📚 AQA Year 13 Geography: Quick-Reference Formula & Theorem Handbook | AQA Year 13 地理公式定理速查手册
This handbook compiles the essential formulas, statistical tests, and models required for AQA A-Level Geography (Year 13). Use it as a rapid revision tool to master quantitative skills, physical processes, and human geography models that appear across your examinations.
本手册汇编了AQA A-Level地理（Year 13）所必需的核心公式、统计检验和模型。将其作为快速复习工具，掌握考试中涉及的定量技能、自然地理过程及人文地理模型。
1. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
The Spearman’s rank test measures the strength and direction of association between two ranked variables. It is non-parametric and suitable for ordinal data or non-linear monotonic relationships. The coefficient rs ranges from -1 (perfect negative) to +1 (perfect positive).
斯皮尔曼等级相关系数检验两个排序变量之间关联的强度和方向。它是非参数检验，适用于顺序数据或非线性单调关系。系数 rs 取值范围从 -1（完全负相关）到 +1（完全正相关）。
rs = 1 – (6 Σ d²) / (n (n² – 1))
Where: d = difference between ranks for each pair, n = number of paired observations. Compare the calculated rs with critical values to test significance at the 95% or 99% confidence level.
其中：d = 每对数据的秩次差，n = 配对观测数量。将计算得到的 rs 与临界值比较，以检验在 95% 或 99% 置信水平下的显著性。
2. Mann-Whitney U Test | 曼-惠特尼U检验
The Mann-Whitney U test compares differences between two independent groups when the dependent variable is ordinal or continuous but not normally distributed. It tests whether the two samples come from the same population.
曼-惠特尼U检验用于比较两个独立组之间的差异，当因变量为顺序变量或连续但不服从正态分布时。它检验两个样本是否来自同一总体。
U₁ = n₁n₂ + (n₁(n₁ + 1) / 2) – R₁
U₂ = n₁n₂ + (n₂(n₂ + 1) / 2) – R₂
Here, n₁ and n₂ are the sample sizes; R₁ and R₂ are the sum of ranks for each group. The smaller U-value (U = min(U₁, U₂)) is compared with the critical U to reject the null hypothesis.
此处，n₁ 和 n₂ 为样本大小；R₁ 和 R₂ 为各组的秩和。取较小的 U 值（U = min(U₁, U₂)）与临界 U 值比较，以拒绝原假设。
3. Chi-Squared Test | 卡方检验
Chi-squared (χ²) is used to test for a significant association between two categorical variables, or to compare observed frequencies with expected frequencies (goodness of fit).
卡方 (χ²) 检验用于检验两个分类变量之间是否存在显著关联，或比较观察频率与期望频率（拟合优度）。
χ² = Σ ((O – E)² / E)
O = observed frequency, E = expected frequency. Degrees of freedom (df) = (number of rows – 1) × (number of columns – 1) for contingency tables. The larger the χ² value, the more likely the difference is significant.
O = 观察频率，E = 期望频率。对于列联表，自由度 (df) = (行数 – 1) × (列数 – 1)。χ² 值越大，差异越可能显著。
4. River Discharge and Water Balance | 河流流量与水平衡
Discharge is the volume of water flowing past a point per unit time, measured in cubic metres per second (cumecs). It is fundamental to flood risk and hydrological analysis.
流量是单位时间内流过某点的水体积，以立方米每秒 (cumecs) 计量，是洪水风险和水文分析的基础。
Q = A × V
Where Q = discharge (m³/s), A = cross-sectional area of the channel (m²), V = mean velocity (m/s).
其中 Q = 流量 (m³/s)，A = 河道横截面积 (m²)，V = 平均流速 (m/s)。
The water balance equation describes the hydrological cycle over a drainage basin: P = Q + E + ΔS, where P = precipitation, Q = runoff, E = evapotranspiration, ΔS = change in storage (soil moisture, groundwater).
水平衡方程描述流域水文循环：P = Q + E + ΔS，其中 P = 降水量，Q = 径流量，E = 蒸发蒸腾量，ΔS = 储量变化（土壤水分、地下水）。
5. Hjulström Curve and Critical Erosion Velocity | 尤斯特罗姆曲线与临界侵蚀速度
The Hjulström Curve illustrates the relationship between particle size and flow velocity needed for erosion, transportation, and deposition. It is not expressed as a single formula but relies on empirical thresholds.
尤斯特罗姆曲线表示粒径与侵蚀、搬运和沉积所需流速之间的关系。它并非单一公式，而是依赖经验阈值。
For fine sand and silt, the critical erosion velocity can be roughly estimated using the Shields criterion; however, in exam contexts, students are expected to interpret the curve and note that larger particles (e.g., gravel) require higher velocities to be entrained, while cohesive clays resist erosion despite higher velocities.
对于细沙和粉砂，临界侵蚀速度可用希尔兹准则粗略估算；但在考试中，学生应能解读曲线并指出：较大颗粒（如砾石）需要更高流速才能起动，而粘性粘土即使流速较高也能抵抗侵蚀。
A simplified approximation for the onset of transport: τ = 0.06 × (ρs – ρw) × g × D, where τ is shear stress, ρs and ρw are sediment and water density, g is gravity, D is grain diameter. However, the Hjulström curve is the standard graphical model.
搬运起动的简化近似为：τ = 0.06 × (ρs – ρw) × g × D，其中 τ 为剪切应力，ρs 和 ρw 为泥沙和水的密度，g 为重力加速度，D 为颗粒直径。但标准图形模型仍是尤斯特罗姆曲线。
6. Sediment Fall Velocity (Stokes’ Law) | 泥沙沉降速度（斯托克斯定律）
Stokes’ Law gives the settling velocity of small spherical particles in a fluid. It is applied in coastal and fluvial contexts to understand sediment deposition.
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