📚 Pre-U AQA Sociology: Formulae & Theorems Quick Reference Handbook | Pre-U AQA 社会学：公式定理速查手册
Sociological research often relies on quantitative methods to identify patterns, test hypotheses, and measure social phenomena. This quick reference handbook brings together the essential formulae and key theoretical propositions you are likely to encounter in the Pre-U AQA Sociology course. From descriptive statistics to inferential tests and demographic rates, each entry is presented with a clear formula and paired explanations to support both exam preparation and independent research design.
社会学研究常依赖定量方法识别模式、检验假设并测量社会现象。这本速查手册汇集了 Pre-U AQA 社会学课程中常见的核心公式与重要理论命题。从描述统计到推断检验，再到人口统计学指标，每个条目均配有清晰的公式和对应的中英文解释，帮助你在备考和独立研究设计中快速查阅。
1. Measures of Central Tendency | 集中趋势测量
The mean is the arithmetic average of all values and is sensitive to extreme scores. It is calculated as the sum of all observations divided by the number of observations.
均值是所有数值的算术平均数，对极端值敏感。其计算方式为观测值总和除以观测个数。
Mean (x̄) = Σx / n
The median is the middle value when data are ordered from smallest to largest. If n is even, the median is the average of the two middle numbers.
中位数是将数据从小到大排序后处于中间位置的值。如果 n 为偶数，则取中间两个数的平均值。
Median position = (n + 1) / 2
The mode is the most frequently occurring value in a data set. A data set may be unimodal, bimodal, or multimodal.
众数是数据集中出现频率最高的值。数据集可以是单峰的、双峰的或多峰的。
In skewed distributions the median is often a better measure of central tendency than the mean, because it is not pulled by outliers.
在偏态分布中，中位数往往比均值更能代表集中趋势，因为它不受异常值的牵引。
2. Measures of Dispersion | 离散趋势测量
The range is the simplest measure of spread: it is the difference between the maximum and minimum values.
极差是最简单的离散度量：即最大值与最小值之差。
Range = xₘₐₓ − xₘᵢₙ
The interquartile range (IQR) measures the middle 50% of data and is less affected by outliers than the range. IQR = Q3 − Q1.
四分位距 (IQR) 测量中间 50% 数据的范围，相比极差更不易受异常值影响。IQR = Q3 − Q1。
IQR = Q₃ − Q₁
Variance quantifies the average squared deviation from the mean. For a sample, the denominator is n − 1 to produce an unbiased estimate.
方差衡量观测值与均值之差的平方的平均值。对于样本，分母为 n − 1，从而得到无偏估计。
s² = Σ(x − x̄)² / (n − 1)
Standard deviation is the square root of variance and returns the spread to the original units of measurement.
标准差是方差的平方根，它将离散程度还原为原始度量单位。
s = √[ Σ(x − x̄)² / (n − 1) ]
When reporting spreads in social surveys, researchers usually provide both the mean and the standard deviation to summarise data distributions.
在报告社会调查的离散特征时，研究者通常同时给出均值与标准差，以概括数据分布。
3. Correlation Coefficients | 相关系数
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. Values range from −1 to +1.
皮尔逊 r 衡量两个连续变量之间线性关系的强度和方向，取值范围从 −1 到 +1。
r = Σ((x − x̄)(y − ȳ)) / √( Σ(x − x̄)² Σ(y − ȳ)² )
A coefficient of +1 indicates a perfect positive linear correlation, −1 a perfect negative correlation, and 0 no linear correlation.
系数为 +1 表示完全正线性相关，−1 表示完全负线性相关，0 表示无线性相关。
Spearman’s rank correlation coefficient (ρ or rₛ) is used when data are ordinal or when the relationship is monotonic but not necessarily linear.
斯皮尔曼等级相关系数 (ρ 或 rₛ) 适用于定序数据或单调但未必线性的关系。
rₛ = 1 − ( 6 Σd² ) / ( n(n² − 1) )
where d is the difference between the ranks of each pair. Spearman’s rho is less sensitive to outliers than Pearson’s r.
其中 d 为每对观测值的秩次之差。斯皮尔曼 ρ 相比皮尔逊 r 对异常值更不敏感。
4. Chi-Squared Test | 卡方检验
The chi-squared test assesses whether there is a statistically significant association between two categorical variables in a contingency table.
卡方检验用于评估列联表中两个类别变量之间是否存在统计上显著的关联。
χ² = Σ( (O − E)² / E )
O stands for the observed frequency and E for the expected frequency under the null hypothesis of independence. Expected frequency for a cell is (row total × column total) / grand total.
O 代表观测频数，E 代表独立虚无假设下的期望频数。单元格的期望频数 = (行总计 × 列总计) / 总计。
The degrees of freedom (df) for a chi-squared test of independence are (number of rows − 1) × (number of columns − 1).
独立性卡方检验的自由度 (df) 为 (行数 − 1) × (列数 − 1)。
A statistically significant chi-squared value (p < 0.05) leads sociologists to reject the null hypothesis and infer an association in the population.
统计上显著的卡方值 (p < 0.05) 使社会学家拒绝虚无假设，推断总体中存在关联。
5. t-Test for Comparing Means | 比较均值的 t 检验
An independent-samples t-test determines whether the means of two unrelated groups differ significantly. It assumes roughly normal distributions and homogeneity of variance.
独立样本 t 检验判断两个不相关组别的均值是否存在显著差异，其假设为近似正态分布和方差齐性。
t = (x̄₁ − x̄₂) / √( sₚ² (1/n₁ + 1/n₂) )
where sₚ² is the pooled variance: sₚ² = ( (n₁−1)s₁² + (n₂−1)s₂² ) / (n₁ + n₂ − 2).
其中 sₚ² 为合并方差：sₚ² = ( (n₁−1)s₁² + (n₂−1)s₂² ) / (n₁ + n₂ − 2)。
Degrees of freedom for this test equal n₁ + n₂ − 2. Sociologists might use a t-test to compare, for example, exam scores of pupils taught with different methods.
该检验的自由度为 n₁ + n₂ − 2。社会学家可能使用 t 检验比较，例如，接受不同教学法学生的考试成绩。
6. Confidence Intervals | 置信区间
A confidence interval gives a range of plausible values for a population parameter, such as the population mean. The 95% confidence interval for a mean is widely used in sociological research.
置信区间给出总体参数（如总体均值）的合理取值范围。均值的 95% 置信区间在社会学研究中被广泛使用。
95% CI for μ = x̄ ± t* × (s / √n)
where t* is the critical value from the t-distribution for n − 1 degrees of freedom (often approximated by 1.96 for large samples).
其中 t* 是自由度为 n − 1 的 t 分布的临界值（大样本时常约为 1.96）。
A narrower interval suggests a more precise estimate. Confidence intervals can also be constructed for proportions and differences between means.
区间越窄表示估计越精确。置信区间还可为比例、均值之差等参数构建。
7. Demographic Rates | 人口统计学率
Crude birth rate (CBR) is the number of live births per 1,000 population in a given year.
粗出生率 (CBR) 是某年每千人口中的活产数。
CBR = (Number of live births / Total population) × 1,000
Crude death rate (CDR) similarly measures deaths per 1,000 population. The rate of natural increase is CBR minus CDR, usually expressed as a percentage.
粗死亡率 (CDR) 类似地衡量每千人口中的死亡数。自然增长率等于粗出生率减去粗死亡率，通常以百分比表示。
Rate of natural increase (%) = ((CBR − CDR) / 1,000) × 100
The total fertility rate (TFR) is the average number of children a woman would bear if she experienced current age-specific fertility rates throughout her childbearing years.
总和生育率 (TFR) 是指若一名妇女按当前各年龄组生育率度过生育期，她平均会生育的孩子数量。
TFR = Σ (Age-specific fertility rates over all ages)
8. Measures of Social Inequality | 社会不平等测量
The Gini coefficient is a summary measure of income inequality derived from the Lorenz curve. A value of 0 represents perfect equality and 1 maximal inequality.
基尼系数是由洛伦兹曲线得出的收入不平等综合度量，0 表示完全平等，1 表示极度不平等。
G = A / (A + B)
where A is the area between the line of perfect equality and the Lorenz curve, and B is the area under the Lorenz curve.
其中 A 是完全平等线与洛伦兹曲线之间的面积，B 是洛伦兹曲线下的面积。
The 20:20 ratio compares the average income of the richest 20% to that of the poorest 20%. Similarly, the Palma ratio is the ratio of the richest 10% to the poorest 40%.
20:20 比率将最富 20% 与最穷 20% 的平均收入进行比较。帕尔马比率则是前 10% 与后 40% 的收入比。
Quantile shares, such as the percentage of total income held by the top 1%, are also common in sociological analyses of stratification.
分位数份额（如前 1% 人口持有的总收入百分比）也常用于社会分层分析。
9. Indices of Social Mobility | 社会流动性指数
Absolute mobility measures the proportion of individuals who move to a different social class compared with their parents.
绝对流动性衡量与父母相比，进入不同社会阶层的个体比例。
Absolute mobility rate = (Number mobile / Total sample) × 100
Relative mobility compares the chances of moving up or down between groups, often expressed through odds ratios in a mobility table.
相对流动性比较不同组别上升或下降的机会，常通过流动表中的优势比表达。
Odds ratio = (A/B) / (C/D)
where A, B, C, D are cell frequencies in a 2×2 table contrasting origin and destination classes.
其中 A、B、C、D 是 2×2 流动表中比较出身阶级与到达阶级的单元格频数。
Sociologists also compute social fluidity indices that separate structural change from genuine openness.
社会学家还计算社会流动性指数，以区分结构性变化与真正的社会开放性。
10. Theoretical Propositions as Formal Statements | 以公式化陈述呈现的理论命题
Some sociological theories can be expressed as formal propositions. Durkheim’s classic insight into suicide argued that suicide rates vary inversely with social integration and regulation.
某些社会学理论可表达为形式化命题。涂尔干关于自杀的经典见解认为，自杀率与社会整合和规范调节水平呈反向变动。
Suicide rate ∝ 1 / (Integration × Regulation)
Marx’s theory of exploitation can be encapsulated in the rate of surplus value, linking sociology to political economy.
马克思的剥削理论可用剩余价值率概括，将社会学与政治经济学联系起来。
Surplus value rate = s / v (surplus value over variable capital)
In exchange theory, Homans proposed that the more often a person’s action is rewarded, the more likely the person will repeat that action. This can be written as a probability statement: P(repeat) ∝ Frequency of reward.
在交换理论中，霍曼斯提出，一个人的行动获得回报越频繁，其重复该行动的可能性越大。这可写为概率陈述：P(重复) ∝ 回报频率。
Rational choice theory assumes actors maximise utility. This is often formalised as U = Σ (pᵢ × uᵢ), where pᵢ is the probability of outcome i and uᵢ its utility.
理性选择理论假设行动者最大化效用，通常形式化为 U = Σ (pᵢ × uᵢ)，其中 pᵢ 是结果 i 的概率，uᵢ 是其效用。
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