📚 Year 12 CAIE Sociology Formula and Theorem Quick Reference Handbook | Year 12 CAIE 社会学：公式定理速查手册
This quick‑reference handbook brings together the essential formulas, statistical tools and foundational theoretical statements you must master for the CAIE AS Sociology course (9699). It is designed for fast revision, linking quantitative method calculations with the ‘big idea’ theorems from key sociological perspectives. Each entry presents the concept in straightforward English, followed immediately by its Chinese explanation, so you can switch between languages while reinforcing your understanding.
这本速查手册汇集了CAIE AS社会学（9699）课程中必须掌握的核心公式、统计工具以及基础理论陈述。它专为快速复习而设计，将定量研究方法的计算与社会学主要视角的“大观念”定理串联起来。每个条目先用简明英文阐述，紧接着提供中文解说，让你可以在双语切换中巩固理解。
1. Mean, Median and Mode | 均值、中位数与众数
The mean is calculated by summing all values and dividing by the total number of cases: Mean = Σx ÷ n. It is sensitive to extreme scores and therefore works best with interval/ratio data that are symmetrically distributed.
均值等于所有数值之和除以个案总数：均值 = Σx ÷ n。它对极端值敏感，因此最适合于对称分布的定距/定比数据。
The median is the middle value when data are ordered from smallest to largest. If there is an even number of cases, the median is the average of the two middle numbers. It is robust against outliers and is preferred for ordinal data or skewed distributions.
中位数是将数据从小到大排序后位于中间的值。如果个案数为偶数，则中位数为中间两个数的平均值。中位数不受极端值影响，适合用于定序数据或偏态分布。
The mode is the most frequently occurring value in a dataset. It is the only measure of central tendency that can be used with nominal data, and a distribution may have one mode (unimodal), two modes (bimodal) or more.
众数是数据集中出现次数最多的值。它是唯一能用于定类数据的集中趋势测量，一个分布可以是单峰的、双峰的甚至多峰的。
2. Range and Interquartile Range | 全距与四分位距
The range is the difference between the highest and the lowest scores: Range = Xmax − Xmin. It gives a quick sense of spread but is heavily affected by a single outlier.
全距是最大值与最小值之差：全距 = 最大值 − 最小值。它让人快速感知数据的离散程度，但极易受单个异常值的影响。
The interquartile range (IQR) overcomes this weakness by focusing on the middle 50 % of cases: IQR = Q3 − Q1, where Q1 is the 25th percentile and Q3 is the 75th percentile. IQR is particularly useful for comparing the spread of different groups when medians are reported.
四分位距（IQR）通过关注中间50%的个案克服了这一弱点：IQR = 第三四分位数 − 第一四分位数。当报告中位数时，IQR特别适合比较不同组别的离散程度。
2. Standard Deviation | 标准差
Standard deviation measures the average distance of each data point from the mean. For a population, the formula is σ = √[ Σ (x − μ)² ÷ N ]. For a sample, the divisor is N−1 to give an unbiased estimate: s = √[ Σ (x − x̄)² ÷ (n−1) ]. A small standard deviation indicates that scores cluster closely around the mean; a large one signals wide dispersion.
标准差衡量每个数据点与均值的平均距离。总体的公式为 σ = √[ Σ (x − μ)² ÷ N ]；样本则除以 N−1 以获得无偏估计：s = √[ Σ (x − x̄)² ÷ (n−1) ]。标准差小说明数据紧密聚集在均值周围，大则表示离散幅度很大。
In sociology, standard deviation is essential when comparing educational attainment or income inequality across social classes. It allows researchers to quantify how much variation exists within a group, not just between groups.
在社会学中，比较不同社会阶级的教育成就或收入不平等时，标准差必不可少。它使研究者能量化组内差异的程度，而不仅仅是组间差异。
4. Pearson’s Correlation Coefficient | 皮尔逊相关系数
The Pearson product‑moment correlation coefficient, r, measures the strength and direction of a linear relationship between two interval‑level variables. The formula is r = Σ (x − x̄)(y − ȳ) ÷ √[ Σ (x − x̄)² Σ (y − ȳ)² ]. Values range from −1 (perfect negative correlation) through 0 (no linear correlation) to +1 (perfect positive correlation).
皮尔逊积矩相关系数 r 测量两个定距变量之间线性关系的强度与方向。公式为 r = Σ (x − x̄)(y − ȳ) ÷ √[ Σ (x − x̄)² Σ (y − ȳ)² ]。数值范围从−1（完全负相关）经0（无线性相关）至+1（完全正相关）。
For CAIE Sociology, you should be able to interpret a given r‑value, for instance recognising that r = +0.8 indicates a strong positive association, while r = −0.3 suggests a weak negative association. Always remember that correlation does not imply causation; a third variable may be responsible for the observed pattern.
在CAIE社会学考试中，你需要能够解释给定的 r 值，例如识别出 r = +0.8 表示强正相关，而 r = −0.3 表示弱负相关。始终牢记：相关并不蕴含因果；第三者变量可能才是观察到的规律背后的原因。
5. Chi‑Squared Test (χ²) | 卡方检验
The chi‑squared test assesses whether there is a statistically significant association between two categorical variables. The test statistic is χ² = Σ [ (O − E)² ÷ E ], where O is the observed frequency and E is the expected frequency under the null hypothesis of no association. A larger χ² value, relative to the degrees of freedom, indicates a smaller probability that the observed association arose by chance.
卡方检验用于评估两个类别变量之间是否存在统计上显著的关联。检验统计量为 χ² = Σ [ (O − E)² ÷ E ]，其中 O 为观察频数，E 为无关联零假设下的期望频数。当自由度不变时，χ² 值越大，表明观察到的关联偶然出现的概率越小。
In AS Sociology, you do not need to calculate χ² from scratch, but you must interpret a given χ² value together with its p‑value. For example, if p < 0.05, sociologists reject the null hypothesis and conclude that a significant relationship exists — say between gender and subject choice.
在AS社会学中，你不需要从头计算 χ²，但必须能够结合 p 值解释给定的 χ² 数值。例如，若 p < 0.05，社会学家便拒绝零假设，认定存在显著关系——例如性别与科目选择之间的关系。
6. Standard Error and Confidence Intervals | 标准误与置信区间
The standard error of the mean quantifies how much a sample mean is expected to vary from the true population mean. It is calculated as SE = σ ÷ √n (or s ÷ √n when the population standard deviation is unknown). A smaller standard error means the sample mean is a more precise estimator of the population parameter.
均值的标准误衡量样本均值预期与真实总体均值之间的偏差程度。计算公式为 SE = σ ÷ √n（当总体标准差未知时用 s ÷ √n）。标准误越小，表示样本均值对总体参数的估计越精确。
A 95 % confidence interval is built around the sample mean: CI = x̄ ± (z × SE). For a 95 % confidence level the z‑value is usually 1.96. If the interval for the difference between two group means does not include zero, researchers conclude the difference is statistically significant.
95%置信区间围绕样本均值构建：置信区间 = x̄ ± (z × SE)。在95%置信水平下，z 值通常为1.96。如果两个组别均值之差的置信区间不包含零，研究者便认为该差异具有统计显著性。
7. Durkheim’s ‘Suicide Theory’ as a Sociological Theorem | 涂尔干的“自杀论”作为社会学定理
Émile Durkheim’s theorem states that suicide rates vary inversely with the degree of social integration and moral regulation. He identified four ideal types: egoistic suicide (too little integration), altruistic suicide (too much integration), anomic suicide (too little regulation) and fatalistic suicide (too much regulation). This is one of the earliest ‘laws’ in sociology because it links a seemingly individual act to social forces.
涂尔干的定理指出，自杀率与社会整合程度及道德规制程度呈反向变化。他确定了四种理想类型：利己型自杀（整合过弱）、利他型自杀（整合过强）、失范型自杀（规制过弱）和宿命型自杀（规制过强）。这是社会学中最早的“定律”之一，因为它将看似个人的行为与社会力量联系起来。
| Type of Suicide | Integration | Regulation |
|---|---|---|
| Egoistic | Low | — |
| Altruistic | High | — |
| Anomic | — | Low |
| Fatalistic | — | High |
8. Marx’s Theorem of Exploitation | 马克思的剥削定理
Karl Marx’s central theorem asserts that surplus value is extracted from the proletariat by the bourgeoisie. The rate of exploitation can be expressed as s′ = s ÷ v, where s is surplus value and v is variable capital (wages). The organic composition of capital is c ÷ v, and a rising organic composition leads to a falling rate of profit, a structural contradiction that Marx predicted would intensify class conflict.
马克思的核心定理断言，剩余价值由资产阶级从无产阶级身上榨取。剥削率可表示为 s′ = s ÷ v，其中 s 为剩余价值，v 为可变资本（工资）。资本有机构成为 c ÷ v，有机构成提高导致利润率下降，这一结构性矛盾被马克思预言将激化阶级冲突。
In the CAIE syllabus, you use this theorem to explain the relationship between base (economic structure) and superstructure (institutions, ideas), and to evaluate the claim that the capitalist state serves the interests of the ruling class.
在CAIE大纲中，你用这一定理来解释基础（经济结构）与上层建筑（制度、观念）之间的关系，并评价资本主义国家为统治阶级利益服务的论断。
9. Merton’s Strain Theory: A Propositional Theorem | 默顿的应变理论：命题式定理
Robert Merton’s theorem links deviance to a disjunction between culturally approved goals and the socially structured means available. The adaptation matrix produces five logical possibilities, each a testable proposition.
默顿的定理将偏差行为与文化认可的目标和制度化手段之间的脱节联系起来。适应矩阵产生了五种逻辑可能性，每一个都是一个可检验的命题。
| Adaptation | Cultural Goals | Institutionalised Means |
|---|---|---|
| Conformity | Accepts | Accepts |
| Innovation | Accepts | Rejects |
| Ritualism | Rejects | Accepts |
| Retreatism | Rejects | Rejects |
| Rebellion | Replaces | Replaces |
Merton’s theorem is widely used to explain why working‑class crime rates may be higher when legitimate opportunities are blocked, making it a cornerstone of subcultural theory.
默顿的定理被广泛用来解释当合法机会受阻时，为什么工人阶级的犯罪率可能更高，这使得它成为亚文化理论的基石。
10. Becker’s Labelling Theorem | 贝克尔的标签定理
Howard Becker’s theorem asserts that deviance is not a quality of the act a person commits, but rather a consequence of the application by others of rules and sanctions to an ‘offender’. The theorem can be summed up as: Deviance = Social Reaction × Label. Once an individual is successfully labelled, a self‑fulfilling prophecy may lock them into a deviant career.
贝克尔定理指出，偏差并非个体所犯行为的固有品质，而是他人将规则与制裁应用于“越轨者”的结果。该定理可概括为：偏差 = 社会反应 × 标签。一旦个体被成功贴上了标签，自我实现的预言便可能将其锁定于偏差生涯中。
For the exam, you should connect this theorem to concepts of master status, moral entrepreneurs and the differential enforcement of the law. This theoretical statement is often treated as a counter‑theorem to positivist explanations of crime.
考试中，你应将这一定理与首要地位、道德创业者以及法律的选择性执行等概念联系起来。这一理论陈述常被视为犯罪实证主义解释的对抗定理。
11. The ‘Operationalisation’ Rule | “操作化”规则
Every concept in sociological research must be operationalised before it can be measured. The formal rule is: Abstract Concept → Defined Dimensions → Measurable Indicators. For example, ‘social class’ might be broken into dimensions of occupation, income and education, each represented by indicators such as the NS‑SEC classification, household income brackets and highest qualification level.
社会学研究中的每一个概念在测量前都必须操作化。形式规则为：抽象概念 → 界定维度 → 可测量的指标。例如，“社会阶级”可划分为职业、收入和教育维度，每个维度由指标代表，如NS-SEC分类、家庭收入区间和最高学历水平。
Operationalisation is itself a methodological theorem: if a concept cannot be operationalised reliably and validly, any ‘finding’ about it lacks rigour. This rule underpins the CAIE requirement to evaluate the strengths and limitations of quantitative and qualitative data.
操作化本身就是一个方法论定理：如果一个概念不能被可靠地、有效地操作化，那么任何关于它的“发现”都缺乏严谨性。这条规则支撑了CAIE对评估定量与定性数据优缺点的要求。
12. Sampling Probability Formulas | 抽样概率公式
A probability sample is built on the principle that every unit in the target population has a known, non‑zero chance of selection. For simple random sampling, the selection probability is p = n ÷ N, where n is the sample size and N is the population size. This formula guarantees that results can be generalised using inferential statistics.
概率抽样建立在这样一个原则之上：目标总体中的每个单元都有已知的、非零的被选几率。对于简单随机抽样，入选概率为 p = n ÷ N，其中n为样本量，N为总体规模。该公式保证了可以使用推论统计对结果进行类推。
Systematic sampling replaces randomness with a fixed interval: k = N ÷ n. After selecting a random start between 1 and k, every k‑th unit is chosen. The sampling fraction f = n ÷ N also helps sociologists assess whether a sample is large enough to minimise sampling error.
系统抽样用一个固定的间隔取代了随机性：间隔 k = N ÷ n。在1到k之间随机选定一个起点后，每隔k个单元选取一个。抽样比 f = n ÷ N 也帮助社会学家评估样本量是否够大以尽量减少抽样误差。
Published by TutorHao | Sociology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导