A-Level Maths: Population and Sample Concepts in Statistics | A-Level 数学：统计中的总体与样本概念

📚 A-Level Maths: Population and Sample Concepts in Statistics | A-Level 数学：统计中的总体与样本概念

In statistics, the ideas of a population and a sample form the foundation of all data collection and inference. Understanding the difference between them allows us to draw meaningful conclusions from limited data.

在统计学中，总体与样本的概念构成了所有数据收集和推断的基础。理解它们之间的区别，使我们能依据有限的数据得出有意义的结论。

1. Defining Population and Sample | 总体与样本的定义

A population is the complete set of all elements or individuals that we are interested in studying. It can be people, objects, measurements, or events. For example, if we want to know the average height of all 16-year-old students in the UK, the population is every 16-year-old student in the UK.

总体是我们感兴趣研究的全部元素或个体的集合。它可以是人、物体、测量值或事件。例如，如果我们想知道英国所有16岁学生的平均身高，总体就是英国每一位16岁学生。

A sample is a subset of the population that is actually selected for investigation. The sample should be representative of the population so that conclusions drawn from the sample can be generalised to the whole population.

样本是从总体中实际选择进行调查的一部分。样本应当能够代表总体，这样从样本得出的结论才能推广到整个总体。

2. The Purpose of Sampling | 抽样的目的

It is often impractical or impossible to measure an entire population. A census, which surveys every member of a population, is costly, time-consuming, and sometimes destructive. Sampling provides a practical alternative.

测量整个总体往往是不现实或不可能的。普查虽然调查总体中的每一个成员，但成本高、耗时长，有时甚至具有破坏性。抽样提供了一种实用的替代方案。

Sampling reduces cost and time.

抽样节省成本和时间。
Sampling allows results to be obtained quickly.

抽样可以快速获得结果。
Sampling is essential when the population is infinite or constantly changing.

当总体是无限或不断变化时，抽样是必不可少的。

3. Random Sampling | 随机抽样

A random sample is one in which every member of the population has an equal chance of being selected. This avoids bias and makes the sample more likely to be representative.

随机样本是指总体中每个成员被选中的概率相同。这样可以避免偏差，使样本更可能具有代表性。

One common method is simple random sampling, where a sampling frame (a list of all population members) is used. Each member is assigned a number, and a random number generator or a table of random digits selects the sample.

一种常用方法是简单随机抽样，即使用抽样框（所有总体成员的名单）。每个成员被分配一个编号，然后利用随机数生成器或随机数表来选取样本。

P(selecting a particular member) = 1 ⁄ N

Here, N is the population size. Every member has the same probability of selection.

其中 N 是总体容量。每个成员被选中的概率相同。

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. A proportional random sample is then taken from each stratum.

分层抽样根据某个特征（如年龄、性别或收入）将总体划分为不同的组，称为层。然后从每一层中按比例进行随机抽样。

The number taken from each stratum is proportional to the stratum’s size in the population. For a stratum of size S in a population of size N, the sample size from that stratum is:

从每层中抽取的数量与该层在总体中的大小成比例。对于总体容量 N 中容量为 S 的层，从该层抽取的样本量为：

Sample size from stratum = (S ⁄ N) × total sample size

This method guarantees that each subgroup is represented in the sample, making it more precise than simple random sampling when the population is heterogeneous.

这种方法保证了每个子群在样本中都有代表，当总体具有异质性时，它比简单随机抽样更精确。

5. Quota Sampling | 定额抽样

Quota sampling is a non-random method where an interviewer selects a pre-determined number of people from each group, according to quotas. The interviewer has freedom to choose who to ask, as long as the quota is filled.

定额抽样是一种非随机方法，访问员根据预定的配额从每个群体中选择一定数量的人。只要配额满足，访问员可以自由选择询问对象。

Quota sampling is faster and cheaper than random methods.

定额抽样比随机方法更快、更便宜。
It does not require a sampling frame.

它不需要抽样框。
However, it is prone to interviewer bias, as the selection is not random.

然而，由于选择不是随机的，它容易受到访问员偏差的影响。

6. Systematic and Cluster Sampling | 系统抽样与整群抽样

Systematic sampling selects every k-th member from a sampling frame after a random start. For example, to obtain a sample of 50 from a population of 1000, we might select every 20th person. This is simple to implement but may be biased if there is a periodic pattern in the list.

系统抽样是在随机起点之后，从抽样框中每隔 k 个成员选取一个。例如，要从1000人中抽取50人，我们可以每隔20人选取一个。这种方法实施简单，但如果名单中存在周期性模式，可能会产生偏差。

Cluster sampling divides the population into clusters, randomly selects some clusters, and then includes all members from the chosen clusters. This is useful when the population is spread over a large geographical area, but it may produce less precise results than other methods.

整群抽样是将总体划分为若干群，随机选择一些群，然后调查所选群中的所有成员。当总体分布在大范围地理区域时很有用，但结果可能不如其他方法精确。

Method	Random?	Key Feature
Simple random	Yes	Equal chance for all
Stratified	Yes	Proportional from each stratum
Systematic	Yes (if random start)	Every k-th member
Cluster	Yes	Whole random clusters
Quota	No	Quotas filled by interviewer

7. Census vs Sample | 普查与样本对比

A census is a survey of the entire population. It gives exact information, but it has serious drawbacks: it is expensive, time-consuming, and often impossible for large or infinite populations. It may also be destructive, for example when quality control requires testing every product.

普查是对整个总体进行的调查。它能提供准确的信息，但有严重缺点：成本高、耗时长，对于庞大或无限的总体通常不可能做到。它还可能具有破坏性，例如质量控制要求测试每一件产品时。

A sample, on the other hand, is smaller and more manageable. The trade-off is that sample results carry sampling error — the difference between the sample estimate and the true population value.

相比之下，样本更小、更易操作。代价是样本结果带有抽样误差——即样本估计值与真实总体值之间的差异。

8. Parameters and Statistics | 参数与统计量

A parameter is a numerical characteristic of a population, such as the population mean μ or the population proportion p. A statistic is a numerical characteristic calculated from a sample, such as the sample mean x̄ or the sample proportion p̂.

参数是总体的数值特征，例如总体均值 μ 或总体比例 p。统计量是根据样本计算出的数值特征，例如样本均值 x̄ 或样本比例 p̂。

Population parameter (μ) → Sample statistic (x̄)

We use statistics as estimators of parameters. For example, the sample mean x̄ is an unbiased estimator of the population mean μ, meaning that on average it equals the true value.

我们用统计量作为参数的估计值。例如，样本均值 x̄ 是总体均值 μ 的无偏估计量，即平均而言它等于真实值。

9. Bias in Sampling | 抽样中的偏差

Bias occurs when a sample systematically over- or under-represents certain parts of the population. A biased sample gives misleading estimates, even if the sample is large.

当样本系统性地高估或低估总体的某些部分时，就会出现偏差。有偏的样本即使容量很大，也会给出误导性的估计。

Common sources of bias include:

常见的偏差来源包括：

Selection bias – some members are more likely to be chosen than others.

选择偏差——某些成员比其他成员更可能被选中。
Non-response bias – selected individuals do not participate.

无响应偏差——被选中的人没有参与。
Sampling frame bias – the list used to draw the sample is incomplete or inaccurate.

抽样框偏差——用于抽取样本的名单不完整或不准确。

10. Sampling Distribution of the Mean | 均值的抽样分布

When we take repeated samples of size n from a population, the sample mean x̄ varies from sample to sample. The distribution of all possible sample means is called the sampling distribution of the mean.

当我们从总体中重复抽取容量为 n 的样本时，样本均值 x̄ 会随样本不同而变化。所有可能样本均值的分布称为均值的抽样分布。

If the population has mean μ and variance σ², then the sampling distribution of x̄ has:

如果总体均值为 μ，方差为 σ²，那么 x̄ 的抽样分布具有：

Mean = μ, Variance = σ² ⁄ n, Standard error = σ ⁄ √n

The standard error measures the spread of sample means around the population mean. It decreases as the sample size n increases.

标准误差衡量样本均值围绕总体均值的离散程度。随着样本量 n 的增大，标准误差减小。

11. The Central Limit Theorem | 中心极限定理

For a sufficiently large sample size (usually n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the shape of the original population distribution. This is the Central Limit Theorem.

当样本量足够大（通常 n ≥ 30）时，样本均值的抽样分布近似正态，无论原始总体分布的形状如何。这就是中心极限定理。

This theorem underpins many statistical inference procedures, allowing us to construct confidence intervals and perform hypothesis tests based on the normal distribution.

该定理支撑了许多统计推断方法，使我们能够基于正态分布构建置信区间并进行假设检验。

12. Key Takeaways | 要点总结

To succeed in A-Level statistics, you must clearly distinguish between population and sample, understand the advantages and disadvantages of different sampling methods, and know how to use sample statistics to estimate population parameters.

要在A-Level统计中取得好成绩，你必须清楚区分总体与样本，理解不同抽样方法的优缺点，并知道如何用样本统计量估计总体参数。

A population is the entire set under study; a sample is a subset.

总体是研究的全部集合；样本是其子集。
Random sampling reduces bias and allows probability-based inference.

随机抽样减少偏差，并允许基于概率的推断。
Stratified sampling ensures representation of subgroups.

分层抽样确保子群得到代表。
Quota sampling is non-random and may be biased.

定额抽样是非随机的，可能产生偏差。
Parameters describe populations; statistics describe samples.

参数描述总体；统计量描述样本。
The central limit theorem justifies using the normal distribution for large samples.

中心极限定理为大样本使用正态分布提供了依据。

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A-Level Maths: Population and Sample Concepts in Statistics | A-Level 数学：统计中的总体与样本概念