📚 Populations and Samples | 总体与样本
In A-Level Statistics, understanding the difference between a population and a sample is the foundation of data collection. This topic covers census and sample surveys, sampling frames, sampling methods, bias, and the notation used to distinguish population parameters from sample statistics.
在A-Level统计中，理解总体与样本的区别是数据收集的基础。本主题涵盖普查与抽样调查、抽样框、各种抽样方法、偏倚以及区分总体参数与样本统计量的符号。
1. Key Definitions: Population, Sample, Census | 核心定义：总体、样本、普查
A population is the whole set of items that are of interest in a statistical investigation. A sample is a subset of the population that is selected for study. A census collects data from every member of the population.
总体是统计调查中感兴趣的全部项目集合。样本是从总体中选出来用于研究的子集。普查则从总体的每一个成员收集数据。
- Population: the entire group being studied. | 总体：被研究的整个群体。
- Sample: a selection of members from the population. | 样本：从总体中选出的一部分成员。
- Census: a survey of every unit in the population. | 普查：对总体中每个单位的调查。
A sample is usually used when a census is too expensive, too slow, or impossible because testing destroys the items.
当普查太昂贵、太慢或因为测试会破坏物品而无法进行时，通常使用样本。
2. Sampling Units and Sampling Frame | 抽样单位与抽样框
A sampling unit is each individual member of the population. A sampling frame is a list of all sampling units from which the sample is selected. For example, if a school wishes to sample Year 12 students, the sampling unit is one student, and the register of all Year 12 students is the sampling frame.
抽样单位是总体中的每一个个体成员。抽样框是列出所有抽样单位的名单，样本从这个名单中选取。例如，如果一所学校想对12年级学生进行抽样，抽样单位是一个学生，而所有12年级学生的名册就是抽样框。
If the sampling frame is incomplete or out of date, the sample may be biased because not every member has a known chance of selection.
如果抽样框不完整或过时，样本可能产生偏倚，因为不是每个成员都有已知的被选机会。
Common sampling frames include electoral rolls, school registers, postcode lists and telephone directories.
常见的抽样框包括选民名册、学校注册表、邮编列表和电话簿。
3. Census vs Sample: Advantages and Disadvantages | 普查与抽样：优缺点
A census provides complete information and avoids sampling error, but it is often too expensive, time-consuming or impractical. A sample is faster and cheaper, but may not fully represent the population.
普查能提供完整的信息并避免抽样误差，但通常太昂贵、耗时或不切实际。抽样更快更便宜，但可能不能完全代表总体。
- Census advantages: completely accurate; no sampling error. | 普查优点：完全准确；无抽样误差。
- Census disadvantages: expensive, time-consuming, can destroy items, large volume of data. | 普查缺点：昂贵、耗时、可能损坏物品、数据量大。
- Sample advantages: quicker, cheaper, less data to process. | 抽样优点：更快、更便宜、处理的数据更少。
- Sample disadvantages: may not be representative; sampling error and possible bias. | 抽样缺点：可能不具代表性；存在抽样误差和可能的偏倚。
In most real-life investigations, sampling is preferred because a census is rarely feasible for large populations.
在大多数实际调查中，抽样更受青睐，因为对大规模总体进行普查很少是可行的。
4. Simple Random Sampling | 简单随机抽样
In a simple random sample, every member of the population has an equal chance of being selected. This can be done using random number generators or drawing names from a hat. It requires a full sampling frame.
在简单随机抽样中，总体的每个成员都有相等的机会被选中。这可以通过随机数生成器或从帽子中抽签来实现。它需要一个完整的抽样框。
Advantages: free from selection bias; easy to understand. Disadvantages: need a sampling frame; may be time-consuming if population is large.
优点：不受选择偏倚影响；易于理解。缺点：需要抽样框；如果总体很大，可能耗时。
A true simple random sample gives every possible sample of the same size an equal chance of being selected, not just every member.
真正的简单随机抽样使每个相同容量的可能样本都有相等的机会被选中，而不仅仅是每个成员有相等机会。
5. Systematic Sampling | 系统抽样
Systematic sampling selects members at regular intervals from an ordered list. Choose a random starting point, then take every kth member, where k = population size ÷ sample size.
系统抽样从有序名单中按固定间隔选取成员。先随机选择一个起点，然后每隔k个成员取一个，其中k = 总体大小 ÷ 样本大小。
k = N / n
It is quick and can be used when a sampling frame is available. However, if the list has a hidden periodic pattern, the sample can become biased.
这种方法很快，当有抽样框时可以使用。但如果名单存在隐藏的周期性规律，样本可能会产生偏倚。
For example, selecting every 10th house on a street where every 10th house is a corner plot would produce a biased sample.
例如，在一条街上每隔10栋选一栋，而每隔10栋恰好是转角房，就会产生有偏倚的样本。
6. Stratified Sampling | 分层抽样
Stratified sampling divides the population into mutually exclusive strata, such as year groups or genders. A simple random sample is taken from each stratum in proportion to the size of the stratum.
分层抽样将总体分为互不重叠的层，例如年级或性别。然后按各层在总体中的比例从每一层进行简单随机抽样。
Number from a stratum = (stratum size / population size) × sample size
This method gives a more representative sample and reduces sampling error, but requires knowledge of population structure and a sampling frame for each stratum.
这种方法能得到更具代表性的样本并减少抽样误差，但需要了解总体结构并且每层都有抽样框。
Stratified sampling is especially useful when some groups are small and might be missed by a simple random sample.
当某些群体较小而可能被简单随机抽样遗漏时，分层抽样尤其有用。
7. Quota Sampling | 配额抽样
Quota sampling involves an interviewer selecting a fixed number of people from different categories, such as age or gender, without using a random selection method. It does not require a sampling frame.
配额抽样是由调查员从不同类别（如年龄或性别）中选取固定数量的人，而不使用随机选择方法。它不需要抽样框。
It is quick and cheap, often used in market research. However, because the interviewer can choose who to interview within each quota, it can introduce interviewer bias.
这种方法快速且成本低，常用于市场调查。然而，由于调查员可以在每个配额内选择访问对象，可能会引入调查员偏倚。
Quota sampling may appear similar to stratified sampling, but the key difference is that quota sampling does not use random selection within each group.
配额抽样可能看起来与分层抽样相似，但关键区别在于配额抽样在每个组内不使用随机选择。
8. Opportunity Sampling | 机会抽样
Opportunity sampling, also called convenience sampling, selects individuals who are readily available at the time of the study, such as the first 30 people who walk past a shop.
机会抽样，又称便利抽样，选择在研究时最容易接触到的人，例如走过商店的前30个人。
It is the easiest and cheapest method, but it is highly likely to be unrepresentative and biased because the sample is not selected randomly.
这是最简单、最便宜的方法，但极可能不具代表性且存在偏倚，因为样本不是随机选取的。
In exam answers, you should recognise that opportunity sampling is a non-random method and therefore cannot be used to make reliable inferences about the population.
在考试答案中，你应该认识到机会抽样是一种非随机方法，因此不能用来对总体做出可靠推断。
9. Bias in Sampling | 抽样中的偏倚
Bias is a systematic error that makes a sample not representative of the population. Sources of bias include a flawed sampling frame, non-random selection, non-response, and leading questions in questionnaires.
偏倚是一种系统误差，导致样本不能代表总体。偏倚的来源包括有缺陷的抽样框、非随机选择、无回应以及问卷中的诱导性问题。
If a sample is biased, conclusions drawn from it are unreliable, even if the sample size is large. Increasing sample size reduces random sampling error, but does not remove bias.
如果样本有偏倚，从中得出的结论不可靠，即使样本量很大。增大样本量可以减少随机抽样误差，但不能消除偏倚。
Non-response bias occurs when a large proportion of selected people do not respond, and their views may differ from those who do respond.
无回应偏倚发生在被选中的大部分人没有回应时，而他们的观点可能与回应者不同。
10. Parameters and Statistics: Notation | 参数与统计量：符号
A parameter is
Published by TutorHao | A-Level 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