📚 Year 13 AQA Geography: Formulas & Theorems Quick Reference Handbook | Year 13 AQA 地理：公式定理速查手册
Mastering the essential formulas and statistical tests is crucial for success in Year 13 AQA Geography, especially for the Non-Examined Assessment (NEA) and data-response questions. This handbook compiles all the key equations, indices, and inferential tests you need to analyse fieldwork data and interpret geographical patterns with confidence.
掌握核心公式和统计检验对于 Year 13 AQA 地理考试至关重要，尤其在非考试评估（NEA）和数据分析题中。本手册汇编了所有关键的方程、指数和推断性检验，帮助你自信地分析实地考察数据并解释地理模式。
1. Measures of Central Tendency & Dispersion | 中心趋势与离散度测量
The mean (x̄), median and mode summarise average values in a dataset, while range, interquartile range (IQR) and standard deviation describe spread. For normally distributed data, the standard deviation is particularly useful in assessing how tightly observations cluster around the mean.
平均值 (x̄)、中位数和众数概括了数据集的中心值，而极差、四分位距 (IQR) 和标准差则描述了离散程度。对于正态分布数据，标准差尤其有助于评估观测值围绕均值的聚集程度。
Key formulas:
核心公式：
x̄ = Σ x / n
s = √[ Σ (x – x̄)2 / (n – 1) ] (sample standard deviation)
IQR = Q3 – Q1, where Q1 is the 25th percentile and Q3 the 75th percentile. Always pair these with graphical displays like box plots.
IQR = Q₃ – Q₁，其中 Q₁ 为第25百分位数，Q₃ 为第75百分位数。务必结合箱线图等图形展示使用。
2. Spearman’s Rank Correlation | 斯皮尔曼等级相关系数
Spearman’s Rank correlation coefficient (rs) measures the strength and direction of a monotonic relationship between two variables that are at least ordinal. In geography it is widely used to test associations such as distance from source and sediment size, or deprivation and health outcomes.
斯皮尔曼等级相关系数 (rs) 用于衡量两个至少为顺序尺度的变量之间单调关系的强度和方向。地理学中广泛用于检验例如距源头的远近与沉积物粒径，或剥夺程度与健康状况等关联。
rs = 1 – (6 Σ d2) / (n3 – n)
where d = difference in ranks for each pair, n = number of paired observations. The value of rs ranges from -1 (perfect negative) to +1 (perfect positive). Compare the calculated rs against a critical value table for the chosen significance level (usually 0.05).
其中 d = 每对数据的等级差值，n = 成对观测值的数量。rs 的取值范围为 -1（完全负相关）至 +1（完全正相关）。将计算出的 rs 与选定显著性水平（通常为0.05）的临界值表进行比较。
3. Mann-Whitney U Test | 曼-惠特尼 U 检验
The Mann-Whitney U test compares two independent samples to determine whether they come from the same population. It is an alternative to the t-test when data are ordinal or not normally distributed, commonly applied in geography to compare infiltration rates on different slopes or retail turnover in two towns.
曼-惠特尼 U 检验用于比较两个独立样本是否来自同一总体。当数据为顺序尺度或不满足正态分布时，它可作为 t 检验的替代方法，常用于比较不同坡地的下渗率或两个城镇的零售额。
The test statistic is calculated as follows:
检验统计量按以下公式计算：
U1 = n1n2 + [n1(n1 + 1)] / 2 – R1
U2 = n1n2 – U1
where n1 and n2 are the sample sizes, R1 is the sum of ranks from sample 1. The smaller of U1 and U2 is compared with the critical value; if U is less than or equal to the critical value, the null hypothesis is rejected.
其中 n₁ 和 n₂ 为样本大小，R₁ 为样本1的秩和。取 U₁ 和 U₂ 中较小的值，与临界值比较；若 U 小于或等于临界值，则拒绝零假设，表明两组存在显著差异。
4. Chi-Squared Test | 卡方 (χ²) 检验
The Chi-squared test assesses whether there is a significant association between two categorical variables. Geography students often use it for land-use surveys, questionnaire responses, or species distribution data. It compares observed frequencies (O) with the frequencies expected under the null hypothesis (E).
卡方检验用于判断两个分类变量之间是否存在显著关联。地理学生常将其用于土地利用调查、问卷回答或物种分布数据。它比较观测频数 (O) 与按零假设期望的频数 (E) 之间的差异。
χ² = Σ [ (O – E)2 / E ]
Degrees of freedom (df) = (number of rows – 1) × (number of columns – 1). If the calculated χ² exceeds the critical value at the chosen p-value (e.g., p = 0.05), the null hypothesis of no association is rejected.
自由度 (df) = (行数 – 1) × (列数 – 1)。若计算出的 χ² 大于选定 p 值（如 p = 0.05）下的临界值，则拒绝无关联的零假设。
5. River Discharge & Hydraulic Geometry | 河流流量与水力几何
Discharge (Q) is the volume of water flowing past a cross-section per unit time. It underpins flood risk models and channel-flow analysis.
流量 (Q) 是单位时间内通过河道横截面的水体体积，是洪水风险模型和河道水流分析的基础。
Q = A × v
where A = cross-sectional area (m²), v = mean velocity (m s⁻¹). The hydraulic radius (R) expresses the efficiency of a channel:
其中 A = 横截面积 (m²)，v = 平均流速 (m s⁻¹)。水力半径 (R) 表示河道的输水效率：
R = A / P
with P = wetted perimeter. A larger hydraulic radius implies less friction and higher flow energy.
P = 湿周。水力半径越大，意味着摩擦力越小、水流能量越高。
6. Drainage Basin Morphometry | 流域形态计量
Quantitative measures of basin shape and network allow comparison between catchments and help predict flood response.
流域形状与河网的数量化测量有助于不同集水区之间的比较，并辅助预测洪水响应。
Drainage density (Dd): Total length of all streams (Σ L) divided by basin area (A).
河网密度 (Dd)：所有河道总长度 (Σ L) 除以流域面积 (A)。
Dd = Σ L / A (units: km / km² or km⁻¹)
Sinuosity index (SI): Measures how much a river meanders.
弯曲度指数 (SI)：衡量河流的蜿蜒程度。
SI = channel length / straight-line valley length
Gradient: Ratio of vertical drop to horizontal distance, often expressed as a percentage.
比降：落差与水平距离之比，常以百分比表示。
Gradient (%) = (vertical drop / horizontal distance) × 100
7. Population Indicators | 人口指标
Crucial for demography questions, these rates allow analysis of population structures and change.
这些比率对于人口学问题至关重要，可用于分析人口结构和变化。
| Indicator / 指标 | Formula / 公式 |
|---|---|
| Crude Birth Rate (CBR) | CBR = (live births / total population) × 1000 |
| Crude Death Rate (CDR) | CDR = (deaths / total population) × 1000 |
| Natural Increase (NI) | NI = CBR – CDR (per 1000) |
| Population Growth Rate (%) | (CBR – CDR + net migration rate) / 10 |
| Dependency Ratio | ((Pop 0–14 + Pop 65+) / Pop 15–64) × 100 |
| Net Migration Rate | (immigrants – emigrants) / total population × 1000 |
Always state the units and whether the rate is expressed per 1000 or as a percentage.
始终要注明单位，并说明比率是以每千人计还是以百分比
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