示增长，负值表示下降。注意当初始值为零时，该公式无意义，使用时需加以甄别。
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