📚 Year 11 SQA Sociology: Quick Reference Handbook of Formulas and Theorems | SQA 社会学公式定理速查手册
Sociology blends qualitative insight with quantitative rigour. This handbook summarises essential statistical formulas and foundational theoretical propositions that every SQA Sociology learner should have at their fingertips. Use it for revision, in-class tasks, and assignment support.
社会学融合了定性洞察与定量严谨。本手册总结了每位 SQA 社会学学习者都应熟记的基本统计公式和核心理论命题，可用于复习、课堂任务及作业支持。
1. Measures of Central Tendency | 集中趋势量数
Mean (μ or x̄): ores and n is the number of scores. The arithmetic average is sensitive to extreme values.
均值 (μ 或 x̄)： μ = Σx ÷ n，其中 Σx 为所有得分之和，n 为数据个数。算术平均数易受极端值影响。
Median: The middle value when data are ordered. For an even number of scores, median = (n÷2th score + (n÷2 + 1)th score) ÷ 2. It is unaffected by outliers.
中位数： 数据排序后位于中间的值。若数据个数为偶数，中位数 = (第 n÷2 个数 + 第 (n÷2 + 1) 个数) ÷ 2。中位数不受异常值影响。
Mode: The most frequently occurring value. A data set can have one mode, more than one mode (bimodal or multimodal), or no mode.
众数： 出现频率最高的值。一组数据可能有一个众数、多个众数（双峰或多峰）或无众数。
2. Range and Interquartile Range | 极差与四分位距
Range: Maximum value − Minimum value. It gives a crude measure of spread but is heavily affected by outliers.
极差： 最大值 − 最小值。它粗略地衡量离散程度，但极易受异常值影响。
Interquartile Range (IQR): tile (25th percentile) and Q₃ is the upper quartile (75th percentile). IQR focuses on the middle 50% of data and resists extreme scores.
四分位距 (IQR)： IQR = Q₃ − Q₁，其中 Q₁ 为下四分位数（第 25 百分位数），Q₃ 为上四分位数（第 75 百分位数）。四分位距关注中间 50% 的数据，能抵抗极端值。
3. Variance and Standard Deviation | 方差与标准差
Variance (σ² or s²): σ² = Σ(x − μ)² ÷ n (population) or s² = Σ(x − x̄)² ÷ (n − 1) (sample). It quantifies the average squared deviation from the mean.
方差 (σ² 或 s²)： σ² = Σ(x − μ)² ÷ n（总体）或 s² = Σ(x − x̄)² ÷ (n − 1)（样本）。它衡量各数据与均值之差的平方的平均值。
Standard Deviation (σ or s): σ = √[Σ(x − μ)² ÷ n] (population); s = √[Σ(x − x̄ the spread to the original units, showing how much scores typically deviate from the mean.
标准差 (σ 或 s)： σ = √[Σ(x − μ)² ÷ n]（总体）；s = √[Σ(x − x̄)² ÷ (n − 1)]（样本）。它将离散程度恢复为原始单位，显示分值通常偏离均值多远。
σ = √( Σ(x − μ)² ÷ n )
4. Pearson's Correlation Coefficient | 皮尔逊相关系数
r = Σ[(x − x̄)(y − ȳ)] ÷ √[Σ(x − x̄)² · Σ(y − ȳ)²]. r ranges from −1 to +1. A positive r indicates that as one variable increases, the other tends to increase; a negative r indicates an inverse relationship. r = 0 suggests no linear correlation.
r = Σ[(x − x̄)(y − ȳ)] ÷ √[Σ(x − x̄)² · Σ(y − ȳ)²]。r 的取值范围为 −1 到 +1。正 r 表示一个变量增大时另一个也倾向于增大；负 r 表示反向关系；r = 0 提示无线性相关。
Remember: correlation does not imply causation. A strong r may reflect a third, confounding variable.
注意：相关不等于因果。强相关系数可能反映第三个混淆变量的作用。
5. Chi-Squared Test (χ²) | 卡方检验
χ² = Σ[(O − E)² ÷ E], where O is the observed frequency and E is the expected frequency under the null hypothesis. This test assesses whether there is a significant association between two categorical variables (e.g., gender and voting intention).
χ² = Σ[(O − E)² ÷ E]，其中 O 为观测频数，E 为原假设下的期望频数。该检验用于判断两个分类变量（如性别与投票意向）之间是否存在显著关联。
ompare χ² to a critical value from the χ² distribution table. If χ² > critical value, reject the null hypothesis at the chosen significance level (e.g., p < 0.05).
列联表的自由度：df = (行数 − 1) × (列数 − 1)。将 χ² 值与 χ² 分布表中的临界值比较。如果 χ² > 临界值，则在选定显著性水平（如 p < 0.05）下拒绝原假设。
6. Sampling Error and Confidence Intervals | 抽样误差与置信区间
Standard Error of the Mean (SE): SE = σ ÷ √n (or s ÷ √n). It measures how much the sample mean is expected to vary from the population mean.
均值的标准误 (SE)： SE = σ ÷ √n（或 s ÷ √n）。它衡量样本均值与总体均值的预期差异程度。
95% Confidence Interval for the Mean: CI = x̄ ± 1.96 × SE. This interval captures the population mean in 95% of repeated samples.
均值的 95% 置信区间： CI = x̄ ± 1.96 × SE。该区间在 95% 的重复采样中包含总体均值。
Larger sample sizes reduce SE, yielding narrower confidence intervals and more precise estimates.
样本量越大，标准误越小，置信区间越窄，估计越精确。
7. Sampling Methods and Their Formulas | 抽样方法及其公式
| Method | Key Formula / Principle |
| Simple Random | Each unit has equal probability: p = 1 ÷ N |
| Stratified | Sample size per stratum = (Stratum proportion × n) |
| Cluster | Randomly select clusters; then survey all or sample within clusters |
| Systematic | Select every kth unit; k = N ÷ n |
Each method has trade-offs in cost, precision, and bias. For SQA assignments, justify your choice with reference to the research question and population frame.
每种方法在成本、精度和偏差方面各有取舍。在 SQA 作业中，请根据研究问题和总体框架说明你的选择理由。
8. Functionalism: The Organic Analogy and Social Facts | 功能论：有机类比与社会事实
Durkheim's Theorem of Social Facts: ted parts (institutions) that work together to maintain stability, much like a biological organism. Each institution performs a function: family → primary socialisation, education → skill transmission and value consensus, religion → moral regulation.
涂尔干的社会事实定理： 社会事实是外在于个体、具有强制力的行动、思维和情感方式。社会是一个由相互关联的部分（制度）组成的系统，它们协同运作以维持稳定，就像生物有机体一样。每个机构都履行某种功能：家庭 → 初级社会化，教育 → 技能传递与价值共识，宗教 → 道德规范。
Key formula in functional analysis: If social institution A fails to perform its function, dysfunction arises, possibly leading to anomie.
功能分析的核心公式：如果社会制度 A 未能履行其功能，就会出现功能失调，甚至导致失范。
9. Conflict Theory: Base–Superstructure Model | 冲突论：基础–上层建筑模型
Marx's Theorem of Class Struggle: The economic base (forces and relations of production) shapes the superstructure (law, politics, ideology). In capitalist society, the bourgeoisie own the means of production; the proletariat sell their labour. Conflict between classes is the motor of historical change.
马克思的阶级斗争定理： 经济基础（生产力与生产关系）决定上层建筑（法律、政治、意识形态）。在资本主义社会，资产阶级拥有生产资料，无产阶级出卖劳动力。阶级之间的冲突是历史变革的动力。
Formula: our − Wages paid. This exploitation creates alienation and class consciousness under certain conditions.
公式： 剩余价值 = 劳动创造的价值 − 支付的工资。这种剥削在特定条件下会导致异化与阶级意识。
10. Symbolic Interactionism: The Looking-Glass Self and the Definition of the Situation | 符号互动论：镜中我及情境定义
Cooley's Theorem: "I am not what I think I am; I am not what you think I am; I am what I think you think I am." The self develops through social interaction and the imagined judgments of others.
库利的镜中我定理： “我不是我认为的我，也不是你认为的我，而是我以为你认为的我。”自我通过社会互动和对他人的想象判断而发展。
Thomas Theorem: "If men define situations as real, they are real in their consequences." Subjective meanings guide action, making social reality a constructed process.
托马斯定理： “如果人们将情境定义为真实的，那么其结果也是真实的。”主观意义引导行动，使社会现实成为建构的过程。
11. Labelling Theory and the Self-Fulfilling Prophecy | 标签理论与自我实现预言
Becker's Labelling Proposition: Deviance is not a quality of the act but a consequence of the application by others of rules and sanctions. Primary deviance is initial rule-breaking; secondary deviance occurs when the individual accepts the deviant label, altering their self-concept.
贝克尔的标签命题： 越轨并非行为本身的属性，而是他人应用规则与制裁的后果。初级越轨是初次违规；次级越轨发生在个体接受越轨标签并改变自我概念时。
r (self-fulfilling prophecy). This sequence is central to SQA discussions of crime and education.
该过程可表述为：标签 → 污名 → 主导身份 → 越轨生涯（自我实现预言）。这一序列是 SQA 犯罪与教育讨论的核心。
12. Weber's Bureaucracy and Rationalisation | 韦伯的官僚制与理性化
Weber's Ideal Type of Bureaucracy: A hierarchical organisation with a clear division of labour, written rules, impersonality, and appointment by merit. Its formula: Rational-legal authority replaces traditional and charismatic authority in modern societies.
韦伯的官僚制理想类型： 具有明确分工、书面规则、非人格化和按绩任命特征的层级组织。其公式为：现代社会中，法理型权威取代传统型权威与魅力型权威。
Rationalisation theorem: The world becomes disenchanted as efficiency, calculability, predictability, and control (the "iron cage") dominate social life. This process can lead to the McDonaldisation of society (Ritzer), where tasks are deskilled and dehumanised.
理性化定理： 随着效率、可计算性、可预测性和控制（“铁笼”）主导社会生活，世界走向祛魅。这一过程可能导致社会的麦当劳化（瑞泽尔），使工作去技能化和非人化。
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