📚 Case Study in Statistics: Practical Exercise | 统计学案例分析：实战演练
In this revision guide, we will walk through a complete statistical investigation. Imagine your school surveyed 30 Year 8 students to find out how many hours they spend reading each week and their latest mathematics exam scores. The goal is to explore whether there is a relationship between reading time and performance in maths. You will act as a data analyst, applying the skills you have learned in Edexcel Year 8 statistics: collecting data, organising frequencies, drawing charts, calculating averages, and making predictions. This hands-on case study will solidify your understanding of statistical concepts and help you ace your exams.
在本复习指南中，我们将完成一次完整的统计调查。假设你的学校调查了 30 名八年级学生，了解他们每周花在阅读上的小时数以及他们最近的数学考试成绩。目的是探究阅读时间与数学表现之间是否存在关系。你将扮演数据分析师，运用在爱德思八年级统计学中学到的技能：收集数据、整理频数、绘制图表、计算平均数以及做出预测。这个动手案例学习将巩固你对统计学概念的理解，帮助你在考试中取得优异成绩。
1. Designing the Survey and Collecting Data | 设计调查并收集数据
Before any analysis can begin, we must decide what data to collect and how to gather it. For this case study, two variables are recorded: the number of hours spent reading per week (a continuous numerical variable) and the mathematics test score as a percentage (also numerical). A simple questionnaire was given to a random sample of 30 Year 8 pupils to avoid bias. Ensuring random sampling is crucial; otherwise, the results may not represent the whole year group. Students were asked to estimate their reading hours honestly and provide their most recent maths percentage.
在分析开始之前，我们必须决定收集哪些数据以及如何收集。在本案例中，记录了两个变量：每周阅读小时数（连续数值变量）和数学测试成绩百分比（也是数值变量）。我们向随机抽取的 30 名八年级学生发放了一份简单问卷，以避免偏差。确保随机抽样至关重要；否则，结果可能无法代表整个年级。要求学生诚实估计阅读时间，并提供最近一次数学成绩百分比。
2. Raw Data Table | 原始数据表
The raw data collected from 30 students is shown in the table below. Each row corresponds to one pupil. The first column gives the number of hours spent reading per week, and the second column gives the corresponding mathematics score out of 100.
从 30 名学生收集的原始数据如下表所示。每一行对应一名学生。第一列是每周阅读小时数，第二列是相应的数学成绩（满分 100）。
| Reading Hours (h) | Maths Score (%) |
|---|---|
| 2 | 45 |
| 5 | 78 |
| 1 | 32 |
| 8 | 92 |
| 3 | 55 |
| 6 | 85 |
| 4 | 68 |
| 7 | 90 |
| 0.5 | 25 |
| 4.5 | 70 |
| 3.5 | 60 |
| 5.5 | 80 |
| 2.5 | 50 |
| 6.5 | 88 |
| 7.5 | 95 |
| 1.5 | 35 |
| 8.5 | 96 |
| 9 | 98 |
| 3 | 58 |
| 4 | 72 |
| 5 | 76 |
| 6 | 84 |
| 7 | 89 |
| 8 | 94 |
| 2 | 48 |
| 1 | 30 |
| 4 | 66 |
| 3 | 62 |
| 5.5 | 81 |
| 0 | 20 |
Take a moment to scan the table. Do you notice any pattern? It seems that students with very low reading hours often have lower scores, but we need proper statistical tools to confirm this.
仔细浏览这个表格。你发现什么规律了吗？似乎阅读时间极低的学生往往分数较低，但我们需要合适的统计工具来验证这一点。
3. Grouped Frequency Distributions | 分组频数分布
To see the spread of reading habits, we group the continuous data into class intervals. Let the classes be 0 ≤ h < 2, 2 ≤ h < 4, 4 ≤ h < 6, 6 ≤ h < 8, and 8 ≤ h ≤ 10. By tallying the raw data, we obtain the following grouped frequency table. This helps us understand how common each range of reading time is among the 30 students.
为了观察阅读习惯的分布，我们将连续数据分组到区间内。令组距为 0 ≤ h < 2, 2 ≤ h < 4, 4 ≤ h < 6, 6 ≤ h < 8 和 8 ≤ h ≤ 10。通过整理原始数据，我们得到下面的分组频数表。这有助于我们理解每个阅读时间区间在 30 名学生中的普遍程度。
| Reading Hours (h) | Frequency (f) |
|---|---|
| 0 ≤ h < 2 | 5 |
| 2 ≤ h < 4 | 7 |
| 4 ≤ h < 6 | 8 |
| 6 ≤ h < 8 | 6 |
| 8 ≤ h ≤ 10 | 4 |
The modal class is 4 ≤ h < 6, since it has the highest frequency of 8. This tells us that the most common weekly reading time is between 4 and 6 hours.
众数组是 4 ≤ h < 6，因为它的频数最高，为 8。这告诉我们，每周最常见的阅读时间在 4 至 6 小时之间。
4. Drawing Bar Charts | 绘制条形图
A bar chart can be drawn to display the grouped frequency data. On the horizontal axis, we write the class intervals; on the vertical axis, the frequency. The height of each bar represents the number of students in that interval. When you sketch this by hand or using software, label the axes clearly and give the chart a title, such as ‘Weekly Reading Hours of Year 8 Students’. Bars must be separated by small gaps because the data is grouped, not categorical.
可以绘制条形图来展示分组频数数据。水平轴上标出组距区间；垂直轴上标出频数。每个条形的高度代表该区间内的学生人数。当你手工或在软件中绘制时，要清楚地标注坐标轴，并为图表加上标题，例如“八年级学生每周阅读小时数”。由于数据是分组而非分类的，条形之间应留有微小间隙。
From the bar chart, you can quickly identify the most frequent range and see how the frequencies taper off towards the extremes. This visual aid makes the distribution pattern clearer than just looking at numbers.
通过条形图，你可以迅速找出最常见的区间，并看到频数如何在两端逐渐减少。这种可视化辅助手段比单纯看数字更能清楚地展现分布模式。
5. Scatter Graphs and Correlation | 散点图与相关性
To investigate the relationship between reading hours and maths scores, we plot a scatter graph. Plot each student as a point, with reading hours on the x-axis and maths score on the y-axis. For example, the first student is plotted at (2, 45), the second at (5, 78), and so on. After plotting all 30 points, you will notice a general trend: as reading hours increase, the maths score tends to rise. This suggests a positive correlation.
为了探究阅读小时数与数学成绩之间的关系，我们绘制散点图。将每个学生表示为一个点，阅读小时数在 x 轴，数学成绩在 y 轴。例如，第一个学生画在 (2, 45)，第二个在 (5, 78)，以此类推。绘制完所有 30 个点后，你会注意到一个大致趋势：阅读小时数增加，数学成绩往往上升。这表明存在正相关。
The points are not perfectly in a straight line, so the correlation is moderate, not strong. You could add a line of best fit by eye, roughly passing through the middle of the points. The line slopes upward, confirming the positive relationship. Correlation does not imply causation, however; we cannot simply say more reading causes higher scores without deeper investigation.
这些点并非完全落在一条直线上，因此相关程度中等，而非强相关。你可以凭目测添加一条最佳拟合线，大致穿过点的中心。这条线向上倾斜，证实了正相关关系。然而，相关性不代表因果关系；未经深入调查，我们不能简单地说多阅读就能导致高分。
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Published by TutorHao | Year 8 统计 Revision Series | aleveler.com
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