📚 Pre-U AQA Geography: Formula & Theorem Quick Reference Handbook | Pre-U AQA 地理：公式定理速查手册
This compact reference brings together the key quantitative formulas, statistical tests, and geographical theorems required for Pre-U AQA Geography. Whether you are conducting fieldwork, analysing population data, or interpreting spatial patterns, these tools provide the foundation for rigorous geographical enquiry. Each entry is presented with a clear formula or concept and a concise explanation in both English and Chinese, allowing you to revise efficiently.
本速查手册汇集了 Pre-U AQA 地理课程所必需的关键定量公式、统计检验方法和地理学定理。无论您是在进行实地调查、分析人口数据，还是解读空间格局，这些工具都为严谨的地理学探究提供了基础。每个条目均给出清晰的公式或概念，并配有简洁的中英文解释，便于高效复习。
1. River Discharge Formula | 河流流量公式
Discharge is the volume of water passing a given point in a river per unit time. It is the product of the cross-sectional area and the mean flow velocity.
流量是单位时间内通过河流某一点的水的体积。它是横截面积与平均流速的乘积。
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（m³ s⁻¹），A = 横截面积（m²），v = 平均流速（m s⁻¹）。
To calculate A, use the width multiplied by average depth at a cross-section. Velocity is typically measured with a flowmeter at 0.6 depth. In exam contexts, always check the units – discharge should end up in cubic metres per second.
计算 A 时，用断面宽度乘以平均水深。流速通常用流速仪在 0.6 倍水深位置测量。在考试中务必检查单位——流量最终单位应为立方米每秒。
2. Drainage Density and Bifurcation Ratio | 排水密度与分叉比
Drainage density expresses how closely spaced stream channels are within a basin. It is the total length of all streams divided by the basin area.
排水密度表示流域内河道的密集程度。它是所有河流的总长度除以流域面积。
Dd = ΣL / A
Dd = drainage density (km km⁻²), ΣL = total stream length (km), A = basin area (km²).
Dd = 排水密度（km km⁻²），ΣL = 河流总长度（km），A = 流域面积（km²）。
Horton’s bifurcation ratio (Rb) describes the ratio of the number of streams of one order to the number of streams of the next higher order.
霍顿分叉比 (Rb) 描述某一等级河流的数量与下一更高等级河流数量的比值。
Rb = Nu / Nu+1
Nu and Nu+1 are the numbers of stream segments of order u and order u+1. Average Rb values typically range between 3 and 5 and are used to infer geological controls on the drainage network.
Nu 和 Nu+1 分别为 u 级和 u+1 级河流的段数。平均 Rb 值通常在 3 至 5 之间，可用于推断地质条件对水系网络的控制。
3. Basic Demographic Formulas | 基本人口统计公式
Crude Birth Rate (CBR) and Crude Death Rate (CDR) are the most fundamental indicators of population change. They are expressed per 1000 of the mid-year population.
粗出生率 (CBR) 和粗死亡率 (CDR) 是衡量人口变化最基础的指标。它们以年中人口每千人的数量表示。
CBR = (B / P) × 1000
CDR = (D / P) × 1000
B = total live births in a year, D = total deaths, P = mid-year population. The Rate of Natural Increase (RNI) is simply:
B = 一年内活产婴儿总数，D = 死亡总数，P = 年中人口。自然增长率 (RNI) 计算方法为：
RNI = CBR − CDR
Population density is calculated as total population divided by total land area (persons per km²). Age dependency ratio compares the dependent population (0–14 and 65+) to the working-age population:
人口密度按总人口除以土地总面积计算（人/km²）。抚养比是将被抚养人口（0–14岁和65岁以上）与劳动年龄人口进行比较：
Dependency Ratio = [(P0–14 + P65+) / P15–64] × 100
4. Migration and Population Change Equation | 迁移与人口变化方程
Net migration is the balance of people moving into and out of a region. Add net migration to natural change to obtain total population change.
净迁移是迁入和迁出某地区人口的差额。将净迁移与自然变动相加，即可得出总人口变化。
Net Migration = I − E
I = immigration, E = emigration. The fundamental population change equation over a period t is:
I = 迁入, E = 迁出。一段时期 t 内基本的人口变化方程为：
Pt = P0 + (B − D) + (I − E)
P0 is the initial population, B births, D deaths, I immigration, E emigration. If the time unit is one year, the annual population growth rate (r) can be expressed as r = [(Pt − P0) / P0] × 100%.
P0 为初期人口，B 出生，D 死亡，I 迁入，E 迁出。若时间单位为一年，年人口增长率 r 可表示为 r = [(Pt − P0) / P0] × 100%。
5. Location Quotient | 区位商
The Location Quotient (LQ) measures the concentration of a particular industry or employment sector in a sub-region compared to a larger reference region. An LQ greater than 1 indicates specialisation.
区位商 (LQ) 衡量某一子区域中特定产业或就业部门相对于更大参照区域的集中程度。LQ 大于 1 表示专业化。
LQ = (Ei / Etotal)local ÷ (Ei / Etotal)national
Ei represents employment in industry i, and Etotal total employment. The local ratio is divided by the national ratio. For example, if local manufacturing employment is 30% and nationally it is 15%, LQ = 2.0, signalling a strong manufacturing cluster.
Ei 表示行业 i 的就业人数，Etotal 表示总就业人数。用当地比值除以全国比值。例如，若当地制造业就业占30%，而全国为15%，则 LQ = 2.0，表明存在明显的制造业集聚。
6. 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. It is widely used in geographical fieldwork when data are ordinal or non-normally distributed.
斯皮尔曼等级相关系数 (rs) 评估两组排序数据之间单调关系的强弱和方向。当数据为顺序变量或非正态分布时，该系数在地理实地调查中被广泛使用。
rs = 1 − (6 ∑d² ) / (n(n² − 1))
d = difference in rank of each pair, n = number of pairs. The value ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation). A critical values table is used to determine statistical significance at the 0.05 or 0.01 level.
d = 每对的等级差，n = 对数。取值介于 −1（完全负相关）至 +1（完全正相关）之间。需查阅临界值表，判断在 0.05 或 0.01 水平上是否具有统计显著性。
7. Chi-squared Test | 卡方检验
The chi-squared test (χ²) compares observed frequencies with expected frequencies to test for an association between categorical variables. It is common in land-use surveys, questionnaire analysis, and ecological transects.
卡方检验 (χ²) 通过比较观测频数与期望频数，检验分类变量之间是否存在关联性。它常用于土地利用调查、问卷分析和生态样带研究。
χ² = Σ ((O − E)² / E)
O = observed frequency, E = expected frequency. Degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). The calculated χ² is compared against a critical value: if it exceeds the critical value, the null hypothesis of no association is rejected.
O = 观测频数，E = 期望频数。自由度 df = (行数 − 1) × (列数 − 1)。将计算出的 χ² 值与临界值比较：若大于临界值，则拒绝无关联的零假设。
8. Mann-Whitney U Test | 曼-惠特尼 U 检验
The Mann-Whitney U test is a non-parametric test that determines whether two independent samples come from the same population. It compares the medians and is appropriate when data do not meet the assumptions of a t-test.
曼-惠特尼 U 检验是一种非参数检验，用于判断两个独立样本是否来自同一总体。它比较中位数，适用于不满足 t 检验假设的情形。
Step 1: Rank all observations from both groups together. Step 2: Sum the ranks for each group (R₁ and R₂). Step 3: Compute U values:
步骤一：将两组所有观测值合并排序。步骤二：求各组的秩和 (R₁ 和 R₂)。步骤三：计算 U 值：
U₁ = n₁n₂ + ½ n₁(n₁ + 1) − R₁
U₂ = n₁n₂ − U₁
n₁ and n₂ are the sample sizes. The test statistic U is the smaller of U₁ and U₂. Compare U with the critical value from tables; a value less than or equal to the critical value leads to rejection of the null hypothesis.
n₁ 和 n₂ 为样本大小。检验统计量 U 取 U₁ 与 U₂ 中的较小值。查阅临界值表，若 U 小于或等于临界值，则拒绝零假设。
9. Gravity Model | 重力模型
The gravity model predicts interaction between two places based on their size and distance. It borrows the concept from physics and is widely used in retail, transport, and migration studies.
重力模型根据两地的规模与距离预测其间的相互作用。它借用了物理学概念，广泛应用于零售、交通和迁移研究。
Iij = k (Pi × Pj) / Dijb
Iij = interaction between place i and j, P = population (or mass, e.g., retail floor space), D = distance, k and b are constants, with b usually close to 2. A simplified proportional statement is I ∝ PiPj / D2.
Iij = 地点 i 与 j 之间的相互作用，P = 人口（或质量，如零售楼面面积），D = 距离，k 和 b 为常数，b 通常接近 2。简化的比例关系为 I ∝ PiPj / D2。
10. Distance Decay Principle | 距离衰减原理
Distance decay describes the effect of distance on spatial interaction – the intensity of flows, communication, or influence decreases as distance increases. This principle underpins the gravity model, central place theory, and many locational decisions.
距离衰减描述了距离对空间相互作用的影响——随着距离增加，流动、交流或影响的强度降低。这一原理是重力模型、中心地理论及众多区位决策的基础。
Although there is no single universal formula, the concept is often expressed as an exponential or power function. In qualitative analysis, it explains why local demand is highest near a service centre and why the market area has a spatial limit.
尽管没有统一公式，该概念常以指数函数或幂函数表示。在定性分析中，它解释了为何靠近服务中心的本地需求最高，以及为何市场区存在空间界限。
11. Statistical Hypothesis Testing and p-value | 统计假设检验与 p 值
In geographical research, a null hypothesis (H0) typically states that there is no significant relationship or difference. The alternative hypothesis (H1) claims a significant effect. The p-value represents the probability of obtaining the observed result, or more extreme, if H0 is true.
在地理学研究中，零假设 (H0) 通常陈述不存在显著关系或差异。备择假设 (H1) 则主张存在显著效应。p 值表示在 H0 为真时，得到当前观察结果或更极端结果的概率。
If p ≤ 0.05, the result is statistically significant at the 5% level, and H0 is rejected. Critical values for tests like Spearman’s rank, χ², and Mann-Whitney U are provided in exam papers; you must compare the calculated statistic against the critical value at the chosen significance level.
若 p ≤ 0.05，则结果在 5% 水平上具有统计显著性，拒绝 H0。斯皮尔曼等级相关、卡方检验和曼-惠特尼 U 检验的临界值会在试卷中给出；必须将计算出的统计量与所选显著性水平下的临界值进行比较。
12. Central Place Theory and Key Constants | 中心地理论与关键常数
Christaller’s Central Place Theory explains the size, spacing, and hierarchy of settlements. Three organising principles determine the number of lower-order centres served by one higher-order centre, expressed by the K value.
克里斯塔勒的中心地理论解释了聚落的规模、间距和等级结构。三种组织原则决定了一个高级中心所能服务的低级中心数量，由 K 值表示。
- K=3 (Marketing principle) – minimises the travel for consumers; one higher centre serves three lower centres.
K=3（市场原则）——最小化消费者出行；一个高级中心服务三个低级中心。 - K=4 (Transport principle) – minimises route lengths; the hierarchical arrangement follows a transport network logic.
K=4（交通原则）——最小化线路长度；等级布局遵循交通网络逻辑。 - K=7 (Administrative principle) – political boundaries create a nesting where one higher centre governs seven lower centres.
K=7（行政原则）——政治边界形成嵌套结构，一个高级中心管辖七个低级中心。
These constants are not formulas in a mathematical sense, but they represent fundamental spatial principles that remain examinable and applicable in analysing service provision and urban hierarchies.
这些常数并非数学意义上的公式，但它们代表了基本的空间法则，在分析服务供给和城市等级时仍具有考核价值与现实意义。
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