Sampling Methods in IB Mathematics | IB数学：抽样方法

📚 Sampling Methods in IB Mathematics | IB数学：抽样方法

In statistics, we often need to draw conclusions about a large group of people or objects, called the population. Testing an entire population is usually too costly, time-consuming, or even impossible, so we select a smaller representative subset called a sample. Sampling allows us to estimate population parameters accurately while saving resources.

在统计学中，我们经常需要对一个很大的群体或对象作出推断，这个群体称为总体。检验整个总体通常成本过高、耗时过长甚至无法实现，因此我们会选取一个较小的、具有代表性的子集作为样本。抽样使我们能够在节省资源的同时较准确地估计总体参数。

1. Why Sampling? | 为什么要抽样？

The main purpose of sampling is to make statistical inference without examining every member of the population. For example, a manufacturer who wants to test the quality of light bulbs cannot check every bulb because the test destroys the product. A sample of bulbs can provide enough evidence to estimate the proportion of defective bulbs in the whole production.

抽样的主要目的是无需检查总体中的每一个成员就能进行统计推断。例如，制造商要检测灯泡质量时，不可能逐一检查所有灯泡，因为测试会破坏产品。抽取少量灯泡作为样本，就足以估计整个生产批次中次品所占的比例。

Population | 总体： the complete set of individuals or objects of interest in a study. | 研究中关注的全部个体或对象的集合。
Sample | 样本： a subset of the population that is actually observed or measured. | 总体中实际被观察或测量的一个子集。
Sampling frame | 抽样框： a list or source from which the sample is drawn. | 用于抽取样本的名单或来源。
Parameter | 参数： a numerical characteristic of the population, such as the population mean μ. | 总体的数值特征，例如总体均值 μ。
Statistic | 统计量： a numerical characteristic of the sample, such as the sample mean x̄. | 样本的数值特征，例如样本均值 x̄。

2. Simple Random Sampling | 简单随机抽样

In simple random sampling, every member of the population has an equal chance of being selected. This is a probability sampling method and is considered the most unbiased if the sampling frame is complete and accurate. Common methods include drawing names from a hat, using random number tables, or using a random number generator.

在简单随机抽样中，总体中的每个成员被选中的概率相同。这是一种概率抽样方法，在抽样框完整且准确的情况下被认为偏差最小。常用方式包括抽签、使用随机数表或随机数生成器。

Advantages | 优点：

Free from selection bias if properly conducted. | 如果实施得当，可以避免选择偏差。
Easy to understand and mathematically sound. | 易于理解，且具有严格的数学基础。

Disadvantages | 缺点：

Requires a complete and up-to-date sampling frame. | 需要完整且最新的抽样框。
May be impractical for very large populations. | 对于非常大的总体可能不切实际。
May accidentally under-represent small subgroups. | 可能偶然地使较小的子群体代表性不足。

3. Systematic Sampling | 系统抽样

Systematic sampling involves selecting every k-th member from an ordered list after a random starting point. If the population size is N and the desired sample size is n, then the sampling interval is given by k = N ÷ n. The first member is chosen at random between 1 and k, and then every k-th member is included.

系统抽样是在有序名单中随机选择一个起点，然后每隔 k 个成员抽取一个。若总体大小为 N，所需样本量为 n，则抽样间隔为 k = N ÷ n。首先在 1 到 k 之间随机选择第一个成员，此后每隔 k 个选取一个。

k = N ÷ n

This method is often faster and easier than simple random sampling because it does not require repeated random number generation. However, if the list has a hidden periodic pattern, the sample may be biased.

这种方法通常比简单随机抽样更快、更容易，因为它不需要反复生成随机数。但是，如果名单存在隐藏的周期模式，样本可能会产生偏差。

4. Stratified Sampling | 分层抽样

Stratified sampling divides the population into distinct subgroups, called strata, based on shared characteristics such as age, gender, or income. A random sample is then taken from each stratum in proportion to its size in the population. This ensures that each important subgroup is represented.

分层抽样根据共同特征（如年龄、性别或收入）将总体划分为不同的子群体，称为层。然后从每一层中按其在总体中所占比例进行随机抽样。这样能确保每个重要子群体都有代表。

If the population size in stratum i is Nᵢ and the total population is N, the sample size from stratum i is:

如果第 i 层的总体大小为 Nᵢ，总体总数为 N，则第 i 层的样本量为：

nᵢ = (Nᵢ ÷ N) × n

where n is the total sample size.

其中 n 为总样本量。

Advantages | 优点： guarantees representation of all strata and often gives more precise estimates than simple random sampling. | 保证所有层的代表性，通常比简单随机抽样给出更精确的估计。

Disadvantages | 缺点： requires accurate information about the population structure. | 需要准确的总体结构信息。

5. Cluster Sampling | 整群抽样

Cluster sampling involves dividing the population into clusters, which are naturally occurring groups such as schools, villages, or city blocks. A random selection of clusters is made, and then either all members of the selected clusters are included, or a random sample is taken within each selected cluster.

整群抽样是把总体划分为若干个群，这些群可以是自然形成的群体，如学校、村庄或居民小区。先随机选取若干群，然后可以调查所选群中的全部成员，也可在每个被选群内再随机抽取一部分成员。

This method is very cost-effective when the population is spread over a large geographic area. However, cluster sampling tends to be less precise than stratified sampling because clusters may not be homogeneous; individuals within a cluster may be similar to each other.

当总体分布在地域广阔的区域时，这种方法成本很低。但是，整群抽样通常比分层抽样精度低，因为群内个体可能彼此相似，群与群之间差异较大，导致样本不一定能完全代表总体。

6. Quota Sampling | 配额抽样

Quota sampling is a non-probability sampling method. The population is divided into groups based on selected characteristics, and interviewers are assigned quotas: a fixed number of people from each group. The interviewer then chooses individuals who meet the quota until the target numbers are reached.

配额抽样是一种非概率抽样方法。首先根据选定的特征将总体分成若干组，并为调查员规定配额：每组需要调查的固定人数。调查员在有空时任意选择符合条件的人，直到达到配额为止。

Because the selection within groups is not random, quota sampling is subject to interviewer bias. It is often used in market research because it is fast and cheap, but it does not allow us to calculate sampling error or confidence intervals reliably.

由于组内选择不是随机的，配额抽样容易受到调查员主观偏好的影响。它常用于市场调研，因为它快速且成本低，但无法可靠地计算抽样误差或置信区间。

7. Convenience Sampling | 便利抽样

Convenience sampling, also called opportunity sampling, selects individuals who are easiest to reach, such as friends, passers-by, or volunteers. This method is extremely easy and inexpensive, but the sample is almost always biased because it is not representative of the whole population.

便利抽样也叫机会抽样，是选择最容易接触到的个体，例如朋友、路人或自愿参与者。这种方法极其简单且成本低，但样本几乎总是有偏差，因为不能代表整个总体。

Example: interviewing shoppers at a single supermarket to estimate the spending habits of all residents in a city is convenience sampling. It ignores people who do not shop there, and results cannot be generalised.

例如，为了估计某市所有居民的消费习惯，只在一家超市门口采访顾客就是便利抽样。这种方法忽视了不去该超市的人，因此结果不能推广到整个总体。

8. Bias and Sampling Errors | 偏差与抽样误差

Sampling bias occurs when the sample is not representative of the population, often due to a flawed sampling frame or a non-random selection method. This leads to estimates that are systematically too high or too low.

抽样偏差是指样本不能代表总体的情况，通常由不完善的抽样框或非随机的选择方法造成。这会导致估计值系统地偏高或偏低。

Common types of error include:

常见的误差类型包括：

Sampling error | 抽样误差： the natural variation between different samples; it decreases as sample size increases. | 不同样本之间的自然波动；样本量增大时误差减小。
Non-sampling error | 非抽样误差： errors unrelated to sampling, such as measurement errors, non-response, or recording mistakes. | 与抽样无关的误差，如测量误差、无响应或记录错误。
Selection bias | 选择偏差： when some members of the population are more likely to be chosen than others. | 当总体中的某些成员比其他成员更容易被选中时产生。
Response bias | 响应偏差： when respondents give inaccurate answers, perhaps due to question wording or pressure. | 受访者提供不准确答案时产生，可能与问题措辞或压力有关。

9. Choosing the Appropriate Method | 选择恰当的抽样方法

In IB Mathematics, you may be asked to decide which sampling method is most suitable for a given context. The choice depends on whether a sampling frame exists, whether time and money are limited, and whether specific subgroups must be represented.

在 IB 数学中，你可能会被要求判断在特定情境下哪种抽样方法最合适。选择取决于是否存在抽样框、时间和资金是否有限，以及是否需要保证特定子群体的代表性。

Method 方法	Probability? 概率?	Typical use 典型用途
Simple random 简单随机	Yes 是	Small homogeneous population 小规模同质总体
Systematic 系统	Yes 是	Ordered lists such as factory production 有序名单，如工厂生产
Stratified 分层	Yes 是	Population with clear subgroups 有明确子群体的总体
Cluster 整群	Yes 是	Large geographic areas 地域分布广泛的总体
Quota 配额	No 否	Market research 市场调研
Convenience 便利	No 否	Pilot studies 初步探索性调查

10. IB Examination Tips | IB 考试要点

In IB exams, sampling methods often appear in Paper 2 as short questions. You may be asked to describe a sampling method, identify possible bias, or justify the use of a particular method in context.

在 IB 考试中，抽样方法常以简答题的形式出现在 Paper 2。你可能需要描述某种抽样方法、识别可能存在的偏差，或者证明在特定情境下使用某种方法的合理性。

Always use statistical terminology such as “population”, “sample”, and “bias” precisely. | 务必准确使用”总体”、”样本”和”偏差”等统计术语。
When calculating a stratified sample, show the proportional formula clearly. | 计算分层抽样时，要清晰写出比例公式。
When comparing methods, mention both strengths and limitations. | 比较方法时，既要说明优点也要说明局限。
Do not confuse “cluster sampling” with “stratified sampling”: clusters are selected randomly, while strata are all represented. | 不要混淆”整群抽样”和”分层抽样”：整群抽样是随机选取群，而分层抽样要求每一层都被包含。

11. Worked Example | 例题

A school has 600 students in four year groups: Year 1: 150, Year 2: 200, Year 3: 120, Year 4: 130. A researcher wants to survey 60 students using stratified sampling. Calculate the sample size from each year group.

某学校共有 600 名学生，分布在四个年级：一年级 150 人、二年级 200 人、三年级 120 人、四年级 130 人。研究者想用分层抽样调查 60 名学生。计算每个年级应抽取的人数。

Using the formula nᵢ = (Nᵢ ÷ N) × n, we get:

使用公式 nᵢ = (Nᵢ ÷ N) × n，可得：

Year 1: (150 ÷ 600) × 60 = 15

Year 2: (200 ÷ 600) × 60 = 20

Year 3: (120 ÷ 600) × 60 = 12

Year 4: (130 ÷ 600) × 60 = 13

The total is 15 + 20 + 12 + 13 = 60, as required.

总数为 15 + 20 + 12 + 13 = 60，满足要求。

12. Summary | 总结

Sampling methods are a core topic in IB Mathematics Statistics. Probability-based methods such as simple random, systematic, stratified, and cluster sampling provide a foundation for unbiased statistical inference. Non-probability methods such as quota and convenience sampling are faster but carry higher risk of bias. Recognising the strengths and weaknesses of each method helps you choose the right one and interpret results critically.

抽样方法是 IB 数学统计内容的核心知识。简单随机、系统、分层和整群抽样等概率方法为无偏统计推断奠定了基础。配额抽样和便利抽样等非概率方法更快，但偏差风险更高。了解每种方法的优势与不足，有助于我们选择合适的方法并批判性地解读结果。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Sampling Methods in IB Mathematics | IB数学：抽样方法