📚 Common Sampling Methods and Their Applications | 常见抽样方法及其适用场景
In statistics, sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population. The choice of sampling method directly affects the accuracy, cost, and practicality of the data collected. This article reviews the most common sampling methods required in A-level Mathematics, highlights their advantages and limitations, and explains when to use each one.
在统计学中，抽样是指从总体中选择一部分个体，用以估计总体特征的过程。抽样方法的选择直接影响所收集数据的准确性、成本与可行性。本文回顾 A-level 数学中常见的抽样方法，分析其优缺点，并说明各自的适用场景。
1. Census vs Sample | 普查与抽样
A census collects data from every member of the population. It provides complete information but is often time-consuming, expensive, and sometimes impossible when the population is infinite or destructive testing is required.
普查是对总体中每一个成员进行数据收集。它能提供完整信息，但往往耗时、成本高，而且在总体无限或需要破坏性测试时可能无法实施。
A sample, on the other hand, is a subset chosen to represent the population. Sampling reduces cost and time, but introduces sampling error. Understanding the trade-off between census and sampling is fundamental to statistical inference.
相比之下，样本是从总体中选出的具有代表性的子集。抽样可以降低成本和缩短时间，但会引入抽样误差。理解普查与抽样之间的取舍是统计推断的基础。
2. Simple Random Sampling | 简单随机抽样
Simple random sampling gives every member of the population an equal and independent chance of being selected. Methods include drawing names from a hat, using random number tables, or using a random number generator on a calculator or computer.
简单随机抽样使总体中每个成员被选中的概率相等且相互独立。实现方法包括抽签、使用随机数表，或利用计算器、电脑的随机数生成器。
This method is unbiased and easy to understand. However, it requires a complete list of the population (a sampling frame), and it may produce a sample that is not representative of specific subgroups, especially with small sample sizes.
该法无偏且易于理解，但需要完整的总体名单（即抽样框），并且在样本量较小时，可能无法充分代表特定的子群。
3. Systematic Sampling | 系统抽样
Systematic sampling involves selecting every kth member from an ordered list, where k is calculated as N / n (population size ÷ sample size). The starting point is chosen randomly from the first k members.
系统抽样是按顺序从名单中每隔 k 个选取一个成员，其中 k 由 N / n（总体大小 ÷ 样本大小）确定。起点从前 k 个成员中随机选择。
This method is simpler and faster than simple random sampling, especially with large populations. However, if the list has a periodic pattern (e.g., every 10th item is defective), systematic sampling can introduce bias.
系统抽样比简单随机抽样更简便快捷，尤其适用于大总体。但如果名单存在周期性规律（例如每隔 10 个产品就有缺陷），系统抽样可能引入偏差。
k = N ⁄ n
4. Stratified Sampling | 分层抽样
Stratified sampling divides the population into non-overlapping groups called strata (e.g., by age, gender, or income). A sample is then taken from each stratum in proportion to the stratum’s size in the population.
分层抽样将总体划分为互不重叠的层（例如按年龄、性别或收入）。然后按照各层在总体中所占比例，从每一层中抽取样本。
The number selected from each stratum is calculated as:
nₕ = (Nₕ ⁄ N) × n
where Nₕ is the population size of stratum h and n is the total sample size. This method ensures that every subgroup is represented, which reduces sampling error and improves precision. It is especially useful when the population has known heterogeneity.
其中 Nₕ 是第 h 层总体大小，n 是总样本量。此法能确保每个子群都有代表，从而减少抽样误差、提高精度。当总体具有已知异质性时尤其适用。
5. Quota Sampling | 配额抽样
Quota sampling is a non-random method where the population is divided into groups (quotas) based on characteristics such as gender or age. The interviewer selects individuals until each quota is filled.
配额抽样是一种非随机方法，它先将总体按性别、年龄等特征划分为若干配额，访谈者不断选取个体，直至每个配额填满。
It is cheap, quick, and does not require a sampling frame. However, the selection within quotas is not random, so the results may be biased and not suitable for rigorous statistical inference.
该方法成本低、速度快，且不需要抽样框。但配额内的选择并非随机，因此结果可能存在偏差，不适合进行严格的统计推断。
6. Cluster Sampling | 整群抽样
Cluster sampling divides the population into natural groups called clusters (e.g., schools, city blocks). A random sample of clusters is chosen, and then all individuals within the selected clusters are included in the sample.
整群抽样将总体划分为自然的群组（如学校、街区）。先随机选取部分群组，然后将所选群组中的所有个体都纳入样本。
This method reduces travel and administrative costs, especially for geographically dispersed populations. But clusters may be internally similar or different from each other, which can lead to larger sampling error than simple random sampling of the same size.
整群抽样能降低交通和管理成本，尤其适用于地域分布广泛的总体。但群组内部可能相似，群与群之间可能存在较大差异，因此比同样本量的简单随机抽样可能产生更大的抽样误差。
7. Convenience Sampling | 方便抽样
Convenience sampling selects individuals who are easiest to reach, such as asking friends, using volunteers, or surveying people in a shopping mall. It is commonly used in pilot studies and exploratory research.
方便抽样选取最容易接触到的个体，例如询问朋友、使用志愿者或在商场中调查路人。它常用于试点研究和探索性研究。
This method is extremely easy and inexpensive, but it is highly susceptible to bias because the sample may not represent the population. It should not be used when the goal is to make accurate generalizations.
该方法极其简便且成本低，但极易受偏差影响，因为样本可能无法代表总体。当目标是做出准确推断时，不应使用该方法。
8. Judgement (Purposive) Sampling | 判断（目的）抽样
Judgement sampling relies on the researcher’s experience and expertise to select individuals who are believed to be most representative or informative. The researcher consciously chooses the sample based on the study’s purpose.
判断抽样依靠研究者的经验和专业知识，选择被认为最具有代表性或信息量的个体。研究者根据研究目的有意选择样本。
When the population is very small or the research is qualitative, judgement sampling can provide valuable insights. However, the results are subjective and cannot be statistically generalised.
当总体很小或研究为定性研究时，判断抽样可提供有价值的见解。但结果具有主观性，无法进行统计推广。
9. How to Choose the Right Method | 如何选择合适的方法
Choosing a sampling method depends on several factors: the sampling frame, the nature of the population, the desired precision, available time and budget, and whether a random method is required for statistical inference.
选择抽样方法需考虑以下因素：是否有抽样框、总体性质、所需精度、时间和预算，以及对统计推断而言是否需要随机方法。
- If a full list is available and the population is homogeneous → use simple random sampling.
- If the population is large and ordered → use systematic sampling.
- If the population has known subgroups that should be represented → use stratified sampling.
- If cost and speed are more important than precision → use quota or cluster sampling.
- If only exploratory insights are needed → use convenience or judgement sampling.
- 如果有完整名单且总体同质 → 使用简单随机抽样。
- 如果总体庞大且有序 → 使用系统抽样。
- 如果总体包含需要代表的已知子群 → 使用分层抽样。
- 如果成本与速度比精度更重要 → 使用配额或整群抽样。
- 如果只需探索性见解 → 使用方便或判断抽样。
10. Common Exam Questions and Pitfalls | 常见考题与易错点
Examiners often ask you to identify the sampling method from a description, explain the advantages and disadvantages, or evaluate whether a given method is appropriate. Common pitfalls include confusing stratified sampling with quota sampling, forgetting that simple random sampling requires a sampling frame, and failing to mention sampling error or bias.
考试中常要求你根据描述识别抽样方法、解释优缺点，或评估给定方法是否合适。常见易错点包括：混淆分层抽样与配额抽样；忘记简单随机抽样需要抽样框；未提及抽样误差或偏差。
Another frequent mistake is describing systematic sampling without mentioning the random start. Always state that the first item must be chosen randomly from the first k items to keep the method unbiased.
另一个常见错误是描述系统抽样时未提及随机起点。务必说明：第一个样本必须从前 k 个中随机选择，以保证方法无偏。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply