📚 Year 12 CCEA Sociology: Formula and Theorem Quick Reference Handbook | Year 12 CCEA 社会学：公式定理速查手册
This handbook brings together the essential statistical formulas and key sociological theorems that every Year 12 CCEA Sociology student should have at their fingertips. From calculating percentage change to understanding Merton's strain theory, it covers the quantitative tools and conceptual frameworks needed to analyse social data and interpret social behaviour. Each entry provides a clear English explanation followed by a direct Chinese translation, making it ideal for revision and bilingual mastery.
本手册汇集了每位 Year 12 CCEA 社会学学生都应掌握的常用统计公式和关键社会学定理。从计算百分比变化到理解默顿的失范理论，它涵盖了分析社会数据和解读社会行为所需的量化工具与概念框架。每一条目都先给出清晰的英文解释，再紧跟对应的中文翻译，非常适合复习和双语巩固。
1. Percentage Change | 百分比变化
Percentage change is used to compare the size of an increase or decrease over time. It is widely applied when examining shifts in crime rates, educational attainment or family structures.
百分比变化用于比较一段时间内的增长或下降幅度，广泛运用于考察犯罪率、教育成就或家庭结构的变化。
Percentage Change = ((New Value - Old Value) ÷ Old Value) × 100%
A positive result indicates a percentage increase, while a negative result shows a percentage decrease. Always express the change relative to the original value to maintain consistency.
正值表示百分比增长，负值表示百分比下降。始终以原始值为基准表示变化，以保持一致性。
2. Mean, Median and Mode | 平均数、中位数与众数
These measures of central tendency summarise a data set with a single typical value, helping sociologists describe patterns in income, household size or exam scores.
这些集中趋势度量用单一的典型值概括数据集，帮助社会学家描述收入、家庭规模或考试成绩的模式。
Mean = Σx ÷ n
The mean is the arithmetic average, calculated by summing all values (Σx) and dividing by the total number of cases (n). It is sensitive to extreme outliers.
平均数是算术平均值，将所有数值相加（Σx）再除以个案总数（n）即得。它对极端离群值敏感。
Median = middle value when data are ordered from smallest to largest
The median splits the distribution into two equal halves and is not affected by outliers, making it preferred for skewed data such as wealth distributions.
中位数是将数据按由小到大排序后处于正中间的值，不受离群值影响，因此适用于财富分配等偏态数据。
Mode = most frequently occurring value
The mode is the value that appears most often. A data set can have one mode (unimodal), two modes (bimodal) or several modes (multimodal).
众数是出现频率最高的数值。数据集可以有一个众数（单峰）、两个众数（双峰）或多个众数（多峰）。
3. Range and Interquartile Range | 全距与四分位数间距
Measures of dispersion tell us how spread out the data are. The range is the simplest measure, while the interquartile range (IQR) focuses on the middle 50% and is more resistant to outliers.
离散程度度量告诉我们数据的分散程度。全距是最简单的指标，而四分位数间距关注中间50%的数据，更能抵抗离群值的干扰。
Range = Maximum Value - Minimum Value
The range gives a quick sense of the spread but can be misleading if a single extreme score exists.
全距可以快速了解数据的分布范围，但如果存在单个极端值，可能产生误导。
IQR = Q₃ - Q₁
Q₁ is the first quartile (25th percentile) and Q₃ is the third quartile (75th percentile). The IQR therefore captures the range of the middle half of the data.
Q₁ 是第一四分位数（第25百分位数），Q₃ 是第三四分位数（第75百分位数）。因此 IQR 反映的是数据中间一半的变动范围。
4. Standard Deviation | 标准差
Standard deviation measures the average distance of each data point from the mean. A small standard deviation indicates that values cluster closely around the mean; a large standard deviation signals greater variability.
标准差衡量各数据点与平均数的平均距离。标准差小说明数值紧密聚集在均值周围；标准差大则表示变异性较大。
σ = √[ Σ(x - μ)² ÷ N ] (population)
s = √[ Σ(x - x̄)² ÷ (n - 1) ] (sample)
When working with a whole population, divide by N. For a sample, divide by (n - 1) to obtain an unbiased estimate. In sociology, sample standard deviation is far more common because we usually work with survey data.
计算总体标准差时除以 N，计算样本标准差时除以 (n - 1) 以获得无偏估计。在社会学中，由于通常使用调查数据，样本标准差更为常见。
5. Pearson Correlation Coefficient | 皮尔逊相关系数
The Pearson correlation coefficient (r) quantifies the strength and direction of a linear relationship between two continuous variables, such as hours of study and exam results or income and health outcomes. It always lies between -1 and +1.
皮尔逊相关系数 (r) 用于量化两个连续变量之间线性关系的强弱和方向，例如学习时间与考试成绩，或收入与健康结果。其取值总是在 -1 到 +1 之间。
r = Σ[(x - x̄)(y - ȳ)] ÷ √[ Σ(x - x̄)² Σ(y - ȳ)² ]
A value close to +1 indicates a strong positive correlation, close to -1 a strong negative correlation, and around 0 suggests no linear relationship. Remember that correlation does not imply causation.
r 接近 +1 表示强正相关，接近 -1 表示强负相关，接近 0 则表明没有线性关系。切记相关并不代表因果。
6. Chi-Square Test of Independence | 卡方独立性检验
Although the full calculation is often performed with software, the chi-square (χ²) statistic is a key tool for testing whether two categorical variables (e.g., gender and educational choice) are associated. Its formula is essential for understanding the logic of inferential statistics in sociological research.
尽管完整的计算常由软件完成，卡方统计量是检验两个类别变量（如性别与教育选择）是否有关联的关键工具。掌握其公式有助于理解社会学研究中推论统计的逻辑。
χ² = Σ[ (O - E)² ÷ E ]
Here O stands for the observed frequency and E for the expected frequency under the null hypothesis of no association. A larger χ² value indicates a larger discrepancy between observed and expected counts, which may reach statistical significance.
其中 O 代表观察频数，E 代表零假设（无关联）下的期望频数。χ² 值越大，说明观察值与期望值之间的差异越大，可能达到统计显著水平。
7. Gini Coefficient | 基尼系数
The Gini coefficient is a widely used measure of income or wealth inequality within a society. It is derived from the Lorenz curve and ranges from 0 (perfect equality) to 1 (maximum inequality).
基尼系数是衡量社会中收入或财富不平等程度的常用指标。它由洛伦兹曲线推导而来，取值在 0（完全平等）至 1（完全不平等）之间。
ween the line of perfect equality and the Lorenz curve, and B is the area below the Lorenz curve.
ᵢ is the cumulative proportion of the population and Qᵢ the cumulative proportion of income.
实际估算中，社会学家常用简化公式：基尼系数 ≈ 1 - Σ(Pᵢ × (Qᵢ + Qᵢ₋₁))，其中 Pᵢ 为累计人口比例，Qᵢ 为累计收入比例。
8. Social Mobility Rate | 社会流动率
Social mobility is often measured by comparing occupational status across generations. Absolute mobility rates express the percentage of individuals who end up in a different social class from their parents.
社会流动通常通过比较代际职业地位来衡量。绝对流动率表示最终进入与父母不同社会阶层的个体所占的百分比。
Rate = (Number of individuals in a different class from parents ÷ Total individuals) × 100%
Relative mobility rates, often calculated using odds ratios, help sociologists distinguish structural change from genuine openness. An odds ratio of 1 indicates no association between parental class and child's class.
相对流动率通常用比值比计算，帮助社会学家区分结构变迁与真正的社会开放度。比值比为 1 表示父母阶级与子女阶级之间没有关联。
9. Merton's Strain Theorem | 默顿的失范定理
Robert Merton’s strain theory explains deviance as a gap between culturally approved goals (e.g., wealth, success) and the legitimate means available to achieve them. Individuals adapt in five ways: conformity, innovation, ritualism, retreatism and rebellion.
罗伯特·默顿的失范理论将越轨行为解释为文化认可的目标（如财富、成功）与可获得的合法手段之间的断裂。个体采取五种适应方式：顺从、创新、仪式主义、退却主义和反叛。
This 'theorem' bed goals - institutionalised means. When means are blocked, strain increases and deviant adaptations become more likely.
这一“定理”犹如社会学公式：角色压力 = 文化规定的目标 - 制度化的手段。当手段受阻，压力增大，越轨适应出现的可能性也随之上升。
10. The Thomas Theorem | 托马斯定理
The Thomas theorem, formulated by W. I. Thomas, states that 'If people define situations as real, they are real in their consequences.' It underlines the importance of subjective definitions in shaping social reality.
由 W. I. 托马斯提出的托马斯定理指出：“如果人们将情境定义为真实的，那么它们的后果也是真实的。” 这强调了主观定义在塑造社会现实中的重要性。
This principle is central to the interactionist perspective. For instance, if a student is labelled as 'troublesome', teachers may treat them accordingly, leading to a self-fulfilling prophecy that makes the label stick.
这一原则是互动论视角的核心。例如，如果一名学生被贴上“捣蛋”的标签，老师可能据此对待他，从而形成自我实现的预言，使标签固化。
11. Becker's Labelling Theorem | 贝克尔的标签定理
Howard Becker’s labelling theory argues that deviance is not a quality of the act a person commits but rather a consequence of the application of rules and sanctions by others. The deviant is one to whom the label has been successfully applied.
霍华德·贝克尔的标签理论认为，越轨并非行为本身的性质，而是他人运用规则和制裁的结果。越轨者就是被成功贴上标签的那个人。
his theorem is essential for understanding how social control can amplify deviance rather than reduce it.
这一过程可以概括为：初次行为 → 社会反应 → 贴标签 → 主导身份 → 越轨生涯。这一定理对于理解社会控制如何可能放大而非减少越轨行为至关重要。
12. Marx's Theorem of Surplus Value | 马克思剩余价值定理
Though more of an economic law, the concept of surplus value is treated as a foundational theorem in conflict sociology. It describes how capitalists extract profit from the labour of workers by paying them less than the value of what they produce.
虽然更接近经济规律，但剩余价值概念被冲突社会学视为一项基础定理。它描述了资本家如何通过支付低于工人生产价值的工资来榨取利润。
Surplus Value = Value produced by labour - Wages paid to labour
According to Marx, this exploitation is the engine of class conflict. The theorem connects directly to discussions of alienation, polarisation and the eventual collapse of capitalism.
根据马克思的观点，这种剥削是阶级冲突的驱动力。该定理直接关系到异化、两极分化以及资本主义最终崩溃等讨论。
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