A-Level Mathematics: Comparison and Selection of Common Sampling Methods | A-Level 数学：常见抽样方法的比较与选择

📚 A-Level Mathematics: Comparison and Selection of Common Sampling Methods | A-Level 数学：常见抽样方法的比较与选择

In A-Level Mathematics, particularly within the Statistics section, one of the fundamental skills is understanding how to collect data effectively. The way we select a sample from a population can significantly influence the reliability and validity of our conclusions. This article provides a comprehensive comparison of the most common sampling methods encountered in the A-Level syllabus, and offers practical guidance on how to choose the most appropriate method for a given scenario.

在 A-Level 数学中，尤其是统计学部分，理解如何有效收集数据是一项基本技能。我们从总体中选择样本的方式会显著影响结论的可靠性和有效性。本文将全面比较 A-Level 考纲中常见的抽样方法，并为如何在特定情境下选择最合适的方法提供实用指导。

1. Key Definitions in Sampling | 抽样中的关键定义

Before comparing sampling methods, it is essential to establish a common vocabulary. A population is the entire set of individuals or objects of interest in a study, while a sample is a subset of that population which is actually observed. A sampling frame is the list of individuals from which the sample is drawn, and a sampling unit is the individual member of the population that could be chosen.

在比较抽样方法之前，有必要建立共同的术语。总体是研究中感兴趣的全部个体或对象的集合，而样本是从该总体中实际观察到的子集。抽样框是用于抽取样本的个体名单，而抽样单元是总体中可能被选中的单个成员。

There are two broad categories of sampling methods: probability sampling, where every member of the population has a known, non-zero chance of being selected, and non-probability sampling, where selection is not entirely random. In the A-Level Mathematics course, we focus primarily on probability sampling methods, as they allow for statistical inference using probabilities.

抽样方法可分为两大类：概率抽样，即总体中每个成员都有已知且非零的概率被选中；以及非概率抽样，即选择并非完全随机。在 A-Level 数学课程中，我们主要关注概率抽样方法，因为它们允许使用概率进行统计推断。

2. Simple Random Sampling | 简单随机抽样

Simple random sampling (SRS) is the most basic and theoretically important sampling method. In this method, every sample of size n from the population has an equal chance of being selected. This is typically achieved using random number generators, random number tables, or by drawing names from a hat.

简单随机抽样（SRS）是最基本且在理论上最重要的抽样方法。在这种方法中，从总体中抽取大小为 n 的每个样本被选中的机会均等。这通常通过使用随机数生成器、随机数表或从帽子中抽签来实现。

There are two variants of SRS: sampling with replacement and sampling without replacement. In sampling without replacement, once an individual is selected, they cannot be selected again. In sampling with replacement, they can be selected more than once. In A-Level Statistics, we often assume sampling without replacement when dealing with finite populations, and the hypergeometric distribution is relevant in such cases.

SRS 有两种变体：有放回抽样和无放回抽样。在无放回抽样中，一旦某个个体被选中，就不能再次被选中。在有放回抽样中，同一个体可以被多次选中。在 A-Level 统计学中，处理有限总体时我们通常假设无放回抽样，此时超几何分布是相关的。

Key Advantages: Simple to understand, unbiased, and allows for straightforward use of probability theory.

关键优点：易于理解，无偏，且可直接应用概率理论。

However, SRS requires a complete and accurate sampling frame, which may be expensive or impossible to obtain for large populations. Additionally, SRS may accidentally produce a sample that is not representative of the population’s structure, particularly in terms of age, gender, or geographic distribution.

然而，SRS 需要完整且准确的抽样框，对于大型总体而言，这可能是昂贵或无法获得的。此外，SRS 可能偶然产生不能代表总体结构的样本，特别是在年龄、性别或地理分布方面。

3. Systematic Sampling | 系统抽样

Systematic sampling involves selecting every k-th member of the population after a random starting point. For example, if a factory produces 1000 items per day and we want a sample of 50, we would select every 20th item. The first item is chosen randomly from the first k items, and then every subsequent k-th item is chosen automatically.

系统抽样是在随机起点之后，选取总体中每第 k 个成员的方法。例如，如果工厂每天生产 1000 件产品，我们想要一个 50 件的样本，我们会每隔 20 件选取一件。第一个物品从前 k 个物品中随机选取，然后每隔 k 个自动选取后续物品。

This method is often used when the population is naturally ordered, such as a production line, a patient list, or a voter register. It is simpler and faster than SRS because it does not require generating a large number of random numbers.

当总体按自然顺序排列时，如生产线、患者名单或选民登记册，这种方法常用。它比 SRS 更简单、更快捷，因为不需要生成大量随机数。

The main danger of systematic sampling is that it can introduce bias if the population has a periodic or cyclical pattern that coincides with the sampling interval k. For instance, if a factory’s production quality drops every 20th item due to a machine cycle, selecting every 20th item would systematically miss or over-represent those defective items.

系统抽样的主要风险是，如果总体存在与抽样间隔 k 重合的周期性或循环模式，则可能引入偏倚。例如，如果工厂的每第 20 件产品因机器周期而质量下降，则每隔 20 件产品取样会系统性地遗漏或过度代表这些次品。

4. Stratified Sampling | 分层抽样

Stratified sampling is a method used to ensure that specific subgroups (strata) within a population are represented proportionally in the sample. First, the population is divided into mutually exclusive strata based on one or more characteristics, such as age group, gender, or income level. Then, a simple random sample is taken from each stratum, with the sample size in each stratum proportional to its size in the population.

分层抽样是一种确保总体中特定子群（层）在样本中按比例得到代表的方法。首先，根据一个或多个特征（如年龄组、性别或收入水平）将总体划分为互斥的层。然后，从每层中抽取简单随机样本，每层的样本量与其在总体中的规模成比例。

The formula for the sample size in each stratum is given by:

每一层的样本量公式如下：

nₓ = (Nₓ / N) × n

where Nₓ is the population size of the stratum, N is the total population size, and n is the total sample size.

其中 Nₓ 是层的总体大小，N 是总总体大小，n 是总样本大小。

Stratified sampling guarantees proportional representation of each stratum, thereby improving the accuracy and representativeness of the sample compared to SRS. It also allows for separate analysis of each subgroup, which can be insightful for comparing different segments of the population.

分层抽样保证每层按比例得到代表，从而与 SRS 相比提高了样本的准确性和代表性。它还允许对每个子群进行单独分析，这对于比较总体的不同细分部分非常有价值。

The main disadvantage is the practical difficulty of dividing the population into strata and knowing the exact size of each stratum. Moreover, if the selected strata are not relevant to the research objective, this method can be inefficient.

主要缺点是将总体分成层并准确知道每层规模在实际操作中存在困难。此外，如果所选择的层与研究目标无关，则此方法可能效率不高。

5. Cluster Sampling | 整群抽样

Cluster sampling involves dividing the population into naturally occurring groups, known as clusters, and then randomly selecting an entire cluster (or several clusters) to be included in the sample. For example, a researcher studying school students might select a random sample of schools and then include all students from those selected schools in the sample.

整群抽样将总体划分为自然形成的群组，称为群，然后随机选择整个群（或几个群）纳入样本。例如，研究学生的研究人员可能随机选择几所学校，然后将这些被选中学校的所有学生纳入样本。

Cluster sampling is particularly economical and practical when the population is geographically dispersed or when obtaining a complete sampling frame is difficult. It reduces the cost of data collection because data collection is concentrated in fewer locations. It is also useful when a list of clusters is easier to obtain than a list of individuals.

当总体在地理上分布广泛或难以获得完整抽样框时，整群抽样特别经济实用。由于数据收集集中在较少的地点，因此降低了数据收集成本。当群列表比个体列表更容易获得时，它也很有用。

However, cluster sampling is generally less accurate than SRS because individuals within a cluster tend to be more similar to each other than to individuals in other clusters. This reduces the diversity of the sample and increases the sampling error, often requiring a larger sample size to achieve the same level of precision.

然而，整群抽样通常不如 SRS 准确，因为群内个体往往比与其他群中的个体更相似。这减少了样本的多样性并增加了抽样误差，通常需要更大的样本量才能达到相同的精度水平。

A key distinction is that stratified sampling ensures each stratum is represented, whereas cluster sampling may omit entire clusters entirely. The latter is often used in multi-stage sampling, where clusters are selected first, and then individuals within those clusters are further sampled.

一个关键区别是分层抽样确保每层都得到代表，而整群抽样可能完全遗漏某些群。后者常用于多阶段抽样，其中先选择群，然后进一步对群内的个体进行抽样。

6. Quota Sampling | 定额抽样

Quota sampling is a non-probability sampling method often used in market research and opinion polls. In this method, interviewers are assigned a quota of participants with specific characteristics (e.g., a certain number of men and women, or a certain number of people from different age groups). Within each quota, the selection of individuals is left to the interviewer’s discretion, and may be based on convenience or availability.

定额抽样是一种非概率抽样方法，常用于市场研究和民意调查。在这种方法中，访问员被分配一定数量的具有特定特征的参与者（例如，一定数量的男性和女性，或一定数量的不同年龄段的人）。在每个配额内，个体的选择由访问员自行决定，可能基于便利性或者可用性。

Quota sampling is highly convenient, quick, and inexpensive compared to probability sampling methods. It ensures that the sample includes a minimum number of individuals from each subgroup of interest, mirroring some of the benefits of stratified sampling.

与概率抽样方法相比，定额抽样非常方便、快速且成本低廉。它确保样本包括每个关注子群的最少数量的个体，镜像了分层抽样的一些优点。

However, because the selection within quotas is not random, quota sampling is subject to interviewer bias and cannot be reliably used for statistical inference. For example, an interviewer may unconsciously choose individuals who look friendly or approachable, leading to a biased sample. The margin of error for such samples cannot be calculated using probability theory.

然而，由于配额内的选择不是随机的，定额抽样容易受到访问员偏倚的影响，且不能可靠用于统计推断。例如，访问员可能无意识地选择看起来友好或平易近人的个体，导致样本有偏。这种样本的误差范围无法使用概率论计算。

7. Convenience and Opportunity Sampling | 便利抽样与机会抽样

Convenience sampling, also known as opportunity sampling, involves selecting individuals simply because they are easy to reach. A common example is a journalist asking passers-by on the street for their opinion, or a teacher using students in their own class as a sample. The sample consists of whoever happens to be available and willing to participate at the time of the study.

便利抽样，又称机会抽样，是指仅仅因为个体容易接触到而进行选择。一个常见例子是记者在街上询问路人的意见，或者老师使用自己的班级学生作为样本。样本由研究发生时碰巧在场且愿意参与的任何人组成。

While this method is extremely quick, cheap, and requires minimal planning, it is highly prone to bias. The resulting sample is often not representative of the wider population, because it excludes individuals who are not at the same location or who do not have the time or inclination to participate. Consequently, findings from convenience samples should be generalised with extreme caution, if at all.

虽然这种方法极其快捷、便宜，并且几乎不需要计划，但非常容易产生偏倚。由此得到的样本往往不能代表更广泛的总体，因为它排除了不在同一地点或没有时间或意愿参与的个体。因此，便利样本的结果应极其谨慎地推广，甚至不应推广。

8. Comparing Sampling Methods | 抽样方法的比较

To choose the best sampling method for a given situation, it is useful to compare them systematically across several key criteria: representativeness, cost, time, complexity, and suitability for statistical inference. The following table summarises these comparisons for the methods discussed above.

要为特定情况选择最佳抽样方法，系统地在几个关键标准上比较它们是有用的：代表性、成本、时间、复杂性和对统计推断的适用性。下表总结了上述方法的比较。

Method 方法	Representativeness 代表性	Cost & Time 成本与时间	Statistical Inference 统计推断	Main Weakness 主要弱点
Simple Random 简单随机	High (in theory) 高（理论上）	Medium / Medium 中 / 中	Yes 可以	Needs full sampling frame; may miss subgroups 需要完整抽样框；可能遗漏子群
Systematic 系统	Medium-High 中高	Low / Low 低 / 低	Yes (if random start) 可以（若有随机起点）	Risk of periodic bias 周期性偏倚风险
Stratified 分层	High 高	High / Medium 高 / 中	Yes 可以	Need precise stratum sizes 需要精确的各层规模
Cluster 整群	Low-Medium 低中	Low / Low 低 / 低	Yes (with caution) 可以（需谨慎）	Low precision; clusters may be homogeneous 精度低；群内可能同质
Quota 定额	Medium (by design) 中等（按设计）	Low / Very Low 低 / 极低	No 不可以	Interviewer bias; no probability basis 访问员偏倚；无概率基础
Convenience 便利	Very Low 极低	Very Low / Very Low 极低 / 极低	No 不可以	Highly biased sample 样本高度有偏

From this comparison, we can see that probability sampling methods (SRS, systematic, stratified, cluster) are generally preferred for research because they allow for unbiased estimation and calculation of sampling error. Non-probability methods (quota, convenience) are useful only for exploratory or low-stakes investigations.

从这种比较中，我们可以看到，概率抽样方法（简单随机、系统、分层、整群）在研究中通常更受青睐，因为它们允许无偏估计和计算抽样误差。非概率方法（定额、便利）仅适用于探索性或低风险调查。

9. How to Select the Right Method | 如何选择合适的方法

The choice of sampling method depends on several practical factors, including the type of population, the available sampling frame, the resources available, the need for precision, and whether the research is exploratory or confirmatory. Below is a structured decision-making process.

抽样方法的选择取决于多个实际因素，包括总体类型、可用的抽样框、可用资源、对精度的需求，以及研究是探索性的还是验证性的。以下是一个结构化的决策过程。

If a complete sampling frame is available and the population is homogeneous: Use simple random sampling. It is straightforward and unbiased.

如果总体是齐性的且存在完整抽样框：使用简单随机抽样。它直接且无偏。
If the population is ordered and a complete frame exists, but time is limited: Use systematic sampling with a random start.

如果总体有序且存在完整抽样框，但时间有限：使用带有随机起点的系统抽样。
If the population consists of distinct subgroups and you want to ensure proportional representation: Use stratified sampling, but only if stratum sizes are known.

如果总体由不同的子群组成，且希望确保按比例代表：使用分层抽样，但前提是已知各层规模。
If the population is naturally divided into groups and a list of groups is easier to obtain than a list of individuals: Use cluster sampling, especially to reduce travel costs.

如果总体自然分为若干群，且获得群名单比获得个体名单更简单：使用整群抽样，尤其是为了减少差旅成本。
If only an approximate, quick, and inexpensive estimate is needed: Quota sampling may be considered, but clearly identify the limitations.

如果只需要大致、快速且廉价的估计：可考虑定额抽样，但应明确其局限性。
If no frame exists and the research is purely exploratory: Convenience sampling might be acceptable for a pilot study, but results must be treated as indicative only.

如果不存在抽样框且研究纯属探索性：便利抽样可能适合试点研究，但结果只能作为参考。

In an exam context, students are often asked to identify the most suitable sampling method for a described scenario. Look for keywords: “random” suggests SRS; “every nth” suggests systematic; “proportion of groups” suggests stratified; “schools/areas as groups” suggests cluster; “quota/interviewer” suggests quota; “passers-by/volunteers” suggests convenience.

在考试中，学生通常需要为描述的情境识别最合适的抽样方法。注意关键词：”random” 表示简单随机；”every nth” 表示系统；”groups proportional” 表示分层；”schools/areas” 表示整群；”quota/interviewer” 表示定额；”passers-by/volunteers” 表示便利。

10. Common Exam Pitfalls | 常见考试陷阱

In A-Level exams, students often lose marks due to imprecise terminology or confusion between similar methods. For instance, it is easy to confuse stratified sampling with quota sampling. The crucial distinction is that stratified sampling uses random selection within each stratum, while quota sampling uses non-random selection within each quota.

在 A-Level 考试中，学生常因术语不精确或混淆相似方法而失分。例如，分层抽样与定额抽样容易混淆。关键区别在于分层抽样在每层内使用随机选择，而定额抽样在每个配额内使用非随机选择。

Another common mistake is writing “random sampling” when the question specifically requires systematic or stratified sampling. Marks may be awarded for identifying the correct method but deducted for lacking justification. It is always important to justify your choice by linking it to the scenario details.

另一个常见错误是当题目特别要求系统抽样或分层抽样时，写下”随机抽样”。识别正确方法可能得分，但缺乏理由会被扣分。始终要将你的选择与场景细节联系起来进行论证。

Students also frequently forget that cluster sampling does not require every cluster to be represented; instead, only randomly selected clusters are included. In contrast, in stratified sampling, every stratum must be represented. This distinction is often tested directly.

学生还常常忘记整群抽样不要求每个群都得到代表；而是只包括随机选择的群。相比之下，在分层抽样中，每个层必须得到代表。这一区别经常被直接测试。

Finally, do not say that quota sampling is “unbiased” or “can be used to calculate error margins”. Since it is non-probability, such statements are incorrect. Similarly, convenience sampling should never be described as a reliable basis for generalisation.

最后，不要说定额抽样”无偏”或”可用于计算误差范围”。由于它是非概率的，此类陈述是不正确的。同样，便利抽样绝不应被描述为可靠概括的基础。

11. Worked Example | 实例演示

Scenario: A market researcher wants to estimate the average weekly spending of customers in a large shopping mall. There are 5 floors in the mall, and the researcher obtains a list of all 200 shops. Instead of interviewing every shop, the researcher randomly selects 10 shops from the list and then interviews every customer who leaves those selected shops during one day.

场景：市场研究人员想估计大型购物中心顾客的平均每周消费额。购物中心有 5 层，研究人员获得了所有 200 家店铺的名单。研究人员没有访问每家店铺，而是从名单中随机选择 10 家店铺，然后在一天内采访这些被选中店铺离开的每位顾客。

Question: Identify the sampling method used and state one advantage and one disadvantage of this method in this context.

问题：指出所使用的抽样方法，并就此背景说明该方法的一个优点和一个缺点。

Solution: The researcher is using cluster sampling, because the shops are the clusters, and once a shop is randomly chosen, all customers within that shop are sampled.

解答：研究人员使用的是整群抽样，因为店铺是群，一旦随机选中某店铺，该店铺内的所有顾客都被抽样。

Advantage: It is more time-efficient and cost-effective than visiting all 200 shops, as data collection is concentrated in only 10 locations.

优点：比访问全部 200 家店铺更省时省钱，因为数据收集集中在仅 10 个地点。

Disadvantage: Customers of one shop may be very similar (e.g., all buying luxury goods), so the sample may not accurately represent the spending patterns of all customers in the mall.

缺点：同一店铺的顾客可能非常相似（例如，都购买奢侈品），因此样本可能无法准确代表购物中心所有顾客的消费模式。

12. Summary and Revision Checklist | 总结与复习清单

To master sampling methods for the A-Level examination, ensure you understand the definition of each method, its procedure, its advantages and disadvantages, and the situations in which it is most appropriate. Practice applying these methods to new scenarios, and pay close attention to the phrasing of exam questions to avoid common errors.

为了在 A-Level 考试中掌握抽样方法，请确保你理解每种方法的定义、步骤、优缺点及其最适用的情境。练习将这些方法应用于新场景，并密切注意考题的措辞，以避免常见错误。

Simple random: every member equally likely; requires sampling frame.

简单随机：每个成员等可能；需要抽样框。
Systematic: every k-th member after a random start; watch for periodic bias.

系统：随机起点后每隔 k 个；注意周期性偏倚。
Stratified: divide into strata and random-sample proportionally from each.

分层：分为层并按比例从每层随机抽样。
Cluster: divide into clusters and randomly choose whole clusters.

整群：分为群并随机选择整个群。
Quota: non-random selection within predetermined quotas; no inference possible.

定额：在预定配额内非随机选择；无法推断。
Convenience: uses readily available participants; highly biased but very cheap.

便利：使用易于获取的参与者；高度有偏但非常便宜。

Always justify your chosen method in the context of the question, and remember that only probability sampling methods can support reliable statistical inference.

始终在问题情境中论证你选择的方法，并记住只有概率抽样方法才能支持可靠的统计推断。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level Mathematics: Comparison and Selection of Common Sampling Methods | A-Level 数学：常见抽样方法的比较与选择