Stratified Sampling and Social Class | 分层抽样与社会阶级

📚 Stratified Sampling and Social Class | 分层抽样与社会阶级

In A-Level Statistics, the way we collect data can fundamentally shape our conclusions. One of the most powerful sampling methods – stratified sampling – comes alive when we study characteristics like social class, where populations naturally divide into distinct groups. Understanding how to draw representative samples across these groups is not only a key exam skill but also an essential tool for social researchers, market analysts and policy makers.

在 A-Level 统计学中，收集数据的方式会在根本上影响我们得出的结论。其中一种最强大的抽样方法——分层抽样——在我们研究如社会阶级这样自然划分为不同群体的特征时显得格外生动。理解如何跨群体抽取有代表性的样本，不仅是关键的考试技能，也是社会研究者、市场分析师和政策制定者的必备工具。

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1. Why Sampling Matters | 为什么抽样如此重要

When investigating a population, it is rarely possible or practical to collect data from every individual. Sampling allows us to select a smaller group – a sample – to represent the whole. If the sample is selected properly, we can use it to make reliable inferences about the population mean, proportion or other parameters without examining every single unit.

在研究一个总体时，很少有可能或现实地收集到每个个体的数据。抽样允许我们选择一个较小的群体——样本——来代表整体。如果样本选择得当，我们就可以用它来对总体均值、比例或其他参数作出可靠的推断，而无需检查每一个个体。

In the context of social class, the population may consist of people from varied economic and educational backgrounds. A poor sampling method could over-represent one class and under-represent another, leading to biased conclusions about spending habits, political views or health outcomes.

在社会阶级的语境下，总体可能由具有不同经济和教育背景的人组成。糟糕的抽样方法可能会过度代表某一个阶级，而低估另一个阶级，从而导致对消费习惯、政治观点或健康结果等得出有偏差的结论。

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2. Population, Sample and Sampling Frame | 总体、样本与抽样框

Before choosing a sampling method, it is crucial to define the population and the sampling frame. The population is the entire set of individuals or items we wish to study. The sampling frame is a list of all members of the population from which the sample is drawn. For an investigation into social class and online shopping, the population might be all adults in a city, while the sampling frame could be the electoral register or a database of postal addresses.

在选择抽样方法之前，定义总体和抽样框至关重要。总体是我们想要研究的全部个体或项目的集合。抽样框是总体所有成员的列表，样本正是从中抽取。对于一项关于社会阶级与网络购物的调查，总体可能是某个城市的所有成年人，而抽样框可以是选民登记册或邮寄地址数据库。

A mismatch between the population and the sampling frame can create undercoverage bias. For instance, if the sampling frame excludes individuals without a permanent address, homeless people – a particular social class segment – would be omitted, skewing results. In Edexcel S1, you will need to identify such sources of bias and suggest improvements.

总体与抽样框之间的不匹配会产生覆盖不足偏差。例如，如果抽样框排除了没有固定地址的个人，那么无家可归者——一个特定的社会阶级群体——就会被遗漏，从而扭曲结果。在 Edexcel S1 中，你需要识别这类偏差来源并提出改进建议。

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3. Simple Random Sampling and Its Limits | 简单随机抽样及其局限性

Simple random sampling gives every member of the population an equal chance of being selected. This can be implemented using random number tables, lottery methods or computer-generated random numbers. While it is free from selection bias in theory, it can fail to produce a sample that reflects the population’s structure, especially when there are distinct subgroups.

简单随机抽样使总体中每个成员都有均等被选中的机会。这可以通过随机数表、抽签法或计算机生成的随机数来实施。虽然理论上它没有选择偏差，但当存在明显的子群体时，它可能无法产生一个反映总体结构的样本。

Imagine a population where 10% belong to the upper class, 40% to the middle class and 50% to the working class. A simple random sample of size 100 might, by chance, include only 5 from the upper class. This would make estimates for the upper class unreliable. Stratified sampling addresses this problem directly.

设想一个总体，其中 10% 属于上层阶级，40% 属于中产阶级，50% 属于工人阶级。一个容量为 100 的简单随机样本可能偶然只包含 5 名上层阶级成员。这将使对上层阶级的估计不可靠。分层抽样直接解决了这个问题。

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4. Systematic Sampling in a Social Context | 社会背景下的系统抽样

Systematic sampling selects every k-th individual from the sampling frame after a random start. For example, every 20th name on a council tax list might be chosen. While simple to implement, it can introduce hidden patterns. If the list is ordered by postcode and postcodes are associated with social class, the sample could accidentally favour one class.

系统抽样在随机起点后，从抽样框中每隔 k 个个体选取一个。例如，可以选取市政税名单上每隔 20 个的名字。虽然实施简单，但它可能引入隐藏的模式。如果名单按邮政编码排序，而邮政编码又与社会阶级相关，那么样本就可能意外地偏袒某一阶级。

In exam questions, you may be asked to explain why systematic sampling is not always appropriate when studying variables like occupation. Answering that the frame may have periodicity linked to social stratification can demonstrate a deeper understanding of sampling bias.

在考试题目中，你可能会被要求解释为什么在研究职业等变量时系统抽样并不总是适合。回答道抽样框可能存在与社会分层相关的周期性，可以展现出对抽样偏差更深入的理解。

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5. Introducing Stratified Sampling | 分层抽样简介

Stratified sampling divides the population into non-overlapping groups called strata (singular: stratum), based on a characteristic that is known to affect the variable under investigation. Then a random sample is taken from each stratum. Social class is a classic stratifying variable because it influences so many aspects of life – from income and education to health and consumption.

分层抽样根据已知会影响被调查变量的特征，将总体划分为互不重叠的群体，称为层（单数：层）。然后从每一层中随机抽样。社会阶级是一个典型的分层变量，因为它影响着生活的方方面面——从收入和教育到健康和消费。

When we use social class as the stratifying variable, we ensure that each class is represented in the sample in proportion to its size in the population. This produces a sample that is a miniature version of the population, reducing variability in estimates and increasing precision.

当我们使用社会阶级作为分层变量时，我们确保每一阶级在样本中按其在总体中的比例得到代表。这就产生了一个总体缩影般的样本，减少估计值的变异性并提升精确度。

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6. Defining Social Class for Stratification | 为社会分层定义社会阶级

In practice, social class can be measured by occupation, income bracket, level of education or a combination of these. The UK’s National Statistics Socio-economic Classification (NS-SEC) divides people into categories such as higher managerial, intermediate, routine occupations, etc. For the purpose of A-Level problems, you may see simpler groupings like upper, middle and working class.

在实践中，社会阶级可以通过职业、收入区间、教育水平或这些指标的组合来衡量。英国国家统计社会经济分类 (NS-SEC) 将人们划分为高层管理、中间、例行职业等类别。就 A-Level 题目而言，你可能会见到更简单的分组，如上等、中等和工人阶级。

It is important that the strata are clearly defined so that every population unit belongs to exactly one stratum. Ambiguity in defining social class – for instance, if part-time workers are placed in different strata by different criteria – could undermine the whole sampling design.

重要的是各层必须被明确定义，使得每个总体单位恰好属于一个层。定义社会阶级时的模糊性——例如，如果兼职工作者按不同标准被归入不同层——可能会破坏整个抽样设计。

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7. Proportional Allocation: The Maths | 比例分配：数学原理

In proportional stratified sampling, the sample size from each stratum is proportional to the size of that stratum in the population. If the total population size is N, the size of stratum i is Nᵢ, and the total sample size is n, then the sample size from stratum i, nᵢ, is given by:

nᵢ = (Nᵢ / N) × n

在按比例的分层抽样中，每层的样本量与该层在总体中的大小成比例。如果总体总大小为 N，第 i 层的大小为 Nᵢ，总样本量为 n，那么第 i 层的样本量 nᵢ 由下式给出：

nᵢ = (Nᵢ / N) × n

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8. Worked Example with Social Class | 社会阶级分层抽样计算示例

Suppose a town has a population of 20 000 adults divided by social class as follows:

Social Class	Population (Nᵢ)
Upper	2 000
Middle	8 000
Working	10 000

假设某镇有 20 000 名成年人，按社会阶级划分如下：

社会阶级	人口 (Nᵢ)
上层	2 000
中层	8 000
工人	10 000

We want a total sample of size n = 400. Then:

n_upper = (2 000 / 20 000) × 400 = 0.1 × 400 = 40

n_middle = (8 000 / 20 000) × 400 = 0.4 × 400 = 160

n_working = (10 000 / 20 000) × 400 = 0.5 × 400 = 200

我们需要一个总样本量 n = 400。那么：

n_上层 = (2 000 / 20 000) × 400 = 0.1 × 400 = 40

n_中层 = (8 000 / 20 000) × 400 = 0.4 × 400 = 160

n_工人 = (10 000 / 20 000) × 400 = 0.5 × 400 = 200

This allocation ensures that the class proportions in the sample match exactly those in the population. Then simple random sampling is used within each class to select the required number of individuals.

这一分配确保样本中的阶级比例与总体完全相同。然后在每个阶级内部使用简单随机抽样选出所需数量的个体。

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9. Advantages of Stratified Sampling | 分层抽样的优点

Stratified sampling produces a sample that is guaranteed to represent all subgroups of interest. The estimates of population parameters are more precise because variability between strata is removed from the overall error. For instance, when comparing political attitudes across social classes, stratified sampling gives reliable estimates for each class while also keeping the overall margin of error small.

分层抽样产生的样本保证能够代表所有感兴趣的亚群体。总体参数的估计值更加精确，因为层间变异性被从总误差中移除。例如，在比较不同社会阶级的政治态度时，分层抽样为每个阶级提供可靠的估计值，同时也使总体误差幅度保持较小。

It is particularly useful when certain strata are small but important – like an upper class comprising only 10% of the population. A simple random sample might miss them entirely, but stratification ensures their voice is included proportionally.

当某层较小但很重要时——比如仅占人口 10% 的上层阶级——这种方法尤为有用。简单随机抽样可能完全遗漏他们，但分层确保了他们的声音按比例被纳入。

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10. Disadvantages and Practical Limitations | 缺点与实际限制

Stratified sampling requires an up-to-date and accurate sampling frame that lists each unit’s stratum. Obtaining a reliable classification of social class for an entire population is expensive and raises privacy concerns. Moreover, defining strata boundaries can be subjective; different sociologists might categorise the same person into different classes.

分层抽样需要一个最新且准确的抽样框，列出每个单位所属的层。获取整个总体可靠的社会阶级分类是昂贵的，并且会引发隐私问题。此外，界定层的界限可能带有主观性；不同的社会学家可能将同一个人归入不同的阶级。

Another drawback is that if the strata are not homogeneous internally, the precision gain is limited. For example, within the ‘middle class’ stratum there may be huge variation in income and lifestyle, meaning that stratification by broad class labels does not fully capture internal variability.

另一个缺点是，如果层内部不是同质的，精确度的提升就有限了。例如，在“中产阶级”这层内部，收入和生活方式可能存在巨大差异，这意味着按宽泛的阶级标签分层并不能完全捕捉内部变异性。

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11. Comparing Sampling Methods | 比较抽样方法

In Edexcel S1, you are expected to compare different sampling techniques. Stratified sampling tends to outperform simple random sampling when the variable of interest is strongly related to the stratifying variable. However, it is more complex and costly. Quota sampling, a non-random alternative, also uses strata but selects individuals by convenience rather than randomness, which introduces biases that stratified random sampling avoids.

在 Edexcel S1 中，你需要比较不同的抽样技术。当感兴趣的变量与分层变量密切相关时，分层抽样通常优于简单随机抽样。然而，它更复杂且成本更高。配额抽样是一种非随机替代方案，同样使用层，但根据方便性而非随机性选取个体，这会引入分层随机抽样所避免的偏差。

A typical exam question might give a scenario about investigating social class and leisure activities, asking you to justify the use of stratified sampling over other methods. Your answer should mention representativeness, precision and the ability to analyse individual strata.

一个典型的考试题目可能会给出一个关于调查社会阶级与休闲活动的情景，要求你论证为何使用分层抽样优于其他方法。你的答案应当提及代表性、精确度以及能够分析单个层的能力。

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12. Exam Tips for Edexcel S1 | Edexcel S1 考试技巧

• Always define the strata clearly when describing the method.

  在描述方法时，一定要明确定义各层。

• When calculating sample sizes, show the formula nᵢ = (Nᵢ/N) × n and substitute carefully. Round sensibly if needed – the total must add up to n.

  计算样本大小时，展示公式 nᵢ = (Nᵢ/N) × n 并仔细代入。如有必要进行合理取整——最终总和必须等于 n。

• Be prepared to critique a given sampling design. If you see a stratified sample where the strata are chosen based on sex but the research question is about social class, point out that a more relevant stratifying variable should be used.

  准备好对给定的抽样设计进行批判。如果你看到一个分层样本的层是按性别选择的，但研究问题是关于社会阶级，要指出应该使用更相关的分层变量。

• Understand that the main advantage of stratification is increased precision, not just ‘fairness’. Mention that it removes between-strata variation from the estimate’s standard error.

  要理解分层的主要优势是提高精度，而不仅仅是“公平性”。提及它从估计值的标准误中移除了层间变异。

• Use the words ‘representative’, ‘proportional’, and ‘stratum’ correctly. Good terminology usage earns marks.

  正确使用“代表性”、“按比例”和“层”这些词。良好的术语使用可以得分。

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