Pre-U Edexcel Sociology: Formula & Theorem Quick-Reference Handbook | Pre-U Edexcel 社会学：公式定理速查手册

📚 Pre-U Edexcel Sociology: Formula & Theorem Quick-Reference Handbook | Pre-U Edexcel 社会学：公式定理速查手册

Quantitative literacy is an essential skill in Pre-U Edexcel Sociology. Whether you are analysing demographic trends, measuring inequality, or interpreting survey data, a solid grasp of the core statistical formulae and theorems enables you to move beyond anecdotal evidence and engage with sociological research critically. This quick-reference handbook brings together the most important sociological formulae, from basic demographic rates to advanced measures of association, presented with worked-style definitions and interpretations relevant to the Edexcel specification. Use it alongside your methods revision to consolidate your understanding of how sociologists quantify social phenomena.

定量素养是 Pre-U Edexcel 社会学的重要技能。无论是分析人口趋势、测量不平等，还是解读调查数据，牢固掌握核心统计公式和定理都能让你超越轶事证据，批判性地参与社会学研究。本速查手册汇集了最重要的社会学公式，从基本的人口统计比率到高级的相关性测量，均附有适合 Edexcel 考纲的实用定义与解读。搭配研究方法复习使用，可帮助你巩固对社会学家如何量化社会现象的理解。

1. Demographic Basics: Crude Birth Rate and Death Rate | 人口基础：粗出生率与粗死亡率

The crude birth rate (CBR) and crude death rate (CDR) are starting points for demographic analysis. They express vital events relative to total population size without adjusting for age structure.

粗出生率（CBR）和粗死亡率（CDR）是人口分析的起点。它们将生命事件与总人口规模相联系，不调整年龄结构。

CBR = (B ÷ P) × 1000

where B is the number of live births in a year and P is the mid-year total population. The multiplier 1000 converts the figure into births per thousand population.

其中 B 为年内活产数，P 为年中总人口。乘数 1000 将数字转换为每千人的出生人数。

CDR = (D ÷ P) × 1000

with D representing deaths in a year. Sociologists use these rates to compare fertility and mortality across societies, though they are ‘crude’ because they ignore population composition.

D 代表年内死亡人数。社会学家用这些比率比较不同社会的生育与死亡水平，但它们是“粗”的，因为忽略了人口构成。

The rate of natural increase (RNI) is derived simply: RNI = CBR − CDR. A positive value indicates population growth without migration.

自然增长率（RNI）可直接推导：RNI = CBR − CDR。正值表示不包括迁移的人口增长。

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2. Fertility Rates: General Fertility Rate and Total Fertility Rate | 生育率：一般生育率与总和生育率

While the CBR relates births to the whole population, the general fertility rate (GFR) refines the measure by focusing on women of childbearing age, typically 15–49 years.

CBR 将出生与总人口关联，而一般生育率（GFR）通过关注育龄女性（通常 15–49 岁）使测量更精细。

GFR = (B ÷ F₁₅₋₄₉) × 1000

Here F₁₅₋₄₉ denotes the female population aged 15–49. GFR is still influenced by the age distribution within that group, so demographers use age-specific fertility rates (ASFR) and the total fertility rate (TFR).

此处 F₁₅₋₄₉ 指 15–49 岁女性人口。GFR 仍受该组内年龄分布影响，因此人口学家使用年龄别生育率（ASFR）和总和生育率（TFR）。

TFR is a synthetic cohort measure. It is the sum of age-specific fertility rates over all reproductive age groups, usually for five-year intervals, multiplied by the interval width:

TFR 是一个合成队列指标，即所有育龄组（通常间隔五年）的年龄别生育率之和乘以间隔宽度：

TFR = Σ (ASFRₐ × 5) / 1000

This yields the average number of children a woman would have if she experienced current age-specific rates throughout her reproductive life. A TFR of 2.1 is often taken as replacement level in developed countries.

这表示若一名妇女按当前年龄别生育率度过育龄期，她将生育的平均子女数。在发达国家，TFR 2.1 通常被视为更替水平。

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3. Marriage, Divorce, and Cohabitation Rates | 结婚率、离婚率与同居率

The crude marriage rate (CMR) and crude divorce rate (CDivR) follow the same logic as demographic rates, but researchers also calculate refined rates to better capture union dynamics.

粗结婚率（CMR）与粗离婚率（CDivR）遵循相同的人口统计逻辑，但研究者还计算精细比率以更好地反映婚恋动态。

CMR = (M ÷ P) × 1000

M is the number of marriages in a year. Similarly, CDivR = (Div ÷ P) × 1000, where Div is the number of divorces. However, sociologists often prefer the divorce rate per 1,000 married persons or the divorce-to-marriage ratio.

M 是年内结婚对数。类似地，CDivR = (Div ÷ P) × 1000，其中 Div 为离婚对数。但社会学家常偏好每千名已婚者的离婚率或离婚结婚比。

The cohabitation rate is typically expressed as the percentage of all couples who are cohabiting or the proportion of never-married adults cohabiting at a given age. It lacks a single standard formula and is often derived from survey estimates.

同居率通常表示为所有伴侣中同居者的百分比，或某一年龄段从未结婚成年人中同居的比例。该指标没有单一标准公式，通常来源于调查估计。

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4. Crime Rate and Victimisation Rate | 犯罪率与受害率

Official crime statistics rely on incidence rates. The basic crime rate normalises the number of offences by population size.

官方犯罪统计依赖发生率。基本犯罪率用案件数除以人口规模进行标准化。

Crime Rate = (Number of recorded crimes ÷ Total population) × 1,000 or 100,000

A multiplier of 100,000 is common for less frequent offences. Victimisation rate, often derived from crime surveys like the Crime Survey for England and Wales (CSEW), expresses the proportion of individuals or households experiencing crime in a reference period.

较少见的犯罪通常使用乘数 100,000。受害率常来自英格兰及威尔士犯罪调查（CSEW）等犯罪调查，表示在参考时期内遭受犯罪侵害的个人或家庭比例。

Victimisation Rate = (Number of victims ÷ Total relevant population) × 100

This measure is crucial for evaluating dark figures of crime – offences unreported to the police.

该指标对评估犯罪黑数（未报警的犯罪）至关重要。

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5. Measures of Central Tendency: Mean, Median, Mode | 集中趋势测量：均值、中位数、众数

Central tendency summarises a dataset in a single typical value. The three main measures serve different purposes in sociological research.

集中趋势用单个典型值概括数据集。三种主要测量在社会学研究中有不同用途。

Mean (x̄) = Σx ÷ n

The arithmetic mean is sensitive to extreme values, making it less suitable for skewed distributions such as income.

算术平均数对极端值敏感，因此不太适用于收入等偏态分布。

Median = middle value in ordered list; if n is even, median = (xₙ/₂ + xₙ/₂₊₁) ÷ 2

The median is resistant to outliers and is often preferred for wealth or income data. The mode is the most frequently occurring value and is especially useful for categorical data such as religious affiliation.

中位数不受极端值影响，常用于财富或收入数据。众数是最常出现的值，尤其适用于宗教信仰等类别数据。

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6. Measures of Dispersion: Range, Variance, Standard Deviation | 离散测量：极差、方差、标准差

Describing the spread of data is as important as describing its centre. The simplest measure is the range, but variance and standard deviation provide more robust summaries.

描述数据的散布情况与描述其中心同等重要。最简单的测量是极差，但方差和标准差能提供更稳健的概括。

Range = xₘₐₓ − xₘᵢₙ

Variance measures the average squared deviation from the mean:

方差测量各数值与均值之差的平方的平均值：

σ² = Σ (x − μ)² ÷ N (population variance)

For a sample, the denominator becomes n − 1 to give an unbiased estimate (s²).

对于样本，分母改为 n − 1 以给出无偏估计（s²）。

Standard deviation σ = √σ²

The standard deviation is expressed in the original units and is essential for interpreting normal distributions. In sociological studies on educational attainment or income, reporting the standard deviation alongside the mean clarifies inequality.

标准差以原始单位表达，是解读正态分布的关键。在有关教育成就或收入的社会学研究中，同时报告标准差与均值可以澄清不平等程度。

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7. Correlation: Pearson’s r | 相关：皮尔逊相关系数

Pearson’s product-moment correlation coefficient quantifies the strength and direction of a linear relationship between two continuous variables. It is widely used in secondary data analysis, for instance when examining links between parental income and educational achievement.

皮尔逊积矩相关系数量化两个连续变量之间线性关系的强度和方向。它广泛用于二手数据分析，例如考察父母收入与教育成就之间的联系。

r = Σ (xᵢ − x̄)(yᵢ − ȳ) ÷ √[Σ (xᵢ − x̄)² Σ (yᵢ − ȳ)²]

The value of r ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear correlation. A correlation does not imply causation, a principle repeatedly stressed in sociological methodology. The coefficient of determination r² indicates the proportion of shared variance.

r 值范围从 −1（完全负相关）到 +1（完全正相关），0 表示无线性相关。相关不等于因果，这是社会学方法论反复强调的原则。决定系数 r² 表示共享方差的比例。

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8. Chi-Square Test for Independence | 独立性卡方检验

When sociologists analyse contingency tables – for example, gender by voting intention – they often use the chi-square (χ²) test to determine whether an association is statistically significant.

当社会学家分析列联表（如性别与投票意向）时，常使用卡方（χ²）检验来判断关联是否具有统计显著性。

χ² = Σ [ (O − E)² ÷ E ]

O represents the observed frequency in each cell, while E is the expected frequency under the null hypothesis of independence, calculated as (row total × column total) ÷ grand total. The resulting χ² statistic is compared against a critical value from the chi-square distribution with degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). A significant result suggests that the variables are associated in the population.

O 代表每个单元格的观察频数，E 为独立性原假设下的期望频数，计算公式为（行合计 × 列合计）÷ 总计。将得到的 χ² 统计量与卡方分布的临界值比较，其自由度（df）=（行数 − 1）×（列数 − 1）。显著结果提示变量在总体中相关。

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9. Gini Coefficient of Inequality | 基尼系数

The Gini coefficient is a prominent measure of income or wealth inequality, derived from the Lorenz curve. It condenses the distribution into a single number.

基尼系数是测量收入或财富不平等的重要指标，源自洛伦兹曲线，将分布浓缩为单一数值。

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. G ranges from 0 (perfect equality) to 1 (maximal inequality). In practice, sociologists and economists calculate G using the formula:

其中 A 为绝对平等线与洛伦兹曲线之间的面积，B 为洛伦兹曲线下的面积。G 的取值范围从 0（完全平等）到 1（极端不平等）。实际上，社会学家与经济学家常使用以下公式计算 G：

G = [2 ÷ (n² × x̄)] Σ i × xᵢ − (n + 1) ÷ n

where individuals are ranked by income (xᵢ from lowest to highest) and n is the total number. A rising Gini coefficient signals widening inequality, a central concern in stratification studies.

其中个体按收入排序（xᵢ 由低到高），n 为总数。基尼系数上升意味着不平等加剧，这是分层研究的核心议题。

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10. Social Mobility: Absolute and Relative Rates | 社会流动：绝对流动率与相对流动率

Social mobility research distinguishes between absolute mobility – the total proportion of individuals moving between class positions – and relative mobility, which compares the chances of movement across different origins.

社会流动研究区分绝对流动（个体在阶级位置间流动的总比例）与相对流动（比较不同出身背景的流动机会）。

Absolute Mobility Rate = (Number of individuals in a different class than their origin) ÷ Total sample × 100

Relative mobility is often expressed as an odds ratio. For a 2 × 2 table with origin class (service vs. working) and destination class (service vs. working), the odds ratio is:

相对流动性通常用比值比表示。对于出身阶级（服务阶级 vs. 工人阶级）和目的阶级（服务阶级 vs. 工人阶级）的 2×2 表格，比值比为：

Odds Ratio = (A × D) ÷ (B × C)

where A = number staying in service class, B = downward mobility, C = upward mobility, D = staying in working class. An odds ratio of 1 indicates equal relative chances (perfect fluidity); values above 1 point to inequality of opportunity.

其中 A 为维持服务阶级的人数，B 为下向流动，C 为上向流动，D 为维持工人阶级。比值比为 1 表示相对机会相等（完全流动性）；大于 1 则表明机会不平等。

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11. Dependency Ratio and Ageing Index | 抚养比与老龄化指数

The dependency ratio is a simple indicator of the economic burden on the productive population, widely used in sociology of the family and social policy.

抚养比是衡量生产性人口经济负担的简明指标，广泛用于家庭社会学和社会政策研究。

Total Dependency Ratio = [(P₀₋₁₄ + P₆₅₊) ÷ P₁₅₋₆₄] × 100

This formula expresses the number of ‘dependent’ young and old persons per 100 people of working age. The old-age dependency ratio isolates the 65+ group: OADR = (P₆₅₊ ÷ P₁₅₋₆₄) × 100. The ageing index compares the older population to the young:

该公式表示每 100 名劳动年龄人口对应的“受抚养”少儿与老年人数。老年抚养比单独考察 65 岁以上群体：OADR = (P₆₅₊ ÷ P₁₅₋₆₄) × 100。老龄化指数则将老年人口与少儿人口对比：

Ageing Index = (P₆₅₊ ÷ P₀₋₁₄) × 100

Values above 100 indicate that the older population outnumbers children, a demographic tipping point with profound sociological implications for intergenerational solidarity and welfare states.

数值超过 100 表明老年人口超过少儿，这是一个对代际团结和福利国家有深远社会学影响的人口转折点。

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12. Poverty Measures: Headcount Ratio and Poverty Gap | 贫困测量：贫困人口比率与贫困差距

Poverty is a multidimensional concept, but quantitative sociologists often start with income-based indices. The headcount ratio is the simplest indicator.

贫困是多维度概念，但定量社会学家常从收入为基础的指数入手。贫困人口比率是最简单的指标。

Headcount Ratio (H) = (Number of people below poverty line ÷ Total population) × 100

This measure ignores the depth of poverty, so the poverty gap index (PGI) is used to capture how far the poor fall below the line, expressed as a percentage of the poverty line:

该指标忽略了贫困深度，因此贫困差距指数（PGI）用来衡量贫困人口距离贫困线的平均缺口，以贫困线的百分比表示：

PGI = (1 ÷ N) Σ [ (z − yᵢ) ÷ z ] for all yᵢ < z

where z is the poverty line and yᵢ is the income of the i-th poor person. The squared poverty gap (FGT measure) weights larger gaps more heavily, reflecting sociological concerns with severe deprivation.

其中 z 为贫困线，yᵢ 为第 i 个穷人的收入。平方贫困差距（FGT 指数）对更大缺口赋予更大权重，反映社会学对严重剥夺的关注。

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