📚 Sampling Methods in A-Level Statistics | A-Level 统计学中的抽样方法
Sampling is a core topic in the Edexcel A-Level Mathematics statistics strand, and it also appears in AQA, OCR and Cambridge specifications under the heading of data collection. It underpins how data are gathered, how conclusions are drawn, and how bias can enter a study before a single calculation is done. This article explains the main sampling methods, their advantages and disadvantages, the vocabulary expected in exam answers, and the reasoning that turns a description into full marks.
抽样是 Edexcel A-Level 数学统计部分的核心主题，在 AQA、OCR 与剑桥考试局的考纲中也以"数据收集"的标题出现。它是数据收集、结论推断以及偏差如何进入研究的基础——在尚未进行任何计算之前，偏差就已经可能产生。本文解释主要的抽样方法、它们的优缺点、考试答案中需要使用的语言，以及把一段描述变成满分答案所需要的推理逻辑。
1. Population, Sample and Census | 总体、样本与普查
In a statistical investigation, the population is the entire collection of individuals, items or measurements that we wish to study. A sample is a subset of the population selected for the actual data collection. A census gathers information from every member of the population without exception. For example, if the aim is to estimate the mean height of 17-year-old students in the UK, the population is all such students; measuring only 500 of them produces a sample.
在统计调查中，总体是我们希望研究的全部个体、项目或测量值的集合。样本是从总体中抽取出来用于实际数据收集的一部分。普查则是在没有遗漏的情况下从总体中的每一个成员收集信息。例如，如果目标是估计英国 17 岁学生的平均身高，那么所有这样的学生构成总体；只测量其中 500 人就得到一个样本。
The population must be defined precisely by the research question, including the geographical area and the time period. A well-defined population might be "all students currently enrolled in Year 12 at a particular sixth-form college", not simply "students". Vague populations make the sample meaningless and are penalised in exam mark schemes.
总体必须由研究问题精确定义，包括地理范围和时间范围。一个定义明确的总体可能是"某所预科学院当前所有在读 12 年级学生"，而不是笼统的"学生"。模糊的总体会使样本失去意义，在考试评分标准中会被扣分。
A census can give complete accuracy for the population at the time of collection, but it is often very expensive, time-consuming, and impractical when the population is large or when the testing destroys the items being tested (for example, testing the lifetime of a light bulb).
普查可以对收集数据时的总体给出完整准确的结果，但它通常非常昂贵、耗时，并且当总体很大、或测试会破坏被测物品时（例如测试灯泡的寿命）不切实际。
- Population: every member of the group being studied.
总体：被研究群体的每一位成员。 - Sample: a subset of the population, used to make inferences.
样本：总体的一部分，用于进行推断。 - Census: data collected from every member of the population.
普查：从总体每一位成员处收集数据。
2. Why Sampling? Advantages over a Census | 为什么要抽样：相对普查的优势
Sampling is usually preferred because it is cheaper, faster and easier to manage than a census. Collecting data from a sample produces less data to process, so analysis can be completed more quickly and the results can be published sooner. When testing is destructive, sampling is the only option: if every item were tested to destruction, nothing would remain to sell or use.
抽样通常更受青睐，因为它比普查更便宜、更快捷、也更易于管理。从样本收集数据会产生更少需要处理的数据，因此分析可以更快完成，结果也能更早公布。当测试具有破坏性时，抽样是唯一的选择：如果每个物品都被测试至损坏，就没有东西可以出售或使用了。
A sample also allows more detailed information to be collected on each individual, since the budget can be concentrated on fewer cases. A census may suffer from non-response and poor data quality because of its sheer scale, whereas a well-run sample can be followed up and checked.
样本还允许对每个个体收集更详细的信息，因为预算可以集中在更少的案例上。普查可能因为规模庞大而遭遇无应答和数据质量差的问题，而运行良好的抽样则可以被追踪和核查。
The main disadvantage of a sample is that it gives only an estimate, so it is subject to sampling error. There is always a chance that the sample is not representative of the population, and the size of the sample determines how much random variation we expect.
样本的主要缺点是它只给出估计值，因此会受到抽样误差的影响。样本总有可能不能代表总体，而样本量决定了我们可以预期多大的随机波动。
| Census 普查 | Sample 样本 |
| Complete, no sampling error. 完整，无抽样误差。 | Gives an estimate with sampling error. 给出带抽样误差的估计。 |
| Expensive, slow, needs huge organisation. 昂贵、缓慢、组织难度大。 | Cheap, quick, easier to manage. 便宜、快速、易于管理。 |
| Impossible if testing is destructive. 若测试具破坏性则不可行。 | Essential if testing is destructive. 若测试具破坏性则必不可少。 |
3. Sampling Frame, Sampling Units and Sampling Error | 抽样框、抽样单元与抽样误差
A sampling frame is a list of all the members of the population, or something that identifies every member, such as a register, a database or a map. The individuals on that list are called sampling units. If the sampling frame is incomplete, some members of the population have no chance of being selected, and the sample is said to suffer from undercoverage.
抽样框是总体所有成员的一份名单，或任何能识别每一位成员的东西，例如名册、数据库或地图。名单上的个体被称为抽样单元。如果抽样框不完整，部分总体成员就没有被抽中的机会，这样的样本就被认为存在"覆盖不足"。
A random sample is one in which every member of the population has an equal chance of being selected, and the selection is made without conscious bias. Importantly, randomness must come from a genuine random mechanism such as a random number generator, a random number table or a lottery; choosing people haphazardly by hand is not random.
随机样本是指总体中每一位成员被抽中的机会均等、且抽取过程没有主观偏见的样本。重要的是，随机性必须来自真正的随机机制，例如随机数生成器、随机数表或抽签；用手随意挑选并不是随机。
Sampling error is the natural variation that arises because we observe only part of the population. It can be reduced by increasing the sample size. Bias, in contrast, is a systematic error that pushes results consistently in one direction; increasing the sample size does not remove bias, it merely repeats the same mistake on a larger scale.
抽样误差是由于我们只观察到总体的一部分而产生的自然波动，可以通过增大样本量来减小。相比之下，偏差是一种系统误差，会把结果持续推向同一方向；增大样本量并不能消除偏差，只会把同样的错误以更大的规模重复一遍。
4. Simple Random Sampling | 简单随机抽样
ry individual also has an equal chance of being chosen. The number of possible samples is N! ÷ (n! × (N − n)!, which shows why the method is defined in terms of samples rather than individuals alone.
在简单随机抽样（SRS）中，从规模为 N 的总体中抽取规模为 n 的每一个可能样本被选中的机会都相同，因此每一个体被选中的机会也相同。可能的样本数量为 N! ÷ (n! × (N − n)!)，这说明为什么该方法的定义要基于"样本"，而不只是"个体"。
The practical procedure is straightforward: number every member of the sampling frame from 1 to N, then use a random number generator, a random number table or a calculator such as the TI-Nspire to select n distinct numbers. Ignore repeats and numbers outside the range.
实际操作很直接：把抽样框中每一位成员从 1 到 N 编号，然后用随机数生成器、随机数表或 TI-Nspire 之类的计算器选出 n 个互不相同的号码，忽略重复的号码和超出范围的号码。
- Advantages: unbiased; simple to describe and justify; every member has a known, equal chance; suitable for any population once a frame exists.
优点：无偏；描述与论证简单；每位成员都有已知且相等的机会；只要存在抽样框，适用于任何总体。 - Disadvantages: needs a complete sampling frame; can be slow and costly for large populations; by chance it may under-represent small subgroups; selected individuals may be widely scattered.
缺点：需要完整的抽样框；对于大总体可能缓慢且昂贵；可能因偶然性而低估小型子群体；被选中的个体可能分布非常分散。
A common exam trap is to claim that SRS always produces a representative sample. It does not: it produces an unbiased sample, but any single sample may still be unusual. It is the procedure, not the outcom
e, that is unbiased.
5. Systematic Sampling | 系统抽样
Systematic sampling takes every kth item from an ordered list. First calculate the sampling interval k by dividing the population size by the required sample size; then choose a random starting point between 1 and k, and select that item plus every kth item after it. This random start is what makes the method a form of random sampling.
系统抽样从有序名单中每隔 k 个抽取一个。首先用总体规模除以所需样本量算出抽样间隔 k；然后在 1 到 k 之间随机选一个起点，抽取该项以及之后每隔 k 个的项目。正是这个随机起点使该方法成为一种随机抽样。
k = N ÷ n
23, 39, 55 and so on. If k is not a whole number, the sampling interval must be adjusted carefully, or the method replaced by simple random sampling.
例如，要从 800 名学生的名单中抽取 50 人，k = 800 ÷ 50 = 16。在 1 到 16 之间随机选一个起点 r，比如 r = 7，则抽取编号为 7、23、39、55 等的学生。如果 k 不是整数，必须谨慎调整抽样间隔，或者改用简单随机抽样。
- Advantages: quick and easy to carry out; cheap; spreads the sample evenly through the whole list; does not require a random number generator for every item.
优点：执行快捷简便；成本低；样本在整个名单上分布均匀；不需要为每个项目都使用随机数生成器。 - Disadvantages: requires an ordered sampling frame; if the list has a periodic pattern that matches k, the sample can be seriously biased; not every possible sample has an equal chance.
缺点：需要有序列的抽样框；如果名单存在与 k 相匹配的周期性模式，样本可能严重偏差；并非每一个可能样本都有相等机会。
The classic periodicity example is a street in which houses alternate between flats and detached houses. Taking every second house could produce a sample consisting only of one type, giving a badly distorted estimate.
典型的周期性例子是一条街上公寓与独栋住宅交替排列。若每隔两栋抽一栋，就可能抽到全部属于同一类型的样本，从而得到严重失真的估计。
6. Stratified Sampling | 分层抽样
Stratified sampling divides the population into non-overlapping groups called strata, based on a characteristic that is likely to affect the variable being measured, such as year group, gender, region or income band. A random sample is then taken from within each stratum, and the results are combined to estimate population quantities.
分层抽样把总体按某个可能影响所测变量的特征（如年级、性别、地区或收入档）划分为互不重叠的组，称为"层"。然后在每一层内部进行随机抽样，最后把结果合并以估计总体特征。
If the strata sizes are N₁, N₂, ..., and the total population is N, with a total sample size n, then proportional allocation gives the number sampled from stratum i as:
如果各层规模为 N₁、N₂、…，总体总数为 N，总样本量为 n，则按比例分配时，第 i 层应抽取的数量为：
nᵢ = (Nᵢ ÷ N) × n
n n₁ = (600 ÷ 1000) × 100 = 60 from Year 12 and n₂ = (400 ÷ 1000) × 100 = 40 from Year 13.
假设一所学院有 600 名 12 年级学生和 400 名 13 年级学生，需要抽取 100 人的样本。则 n₁ = (600 ÷ 1000) × 100 = 60，即从 12 年级抽取 60 人；n₂ = (400 ÷ 1000) × 100 = 40，即从 13 年级抽取 40 人。
- Advantages: guarantees representation of every stratum; usually more precise than simple random sampling because variation between strata is removed; allows comparisons between strata.
优点：保证每一层都有代表；由于消除了层间变异，通常比简单随机抽样更精确；允许在各层之间进行比较。 - Disadvantages: requires a sampling frame with stratum information; more complicated and time-consuming to organise; requires knowledge of the size of each stratum.
缺点：需要带有分层信息的抽样框；组织更复杂、更耗时；需要知道每一层的大小。
Note carefully that stratified sampling is a random method, but it is not a simple random sample: some possible samples of size n cannot occur, because each stratum must be sampled.
务必注意：分层抽样是随机方法，但它不是简单随机样本：某些规模为 n 的样本根本无法出现，因为每一层都必须被抽到。
7. Quota Sampling | 配额抽样
Quota sampling is a non-random method in which the researcher decides, in advance, how many people from each category (for example, 40 women and 40 men, or 30 people in each of four age bands) must be interviewed. The interviewer then chooses respondents who fit those quotas until each quota is filled.
配额抽样是一种非随机方法。研究者事先决定每个类别需要采访多少人（例如 40 名女性和 40 名男性，或四个年龄段各 30 人）。访谈员随后自行选择符合这些配额的对象，直到每个配额填满。
Because the final choice of who is interviewed is left to the interviewer, quota sampling is quick and does not require a sampling frame, but the selection is subjective and vulnerable to interviewer bias. Interviewers naturally approach people who look approachable, which can systematically exclude certain groups.
由于最终采访谁由访谈员决定，配额抽样速度快，且不需要抽样框，但选择过程是主观的，容易受到访谈员偏见的影响。访谈员天然会接近那些看起来容易接近的人，这可能系统性地排除某些群体。
- Advantages: cheap and fast; no sampling frame needed; ensures that chosen categories appear in the required numbers.
优点：成本低、速度快；不需要抽样框；确保选定的类别以要求的数量出现。 - Disadvantages: non-random, so results may be biased; interviewer choice introduces subjectivity; no valid way to calculate sampling error or confidence intervals; cannot be used for reliable statistical inference.
缺点：非随机，结果可能有偏；访谈员的选择带来主观性；无法有效计算抽样误差或置信区间；不能用于可靠的统计推断。
Examiners often ask you to compare quota sampling with stratified sampling. The key difference is that in stratified sampling the people within each stratum are chosen at random, whereas in quota sampling they are chosen by the interviewer.
考官常要求你把配额抽样与分层抽样作比较。关键区别在于：分层抽样在每一层内部是随机抽取的，而配额抽样是由访谈员自行挑选的。
8. Opportunity (Convenience) Sampling | 便利抽样（偶遇抽样）
Opportunity sampling, also called convenience sampling, selects whichever members of the population are easiest to reach at the time. Interviewing the first 20 people who walk into the school canteen, or surveying friends on social media, are typical examples.
便利抽样（也称偶遇抽样）选择在当时最容易接触到的总体成员。采访最先走进学校食堂的 20 个人，或在社交媒体上调查朋友，都是典型的例子。
- Advantages: very quick, cheap and simple; useful for pilot studies and for gathering rough preliminary impressions.
优点：非常快速、便宜、简单；适用于预调查和获取粗略的初步印象。 - Disadvantages: highly unrepresentative; strong selection bias because the sample depends on who happens to be present; results cannot be generalised to the population.
缺点：代表性很差；由于样本取决于碰巧在场的人，存在强烈的选择偏差；结果无法推广到总体。
A related non-random method is self-selecting sampling, where individuals volunteer to take part, for example in an online poll, a phone-in vote or a website questionnaire. People who volunteer usually feel strongly about the issue, so the sample is biased towards extreme opinions.
与之相关的非随机方法是自选抽样，即个人自愿参与，例如在线投票、电话投票或网站问卷。自愿参与的人通常对该议题感受强烈，因此样本会偏向极端意见。
The common feature of opportunity and self-selecting sampling is that the probability of selection is unknown, so no measure of sampling error can be attached to the results. This is a sentence worth memorising for explanation questions.
便利抽样与自选抽样的共同特征是：被选中的概率是未知的，因此无法为结果给出任何抽样误差的度量。这句话值得在解释题中背下来。
9. Cluster Sampling and Related Designs | 整群抽样与相关设计
Cluster sampling divides the population into groups called clusters, usually based on geography or existing institutions such as schools, hospitals or streets. A random sample of clusters is selected, and then all, or a random subsample, of the members within the chosen clusters are surveyed. If a random subsample is taken within clusters, the design is called multistage sampling.
整群抽样把总体分成若干"群"，通常依据地理区域或既有机构（如学校、医院、街道）划分。先随机抽取若干群，然后调查被选中群内全部成员或其中一个随机子样本。如果在群内再抽取随机子样本，这种设计称为多阶段抽样。
Cluster sampling is very useful when a complete list of individuals is unavailable but a list of groups exists, and when travelling between distant individuals would be impractical. For instance, a national survey might randomly select 30 schools and then test every pupil in Year 11 at those schools.
当没有个体的完整名单但存在群体名单时，以及在不同个体之间长途奔波不现实时，整群抽样非常有用。例如，一项全国性调查可能随机抽取 30 所学校，然后测试这些学校所有 11 年级的学生。
- Advantages: cheap and convenient; no complete list of individuals is needed; useful for geographically spread populations.
优点：便宜、方便；不需要个体的完整名单；适用于地理上分散的总体。 - Disadvantages: less accurate than simple random sampling, because individuals within a cluster tend to be similar to each other; the effective sample size is smaller than the number of people surveyed; requires more clusters to achieve the same precision.
缺点：精确度低于简单随机抽样，因为同一群内的个体往往彼此相似；有效样本量小于受访人数；要达到同等精度需要抽取更多的群。
Another non-random method is judgement sampling, where an expert deliberately selects units believed to be typical. It is quick and can be informed, but it is subjective and its reliability depends entirely on the expert.
另一种非随机方法是判断抽样，由专家刻意挑选被认为具有代表性的单位。它速度快、可能有专业依据，但主观性强，其可靠性完全取决于该专家。
10. Bias, Reliability and the Effect of Sample Size | 偏差、可靠性与样本量的影响
Bias is a systematic distortion that consistently pushes an estimate away from the true value. Common sources include an incomplete sampling frame (undercoverage), non-response bias, interviewer bias, leading or badly worded questions, and self-selection. Bias is a design fault and cannot be cured by collecting more data.
偏差是一种系统性的扭曲，会把估计值持续推离真值。常见来源包括抽样框不完整（覆盖不足）、无应答偏差、访谈员偏见、诱导性或措辞不当的问题，以及自选偏差。偏差是设计上的缺陷，无法通过收集更多数据来治愈。
Non-response bias is worth extra attention. If a postal questionnaire about exercise is returned mainly by people who exercise regularly, the estimate of average exercise time will be too high, no matter how large the original mailing was.
无应答偏差值得特别关注。如果一份关于运动的邮寄问卷主要由经常运动的人寄回，那么无论最初寄出多少份，平均运动时间的估计值都会偏高。
Increasing the sample size reduces sampling error and makes the estimate more reliable, but it does not remove bias. The standard error of the sample mean illustrates the point:
增大样本量可以减小抽样误差，使估计更可靠，但并不会消除偏差。样本均值的标准误可以说明这一点：
SE = σ ÷ √n
Because of the square root, quadrupling the sample size only halves the standard error. This is why sample size decisions involve a trade-off between precision and cost, and why exam answers should mention both.
由于有平方根，样本量增加到四倍，标准误只会减半。这就是为什么样本量的决定需要在精度与成本之间做权衡，也是为什么考试答案应同时提到这两方面。
11. Choosing the Right Method and Exam Technique | 选择合适的方法与答题技巧
Exam questions typically provide a scenario and ask you to name a suitable sampling method, justify it, and give one advantage and one disadvantage. The method must match the practical constraints described: what lists are available, how the population is structured, and how much time and money are available.
考试题通常给出一个情境，要求你写出合适的抽样方法、说明理由，并给出一条优点和一条缺点。所选方法必须与情境中描述的实际限制相匹配：有哪些名单可用、总体如何构成、时间和经费有多少。
- A complete numbered list is available: simple random sampling.
有完整的编号名单：简单随机抽样。 - The list is long and already ordered, and speed matters: systematic sampling.
名单很长且已有排序，且要求速度：系统抽样。 - The population falls naturally into groups
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