📚 IGCSE Cambridge Statistics: Case Study Practice | IGCSE Cambridge 统计：案例分析实战演练
This article walks through a complete IGCSE Statistics case study: investigating whether students' online revision hours are linked to their mock examination scores. It shows how to plan, collect, organise, display, analyse and interpret data in the way Cambridge examiners expect.
本文通过一个完整的 IGCSE 统计案例，调查学生在线复习小时数与模拟考试成绩之间是否存在关联。文章将展示如何按照剑桥考官期望的方式规划、收集、整理、展示、分析和解释数据。
1. Understanding the Scenario | 理解案例背景
A school wants to know whether the number of hours a Year 11 student spends on online revision affects the percentage score in a statistics mock exam. The variables are quantitative and bivariate: revision hours (x) and mock score (y). A possible hypothesis is that students who revise more tend to score higher.
某所学校想了解 11 年级学生在线复习的小时数是否会影响统计模拟考试的百分制成绩。变量是定量且双变量的：复习小时数 (x) 与模拟成绩 (y)。一个可能的假设是，复习时间更多的学生往往得分更高。
Before doing any calculations, always identify the population, the sample, the explanatory variable and the response variable. In this example, the population is all Year 11 statistics students in the school, the explanatory variable is revision hours, and the response variable is mock score.
在进行任何计算之前，始终要先明确总体、样本、解释变量和响应变量。在本例中，总体是该校全体 11 年级统计学生，解释变量是复习小时数，响应变量是模拟成绩。
- Population: all Year 11 statistics students
- Explanatory variable: hours of online revision
- Response variable: mock score (%)
- Main question: is there a positive linear association?
对应的中文要点：
- 总体：全体 11 年级统计学生
- 解释变量：在线复习小时数
- 响应变量：模拟成绩（百分制）
- 主要问题：是否存在正的线性相关？
2. Planning the Investigation | 规划调查
The alternative hypothesis is that there is a positive correlation, written H₁: ρ > 0.
清晰的计划能把模糊的问题转化为可检验的统计问题。在本案例中，原假设是总体中不存在相关，记作 H₀: ρ = 0。备择假设是存在正相关，记作 H₁: ρ > 0。
Because the data are observational, we cannot automatically say that revision time causes higher scores. Confounding variables such as prior attainment, attendance, number of practice papers attempted, and home study environment may also affect the mock result.
由于数据是观察性的，我们不能自动得出复习时间导致成绩提高的结论。先前成绩、出勤率、练习卷完成数量以及家庭学习环境等混杂变量也可能影响模拟考试结果。
A suitable investigation plan is: use a stratified random sample, record revision hours and scores, draw a scatter diagram, calculate the product moment correlation coefficient, and fit a regression line if the correlation is strong enough.
一个合适的调查计划是：采用分层随机抽样，记录复习小时数和成绩，绘制散点图，计算积矩相关系数，并在相关性足够强时拟合回归直线。
3. Data Collection and Sampling | 数据收集与抽样
The sample must represent the Year 11 cohort fairly. A simple random sample gives every student an equal chance of selection, but it can accidentally produce an unbalanced gender mix. A stratified random sample reduces this risk by sampling proportionally from each gender or class group.
样本必须公平地代表 11 年级群体。简单随机抽样让每名学生被选中的机会相等，但可能偶然导致性别比例失衡。分层随机抽样按性别或班级等分组按比例抽样，可以降低这种风险。
In this case, the school has 90 boys and 60 girls in Year 11. If a sample of 30 is chosen, proportional allocation gives 18 boys and 12 girls. Within each gender group, students are selected using random numbers.
在本案例中，该校 11 年级有 90 名男生和 60 名女生。如果抽取 30 名学生，按比例分配应抽取 18 名男生和 12 名女生。在每个性别组内，使用随机数进行选择。
Self-selected online polls are not suitable because they often attract only highly motivated students. This creates response bias and makes the sample unrepresentative.
自愿参与的在线投票并不合适，因为这种调查往往只吸引学习积极性很高的学生。这会造成响应偏差，使样本不具代表性。
4. Organising Raw Data | 整理原始数据
After collection, raw data should be arranged into grouped frequency tables so patterns can be seen. The table below summarises the revision hours for the 30 sampled students.
收集数据后，应将原始数据整理成组频数表，以便观察规律。下表汇总了 30 名抽样学生的复习小时数。
| Revision hours, x | Frequency, f | Cumulative frequency, CF |
|---|---|---|
| 0 ≤ x < 2 | 4 | 4 |
| 2 ≤ x < 4 | 8 | 12 |
| 4 ≤ x < 6 | 10 | 22 |
| 6 ≤ x < 8 | 6 | 28 |
| 8 ≤ x < 10 | 2 | 30 |
The continuous data are grouped into equal class widths of 2 hours. The cumulative frequency column is built by adding each frequency to the previous total.
连续数据被分为宽度相等的组，组距为 2 小时。累积频数列通过将每个频数加到前一个总数上得到。
5. Visualising the Data | 数据可视化
A histogram is appropriate for continuous data. Since the class intervals have equal width, the frequency density is the same as the frequency. The area of each bar represents the frequency in that interval.
直方图适用于连续数据。由于这些组的组距相等，频率密度与频数相同。每个条形图的面积代表该组的数据频数。
e 4 ≤ x < 6 class, so the median is calculated as 4 + ((15 − 12) / 10) × 2 = 4.6 hours.
累积频数曲线可用于估计中位数和四分位数。对于 30 名学生，中位数位置是第 15 个值，位于 4 ≤ x < 6 这一组，因此中位数为 4 + ((15 − 12) / 10) × 2 = 4.6 小时。
22.5 − 22) / 6) × 2 = 6.167 hours. The interquartile range is 6.167 − 2.875 = 3.292 hours.
下四分位数位置是第 7.5 个值，Q₁ = 2 + ((7.5 − 4) / 8)
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