📚 Data Collection | 数据收集
In any statistical investigation, the way data is collected determines the quality and reliability of the conclusions drawn. This article covers the fundamental concepts of data collection required for the Edexcel A-Level Mathematics specification, including populations, samples, sampling methods, types of data, and potential sources of bias. A firm grasp of these ideas is essential for designing studies and critically evaluating statistical claims.
在任何统计调查中，数据的收集方式决定了所得结论的质量和可靠性。本文涵盖爱德思 A-Level 数学大纲要求的数据收集基本概念，包括总体、样本、抽样方法、数据类型以及可能的偏差来源。牢固掌握这些概念对于设计研究方案和批判性评价统计结论至关重要。
1. Populations and Samples | 总体与样本
A population is the entire set of individuals or items that we wish to study. A sample is a subset of the population selected for the actual investigation. Conducting a census (measuring every member of the population) is often impractical due to time, cost, or accessibility, so sampling is used to make inferences about the population. The main advantage of a sample over a census is that it is quicker and cheaper, but it may introduce sampling error if the sample is not representative.
总体是我们希望研究的全部个体或项目的集合。样本是从总体中选出用于实际调查的一个子集。进行普查（测量总体中的每一个成员）通常因为时间、成本或可及性而不现实，因此使用抽样来对总体进行推断。样本相较于普查的主要优点是更快捷、成本更低，但如果样本不具有代表性，则可能引入抽样误差。
2. Sampling Frames | 抽样框
A sampling frame is a list or map identifying every unit in the population from which a sample may be drawn. In an ideal scenario, the sampling frame perfectly matches the population of interest. Differences between the sampling frame and the population can lead to undercoverage or overcoverage, causing bias. For example, using a telephone directory as a sampling frame excludes people without a listed number, so the sample may not fully represent the wider population.
抽样框是一份列出或标识总体中每一个单元的清单或地图，样本可以从中抽取。在理想情况下，抽样框与研究的目标总体完全吻合。抽样框与总体之间的差异会导致覆盖不足或覆盖过度，从而造成偏差。例如，使用电话簿作为抽样框会排除没有列入号码的人，因此样本可能无法充分代表更广泛的总体。
3. Random vs Non-Random Sampling | 随机抽样与非随机抽样
Sampling methods are broadly divided into random (probability) sampling and non-random (non-probability) sampling. In random sampling, every member of the population has a known, non-zero chance of being selected, which helps to reduce selection bias and allows the use of probability theory to estimate sampling errors. Non-random sampling does not give all members a known chance of selection and is more prone to bias, though it is often cheaper and faster. The choice between them depends on the aim of the study and available resources.
抽样方法大致分为随机（概率）抽样和非随机（非概率）抽样。在随机抽样中，总体中每个成员都有已知且非零的选中概率，这有助于减少选择偏差，并允许使用概率论估算抽样误差。非随机抽样并不给予所有成员已知的选中机会，更容易产生偏差，但通常更经济、更快捷。在两者之间的选择取决于研究目的和可用资源。
4. Simple Random Sampling | 简单随机抽样
Simple random sampling (SRS) gives every possible sample of a given size an equal chance of being selected. In practice, this requires a complete and accurate sampling frame. Each unit is assigned a number, and a random number generator or lottery method is used to select the required number of units. SRS is unbiased and straightforward, but it can be impractical when the population is large and geographically dispersed, and it may not guarantee representation of specific subgroups.
简单随机抽样使给定样本量的每一个可能样本被选中的机会均等。在实践中，这要求有一个完整且准确的抽样框。每个单元被分配一个编号，然后通过随机数生成器或抽签方法选出所需数量的单元。简单随机抽样是无偏且直观的，但当总体庞大且地理分布分散时难以实施，而且它不一定能保证特定子群体的代表性。
5. Stratified Sampling | 分层抽样
In stratified sampling, the population is divided into mutually exclusive strata based on a characteristic relevant to the study (e.g. age, gender, income level). A simple random sample is then taken from each stratum. The number sampled from each stratum is often proportional to the stratum's size in the population, calculated as
nₕ = n × (Nₕ / N)
where n is the total sample size, Nₕ is the stratum size, and N is the population size. Stratified sampling ensures that key subgroups are adequately represented, giving more precise estimates, but it requires detailed population information and can be more complex to organise than SRS.在分层抽样中，总体根据与研究相关的特征（如年龄、性别、收入水平）被分成互斥的层。然后从每一层中进行简单随机抽样。从每一层抽取的样本量常与层在总体中的规模成比例，计算公式为
nₕ = n × (Nₕ / N)
其中 n 是总样本量，Nₕ 是层规模，N 是总体规模。分层抽样能确保关键子群体得到充分代表，给出更精确的估计，但需要详细的总体信息，组织起来比简单随机抽样更复杂。6. Systematic Sampling | 系统抽样
ze N by the desired sample size n: k = N / n. A random starting point between 1 and k is chosen, and then every k-th unit is selected. This method is simpler and quicker than SRS, particularly when dealing with large populations. However, it can introduce bias if there is a hidden pattern or periodicity in the list that coincides with the sampling interval.
系统抽样按照相等的间隔从抽样框中选取单元。抽样间隔 k 由总体规模 N 除以期望样本量 n 得出：k = N / n。在 1 到 k 之间随机选择一个起点，然后每隔 k 个单元选出一个样本。此方法比简单随机抽样更简单快捷，尤其适用于处理大量总体的情形。然而，如果列表中存在与抽样间隔重合的隐蔽模式或周期性，就可能引入偏差。
7. Quota Sampling | 配额抽样
Quota sampling is a non-random method where the researcher sets quotas for different subgroups, often based on known proportions in the population (e.g. 50% male, 50% female). Interviewers then select participants non-randomly to fill these quotas until the required numbers are reached. It is widely used in market research because it is cheap and fast. The major drawback is that selection within quotas is subjective, so the sample may be biased and inference to the population can be unreliable.
配额抽样是一种非随机方法，研究者为不同子群体设定配额，通常基于总体中已知的比例（如 50% 男性，50% 女性）。访问员随后非随机地选择参与者来填满这些配额，直到达到所需数量。配额抽样在市场研究中广泛使用，因为它成本低、速度快。其主要缺陷在于配额内的选择带有主观性，因此样本可能有偏，对总体的推断不可靠。
8. Opportunity (Convenience) Sampling | 便利（机会）抽样
Opportunity sampling, also called convenience sampling, involves selecting individuals who are readily available to the researcher, such as friends, family, or passers-by in a particular location. It is the easiest and least expensive form of sampling, often used for pilot studies or quick polls. The results, however, are highly prone to bias because the sample is unlikely to be representative of any meaningful population, making generalisation risky.
便利抽样又称机会抽样，是指选取研究者容易接触到的个体，如朋友、家人或特定地点的过往行人。这是最容易、成本最低的抽样形式，常用于初步研究或快速民意调查。然而，其结果极易出现偏差，因为样本不太可能代表任何有意义的总体，泛化风险很高。
9. Types of Data: Qualitative and Quantitative | 数据类型：定性数据与定量数据
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe attributes or categories, such as eye colour, favourite sport, or type of car. These are often non-numerical but can be coded into numbers for analysis. Quantitative data result from counting or measuring and consist of numerical values upon which arithmetic operations can be performed, such as height, mass, or test scores.
数据可分为定性（分类）数据和定量（数值）数据。定性数据描述属性或类别，如眼睛颜色、最喜欢的运动或汽车类型。这些数据通常是非数值的，但可以编码为数字进行分析。定量数据来自计数或测量，由可执行算术运算的数值组成，例如身高、体重或考试分数。
10. Discrete and Continuous Data | 离散数据与连续数据
Quantitative data can be further split into discrete and continuous. Discrete data arise from counting and can only take distinct, separate values – often integers. Examples include the number of students in a class or the number of cars passing a junction in an hour. Continuous data arise from measuring and can take any value within a given range. Height, time, and temperature are continuous variables. The distinction is important because it determines which statistical methods and graphical displays are appropriate.
定量数据可进一步分为离散数据和连续数据。离散数据来自计数，只能取特定的、分隔开的值——通常是整数。例如班级中的学生人数或一小时内通过路口的汽车数量。连续数据来自测量，可以在给定范围内取任意值。身高、时间和温度都是连续变量。这一区分之所以重要，是因为它决定了哪些统计方法和图形展示方式才是合适的。
Summary of data types:
| Data Type | Subtype | Examples |
|---|---|---|
| Qualitative | Categorical | Colour, brand, gender |
| Quantitative | Discrete | Shoe size, number of children |
| Continuous | Weight, volume, length |
11. Data Collection Methods | 数据收集方法
The three primary methods of collecting data are experiments, surveys, and observational studies. In an experiment, the researcher actively manipulates one or more variables to observe the effect on a response variable, allowing cause-and-effect conclusions to be drawn if properly controlled. Surveys use questionnaires or interviews to collect self-reported information from a sample; they are efficient for gathering large amounts of data but rely on honest and accurate responses. Observational studies simply record information about subjects without interference, which is useful when experimentation is unethical or impractical, though they cannot establish causation.
数据收集的三种主要方法是实验、调查和观察研究。在实验中，研究者主动操控一个或多个变量，观察其对响应变量的影响，如果控制得当，可以得出因果关系的结论。调查使用问卷或访谈从样本中收集自我报告的信息；这种方法能高效地获取大量数据，但依赖于受访者的诚实和准确回答。观察研究仅是不加干预地记录有关对象的信息，当实验不符合伦理或不可行时，观察研究最为有用，但无法确立因果关系。
12. Sources of Bias in Data Collection | 数据收集中的偏差来源
Bias refers to systematic errors that cause the sample to misrepresent the population. Common sources include selection bias, where the sampling method consistently favours certain groups; non-response bias, when a significant proportion of the chosen sample refuses to participate or cannot be contacted; measurement bias, stemming from inaccurate instruments or leading questions; and voluntary response bias, where only those with strong opinions participate. Awareness of these biases helps in designing better studies and in critically interpreting statistical results.
偏差是指导致样本无法准确代表总体的系统性错误。常见来源包括选择偏差，即抽样方法一贯偏向某些群体；无应答偏差，当被选样本中有很大比例的人拒绝参与或无法联系时产生；测量偏差，源于不准确的工具或引导性问题；以及自愿响应偏差，即只有持有强烈观点的人才会参与。认识到这些偏差有助于设计更好的研究，并有批判性地解读统计结果。
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