IGCSE AQA Statistics: Case Study Practice | IGCSE AQA 统计：案例分析实战演练

📚 IGCSE AQA Statistics: Case Study Practice | IGCSE AQA 统计：案例分析实战演练

Welcome to this case study practice session designed for IGCSE AQA Statistics. Through a realistic scenario involving survey data from secondary school students, we will walk through key statistical techniques, from data collection and presentation to probability, correlation, and interpretation. This hands-on guide will help you apply your statistical knowledge in exam-style questions, and every method is demonstrated on a small, fully worked data set so that you can follow each calculation step by step.

欢迎来到本次 IGCSE AQA 统计案例分析实战演练。我们将通过一个真实的中学学生调查数据场景，逐步演练从数据收集、呈现到概率、相关性以及结果解读等关键统计技术。本实操指南将帮助你运用统计学知识应对考试题型。文中每一个方法都会在一个完整的小型数据集上逐步演算，方便你跟着一起计算、一起思考。

1. Case Scenario and Data Set | 案例情景与数据集

A school conducts a survey of 50 randomly selected Year 11 students to investigate the relationship between the number of hours spent on physical exercise per week and their latest mathematics test score (%). The data collected also includes each student's gender. The full data set therefore has three variables: exercise hours, mathematics score and gender. In this article we use the first ten entries, shown in Table 1, for all the worked examples, because the arithmetic is easy to check by hand while the statistical ideas remain exactly the same as they would be with all 50 students.

某学校随机抽取了 50 名 11 年级学生进行调查，旨在研究每周体育锻炼小时数与最近一次数学考试成绩（百分制）之间的关系。收集的数据还包括每位学生的性别。因此完整数据集包含三个变量：锻炼小时数、数学成绩和性别。本文所有演算示例均使用表 1 中显示的前 10 条记录，原因是这些数据的运算便于手工核对，而其中的统计思想与使用全部 50 名学生完全相同。

Table 1: Raw data (first 10 entries shown).

表 1：原始数据（仅显示前 10 条）。

In an exam, the first job is always to identify each variable and its type, because this determines which charts, averages and probability methods are appropriate. Writing this down takes ten seconds and prevents the most common loss of marks.

在考试中，第一件事永远是识别每个变量及其类型，因为它决定了应该使用哪种图表、平均数和概率方法。把这个步骤写下来只需十秒钟，却能避免最常见的失分。

• Explanatory (independent) variable: exercise hours per week - quantitative, discrete. | 解释变量（自变量）：每周锻炼小时数——定量、离散型。
• Response (dependent) variable: mathematics test score (%) - quantitative, continuous. | 响应变量（因变量）：数学考试成绩——定量、连续型。
• Gender: qualitative (categorical), nominal, with two categories. | 性别：定性（分类）变量，名义型，含两个类别。

2. Types of Data and Sampling | 数据类型与抽样

Before any calculation, a statistician asks two questions: where did the data come from, and how representative is it? Data collected directly by the person analysing it, such as the survey in this case study, is primary data. Data taken from someone else's report, a website or a textbook is secondary data. Primary data is more reliable for the exact question you are asking, but it is slower and more expensive to collect; secondary data is quick but may have been gathered for a different purpose, with different definitions and different rounding.

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在做任何计算之前，统计学家会先问两个问题：数据从哪里来？它具有代表性吗？由分析者本人直接收集的数据（如本案例中的调查）称为一手数据；取自他人报告、网站或教材的数据称为二手数据。对于你所要研究的问题而言，一手数据更可靠，但收集更慢、成本更高；二手数据获取快捷，但可能原本是为其他目的收集的，定义与舍入方式也可能不同。

The survey uses a sample of 50 students rather than the whole year group, so the method of selection matters enormously. A sample should be representative of the population (all Year 11 students in the school). If the school instead asked only the students who attend football club, the sample would be biased towards high exercise levels and would almost certainly overestimate the link between exercise and mathematics performance.

该调查使用的是 50 名学生的样本，而不是整个年级，因此选择方法至关重要。样本应当代表总体（该校全体 11 年级学生）。如果学校只询问足球社团的学生，样本就会偏向锻炼时间较长的人群，几乎必然会高估锻炼与数学表现之间的联系。

• Simple random sampling: every student has an equal chance of selection, for example using a random number generator on the register. | 简单随机抽样：每名学生被选中的机会相同，例如用随机数生成器在名册中抽取。
• Systematic sampling: choose every 5th name on an alphabetical list after a random start. | 系统抽样：在字母顺序名册上随机起点后每隔 5 人抽取。
• Stratified sampling: split the population into strata (for example year group or gender) and take a number from each stratum in proportion to its size. | 分层抽样：将总体分为若干层（如年级或性别），按各层规模比例抽取。
• Quota sampling: the interviewer fills fixed quotas, for example 25 boys and 25 girls, choosing whoever is available. | 配额抽样：调查者按固定配额（如 25 名男生、25 名女生）自行选择可接触到的对象。
• Opportunity (convenience) sampling: whoever is easiest to reach, for example a class the teacher already teaches. | 便利抽样：选择最容易接触到的人，例如任课教师所带的班级。

Quota and opportunity sampling are quick but non-random, so the results may be unrepresentative and it is not possible to give a reliable estimate of the likely error. AQA questions frequently ask you to name the sampling method used and to give one advantage and one disadvantage, so learn them as pairs.

配额抽样和便利抽样速度较快，但并非随机，因此结果可能不具代表性，也无法给出可靠的误差估计。AQA 的考题经常要求你指出所用的抽样方法并说明一个优点和一个缺点，因此请成对记忆。

3. Organising Data: Frequency Tables and Grouping | 数据整理：频数表与分组

Raw lists of numbers are hard to interpret, so the next step is to organise the data into a frequency table. When a variable takes many different values, or is continuous, we group it into class intervals of equal width. Equal widths are important because unequal widths make bar heights and frequencies impossible to compare fairly.

原始数字列表难以解读，因此下一步是把数据整理成频数表。当变量取值很多或为连续型时，我们将其分成宽度相等的组距。组距相等很重要，因为宽度不等的区间会使条形高度与频数无法公平比较。

Table 2 groups the ten students' exercise hours into five intervals and uses the midpoint of each interval to estimate the mean. For grouped data the exact values are lost, so the midpoint is used as a stand-in for every value in that class.

表 2 把十名学生的锻炼小时数分成五个区间，并用每个区间的组中值来估计平均数。对于分组数据，原始数值已丢失，因此用组中值代表该组内的每一个取值。

Exercise hours 锻炼小时	Tally 划记	Frequency f 频数	Midpoint x 组中值	f × x	Cumulative f 累计频数
0 - 1	||	2	0.5	1.0	2
2 - 3	||	2	2.5	5.0	4
4 - 5	|||	3	4.5	13.5	7
6 - 7	||	2	6.5	13.0	9
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