📚 Sampling Methods & Experimental Design | 采样方法与实验设计
In A-Level Mathematics, statistics begins with two practical questions: how do we collect data, and how do we know the data are reliable? Sampling methods determine which members of a population we observe, while experimental design determines whether the conclusions we draw are valid. This article covers both topics in the detail required for AS and A-Level statistics, including the strengths and weaknesses of every method and the vocabulary examiners expect you to use.
在 A-Level 数学中，统计学始于两个实际问题：我们如何收集数据，以及如何知道数据是可靠的？抽样方法决定了我们观察总体中的哪些成员，而实验设计决定了我们得出的结论是否有效。本文将按照 AS 与 A-Level 统计部分的考试要求，详细讲解这两大主题，包括每种方法的优点与缺点，以及考官期待你使用的专业词汇。
1. What Is Sampling? | 什么是抽样？
The population is the entire set of people or items about which information is required. A census is a survey that collects data from every single member of the population. A sample is a subset of the population that is actually observed, and the process of selecting that subset is called sampling.
总体（population）是我们需要获取信息的全部人员或物品的集合。普查（census）是指对总体中的每一个成员都收集数据的调查。样本（sample）是总体中实际被观察的一部分，而选择这一部分的过程称为抽样（sampling）。
Sampling is used in nearly all real surveys because it is faster and cheaper than a census. A carefully chosen sample can provide reliable estimates if it is representative of the population. In contrast, a poorly chosen sample can produce results that are systematically wrong, no matter how large it is.
几乎所有真实调查都使用抽样，因为它比普查更快、更便宜。如果样本选取得当并且能够代表总体，它就能提供可靠的估计。相反，如果样本选择不当，即使样本很大，结果也可能系统性出错。
- Population: the whole set of items of interest. | 总体：所关心的全部对象的集合。
- Census: data collected from every member of the population. | 普查：从总体中每一个成员收集数据。
- Sample: a subset of the population chosen for observation. | 样本：从总体中选出来进行观察的一部分。
- Sampling frame: a list of all members of the population from which the sample is drawn. | 抽样框：从中抽取样本的总体全体成员的名单。
2. Simple Random Sampling | 简单随机抽样
In simple random sampling (SRS), every member of the population has an equal chance of being selected, and each choice is independent of the others. When a complete sampling frame is available, SRS is the gold standard because it removes selection bias entirely.
在简单随机抽样（SRS）中，总体中的每个成员被选中的机会相等，而且每次选择彼此独立。当存在完整的抽样框时，简单随机抽样是黄金标准，因为它完全消除了选择偏差。
There are three standard ways to carry out simple random sampling: use a random number generator on a calculator or computer, use a table of random digits, or use a lottery method such as drawing numbered slips from a hat. In each case every member must be given a unique number before selection.
实施简单随机抽样有三种标准方式：使用计算器或计算机上的随机数生成器、使用随机数表，或使用抽签法（例如从帽子中抽取编号纸条）。在每种方式中，抽取前都必须为每个成员分配唯一的编号。
A key distinction is between sampling with replacement and sampling without replacement. With replacement, the same member may appear in the sample more than once; without replacement, each member can be chosen at most once. In most surveys, sampling is carried out without replacement.
一个关键区别是放回抽样与不放回抽样。放回抽样中，同一个成员可能多次出现在样本里；不放回抽样中，每个成员最多只能被选中一次。在大多数调查中，抽样都是不放回的。
- Advantages: unbiased, simple to describe, and every member has an equal chance of selection. | 优点：无偏、易于描述，且每个成员被选中的机会均等。
- Disadvantages: requires a complete sampling frame; may miss small subgroups; can be expensive for large populations. | 缺点：需要完整的抽样框；可能遗漏较小的子群；对大规模总体而言成本较高。
3. Systematic Sampling | 系统抽样
In systematic sampling, members of the population are selected at regular intervals from a list. To find the interval k, divide the population size N by the desired sample size n.
在系统抽样中，我们从名单上按固定间隔选取总体的成员。计算间隔 k 的方法是用总体规模 N 除以所需样本量 n。
k = N ÷ n
For example, if a school has 1000 students and we need a sample of 100, then k = 1000 ÷ 100 = 10. We choose a random starting point between 1 and 10, select that student, and then select every tenth student after that until the sample is complete.
例如，如果一所学校有1000名学生，我们需要抽取100人的样本，那么 k = 1000 ÷ 100 = 10。我们在1到10之间随机选择一个起点，选取该学生，然后每隔10人选取一名学生，直到样本完成。
A major risk with systematic sampling is periodicity. If the sampling list has a repeating pattern, such as every tenth item sharing a particular property, the sample will be biased even though the procedure is partly random.
系统抽样的一大风险是周期性（periodicity）。如果抽样名单存在重复模式，例如每隔十个项目就具有某种共同特征，那么即使抽样过程部分随机，样本也会出现偏差。
- Advantages: quick to carry out, no random number generator needed after the starting point, and the sample is spread evenly across the list. | 优点：操作迅速，确定起点后无需随机数生成器，且样本在名单上分布均匀。
- Disadvantages: biased if the list has periodicity; the sample may still not reflect the structure of the population. | 缺点：如果名单存在周期性则会产生偏差；样本仍可能无法反映总体结构。
4. Stratified Sampling | 分层抽样
Stratified sampling divides the population into distinct groups called strata, based on a characteristic such as age, gender, or income level. A random sample is then taken from each stratum in proportion to the size of that stratum in the population.
分层抽样根据年龄、性别或收入水平等特征，将总体划分为若干不同的层（strata）。随后按照每一层在总体中所占的比例，从各层中分别进行随机抽样。
The sample size from each stratum is calculated using the formula below.
每层应抽取的样本量按以下公式计算。
nᵢ = (Nᵢ / N) × n
where nᵢ is the sample size from stratum i, Nᵢ is the population size of stratum i, N is the total population size, and n is the total sample size.
其中 nᵢ 为第 i 层的样本量，Nᵢ 为第 i 层的总体规模，N 为总体总规模，n 为总样本量。
For example, a college has 600 boys and 400 girls. A stratified sample of 50 students by gender gives boys = (600 ÷ 1000) × 50 = 30 and girls = (400 ÷ 1000) × 50 = 20. Each gender is therefore represented in exactly the same proportion as in the whole college.
例如，某学院有600名男生和400名女生。按性别抽取50人的分层样本：男生 = (600 ÷ 1000) × 50 = 30 人，女生 = (400 ÷ 1000) × 50 = 20 人。因此每个性别在样本中的比例与全校中的比例完全相同。
- Advantages: guarantees representation of every stratum, so results are more representative than SRS for a mixed population. | 优点：保证每一层都被代表，因此在混合总体中比简单随机抽样更有代表性。
- Disadvantages: requires detailed information about the population structure and a complete sampling frame for each stratum. | 缺点：需要关于总体结构的详细信息，以及每一层的完整抽样框。
5. Quota Sampling | 配额抽样
Quota sampling is a non-random method in which interviewers are told to collect a fixed number, or quota, of responses from each category of the population. The categories are usually based on characteristics such as age, gender, or region, and the quotas are set in proportion to the population.
配额抽样是一种非随机方法：访问员被告知需要从总体的每个类别中收集固定数量（即配额）的答复。类别通常基于年龄、性别或地区等特征，而配额则按总体比例设定。
The crucial difference from stratified sampling is that, within each quota, the interviewer chooses who to interview. There is no random selection. This makes quota sampling much cheaper and faster, and it does not require a sampling frame at all.
配额抽样与分层抽样的关键区别在于：在每个配额内部，由访问员自行决定访问谁，不存在随机选择。这使得配额抽样更便宜、更快捷，并且完全不需要抽样框。
- Advantages: fast, cheap, no sampling frame needed, and ensures all major groups are represented. | 优点：快速、便宜、不需要抽样框，并且确保所有主要群体都有代表。
- Disadvantages: subject to interviewer bias; not random, so results cannot be relied on for formal probability statements. | 缺点：容易受访问员偏差影响；不是随机方法，因此不能依据其结果进行正式的概率推断。
6. Opportunity Sampling | 机会抽样
Opportunity sampling, also called convenience sampling, uses whichever members of the population are available and willing at the time of the survey. A common example is stopping people in a shopping centre or asking classmates to complete a questionnaire.
机会抽样（又称便利抽样）使用调查当下恰好可获得的总体成员。常见的例子是在购物中心拦下路人，或请同学填写问卷。
This method is extremely easy to use, but the sample is almost always unrepresentative. Only the people present in one place at one time can be included, so generalising the results to the whole population is not statistically justified.
这种方法使用极其简便，但样本几乎总是缺乏代表性。因为在同一时间同一地点出现的仅是部分人，所以将结果推广到整个总体在统计上是不成立的。
- Advantages: very quick, inexpensive, and requires no sampling frame. | 优点：非常快速、成本低，且不需要抽样框。
- Disadvantages: highly biased, likely to miss large parts of the population, and cannot be used for reliable inference. | 缺点：偏差很大，可能遗漏总体中的大部分成员，且不能
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