AS AQA Statistics Topic Test: Data, Probability & Random Variables | AS AQA 统计学专题测试：数据、概率与随机变量

📚 AS AQA Statistics Topic Test: Data, Probability & Random Variables | AS AQA 统计学专题测试：数据、概率与随机变量

The statistics component of the OxfordAQA International AS Mathematics 9660 specification is one of the most accessible and rewarding areas for students who grasp the core ideas of data collection, presentation, probability and random variables. This topic test revision guide covers every major concept you need to master, with formulas, worked examples and exam-style guidance.

牛津AQA国际AS数学9660考纲中的统计学部分是学生最容易拿分且回报率最高的领域之一，只要掌握了数据收集、数据呈现、概率和随机变量的核心概念。本专题测试复习指南涵盖你需要掌握的所有主要概念，包含公式、例题和考试策略指导。

1. Sampling Methods | 抽样方法

Sampling is the process of selecting a subset of individuals from a population in order to estimate characteristics of the whole population. A census collects data from every single member of the population, whereas a sample collects data from a chosen subset. In the exam, you must understand when each is appropriate and be able to compare their advantages and disadvantages.

抽样是从总体中选择一部分个体以估计整个总体特征的过程。普查（census）从总体的每一个成员收集数据，而抽样（sample）则从选定的子集中收集数据。在考试中，你必须理解每种方法的适用场景，并能比较它们的优缺点。

Simple random sampling: every member of the population has an equal chance of being selected. This can be done using random number tables, calculators, or computer-generated random numbers. The main advantage is that it removes selection bias; the main disadvantage is that a complete list of the population (a sampling frame) is required.

简单随机抽样：总体中每个成员被选中的机会均等。可以使用随机数表、计算器或计算机生成的随机数来实施。主要优点是没有选择偏倚；主要缺点是需要一个完整的总体名单（抽样框）。

Systematic sampling: every k-th member of the population is selected after a random starting point, where k = population size ÷ sample size. This is easier and quicker than simple random sampling, but can be biased if the population list has a periodic pattern.

系统抽样：在随机起点之后，每隔k个成员选取一个，其中k = 总体大小 ÷ 样本大小。这比简单随机抽样更方便、更快捷，但如果总体列表存在周期性规律，则可能产生偏倚。

Stratified sampling: the population is divided into strata (groups), and the number of individuals selected from each stratum is proportional to the size of that stratum in the population. The formula is: sample size from a stratum = (stratum size ÷ population size) × overall sample size. This ensures each group is represented proportionally and reduces sampling bias.

分层抽样：将总体划分为若干层（组），从每层中抽取的人数与该层在总体中所占比例成正比。公式为：某层抽样数 =（该层人数 ÷ 总体人数）× 总样本量。这确保了每个群体按比例得到代表，减少了抽样偏倚。

Quota sampling: an interviewer selects a pre-determined number of individuals from each group until the quota is filled. No sampling frame is needed, making it fast and cheap, but it is non-random and can introduce interviewer bias.

配额抽样：访问员从每个组中选择预先确定数量的人，直到配额填满。这种方法不需要抽样框，快捷且成本低，但它是非随机的，可能引入访问员偏倚。

Opportunity sampling: selecting individuals who are conveniently available at the time of the study, such as asking people in the street. This is the fastest and cheapest method, but also the least reliable and most prone to bias.

方便抽样：选择在研究时方便获得的个体，例如在街上询问路人。这是最快、最便宜的方法，但也是最不可靠、最容易产生偏倚的方法。

In the exam, you may be asked to identify which sampling method was used, justify its use, or suggest how to reduce bias. Remember the key trade-off: random methods (simple random, systematic, stratified) reduce bias but require more preparation; non-random methods (quota, opportunity) are quicker and cheaper but less representative.

考试中，你可能会被要求识别使用了哪种抽样方法、证明其合理性，或提出减少偏倚的建议。记住关键的取舍：随机方法（简单随机、系统、分层）减少偏倚但需要更多准备工作；非随机方法（配额、方便）更快更便宜但代表性较差。

2. Data Presentation: Stem-and-Leaf and Histograms | 数据呈现：茎叶图与直方图

Once data is collected, it must be presented clearly and appropriately. Stem-and-leaf diagrams and histograms are two essential methods you must be able to construct and interpret in the AS AQA topic test.

数据收集后必须清晰、恰当地呈现。茎叶图和直方图是你在AS AQA专题测试中必须能够绘制和解读的两种基本方法。

For a stem-and-leaf diagram, write the stem on the left of a vertical line and the leaves on the right. Always include a key, for example “3 | 2 means 3.2”. The diagram preserves the original data values while showing the shape of the distribution, allowing you to quickly read off the mode, median and range. When comparing two datasets, use a back-to-back stem-and-leaf diagram.

对于茎叶图，将茎写在竖线左侧，叶写在右侧。始终包含图例，例如”3 | 2 表示 3.2″。茎叶图在展示分布形态的同时保留了原始数据值，可以快速读出众数、中位数和极差。当比较两组数据时，使用背对背茎叶图。

A histogram is used for continuous data. The key difference from a bar chart is that the area of each bar is proportional to the frequency, not the height. The vertical axis is therefore frequency density:

直方图用于连续数据。它与条形图的关键区别在于：每个条形的面积与频率成正比，而不是高度。因此纵轴是频率密度（frequency density）：

Frequency density = Frequency ÷ Class width

频率密度 = 频率 ÷ 组距

For example, if 20 students scored between 10 and 15 marks in

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