📚 Case Study Practice: Hands-on Statistics Exercises | 案例分析实战演练
In Year 8 Statistics, applying knowledge to real-life scenarios is the best way to build confidence. This article presents a complete case study where you will organise data, draw graphs, calculate averages and range, and then interpret your findings—just like a real statistician.
在八年级统计课程中，将知识应用到现实生活场景是建立信心的最佳方式。本文将呈现一个完整的案例研究，你将整理数据、绘制图表、计算平均数与极差，然后解释你的发现——就像真正的统计学家一样。
1. Introduction to the Case Study | 案例简介
sfaction rating (from 1 = very dissatisfied to 5 = very satisfied) and their favourite fruit. Your task is to analyse this data as part of a statistics project.
学校食堂想了解学生的满意度和最喜欢的水果。一位八年级学生调查了 30 位同学，记录了他们从 1（很不满意）到 5（很满意）的满意度评分以及他们最喜欢的水果。你的任务是作为统计项目的一部分分析这些数据。
This case study mirrors the CAIE approach: collect, organise, display, and interpret data. By working through each step, you will reinforce key Year 8 skills and learn to think like a data analyst. The final output will be a clear, evidence-based report that the canteen can use.
这个案例研究反映了 CAIE 的方法：收集、整理、展示和解释数据。通过逐步实践，你将巩固八年级的关键技能，并学会像数据分析师一样思考。最终成果将是一份清晰、基于证据的报告，可供食堂使用。
2. Collecting the Data | 收集数据
The student used a simple questionnaire to gather two variables from each classmate: satisfaction score (numerical, discrete) and favourite fruit (categorical). Good data collection avoids bias by asking neutral questions and ensuring a representative sample, even if it is a convenience sample of classmates.
该学生使用了一份简单的问卷从每位同学那里收集两个变量：满意度分数（数值型、离散）和最喜欢的水果（类别型）。良好的数据收集通过提出中立问题并确保样本具有代表性来避免偏差，即使这是一个便利抽样的同学样本。
The raw data for satisfaction scores is shown below (30 responses):
4, 3, 2, 5, 4, 3, 3, 4, 2, 1, 3, 4, 5, 4, 3, 2, 4, 3, 4, 5, 3, 2, 1, 4, 3, 4, 2, 3, 4, 5
满意度评分的原始数据如下（30 个回复）：
4, 3, 2, 5, 4, 3, 3, 4, 2, 1, 3, 4, 5, 4, 3, 2, 4, 3, 4, 5, 3, 2, 1, 4, 3, 4, 2, 3, 4, 5
The favourite fruit data was recorded as a tally before being counted: Apple 8, Banana 5, Orange 7, Grape 6, Strawberry 4. Always check that the counts sum to the sample size (30) to avoid errors.
最喜欢的水果数据先被记录为计数符号，然后计数：苹果 8，香蕉 5，橙子 7，葡萄 6，草莓 4。务必检查计数之和是否等于样本量（30），以避免错误。
3. Organising Data into Frequency Tables | 将数据整理成频数表
A frequency table turns messy raw data into a clear summary. For the satisfaction scores, list each possible score from 1 to 5 and count how many times it appears. This helps us see the distribution at a glance.
频数表能将杂乱的原始数据转变为清晰的汇总。对于满意度分数，列出从 1 到 5 的每个可能分数，并统计其出现的次数。这有助于我们一眼看清分布情况。
| Satisfaction score | Tally | Frequency |
| 1 | || | 2 |
| 2 | ||||| | 6 |
| 3 | ||||| ||| | 8 |
| 4 | ||||| |||| | 9 |
| 5 | ||||| | 5 |
| Total | 30 |
For the categorical fruit data, a frequency table simply lists each fruit and its count. You can add a relative frequency column (e.g. fraction or percentage) to make comparisons easier, but here frequencies are small enough to work with directly.
对于类别型的水果数据，频数表只需列出每种水果及其计数。你可以添加一个相对频数列（如分数或百分比）以便比较，但这里的频数足够小，可以直接使用。
| Fruit | Frequency |
| Apple | 8 |
| Banana | 5 |
| Orange | 7 |
| Grape | 6 |
| Strawberry | 4 |
4. Choosing the Right Visual Display | 选择合适的可视展示
Different data types call for different graphs. For categorical data like favourite fruit, a bar chart or a pie chart works well to compare frequencies. For discrete numerical data like satisfaction scores, a bar chart (with gaps) is appropriate, or a vertical line graph could be used. We avoid histograms here because the data is not continuous.
不同的数据类型需要不同的图表。对于类别型数据，如最喜欢的水果，条形图或饼图能很好地比较频数。对于离散的数值型数据，如满意度评分，条形图（条间有间隔）是合适的，也可以使用垂直线图。这里我们避免使用直方图，因为数据不是连续的。
- Use a bar chart to compare counts of satisfaction scores — each bar represents a score from 1 to 5.
- Use a pie chart to show the proportion of each fruit choice — each slice corresponds to a fruit's frequency.
- 使用条形图比较满意度评分的计数——每个条形代表 1 到 5 的一个分数。
- 使用饼图显示每种水果选择的比例——每个扇区对应一种水果的频数。
each sector by multiplying the fraction by 360°. For example, Apple: (8/30) × 360° = 96°.
务必给坐标轴贴上标签并为图表加上标题。对于饼图，计算每个扇区的角度需用分数乘以 360°。例如，苹果：(8/30) × 360° = 96°。
5. Drawing Bar Charts and Pie Charts | 绘制条形图和饼图
To construct the satisfaction bar chart, draw a horizontal axis for the score (1 to 5) and a vertical axis for frequency (0 to 10). Draw bars of equal width with small gaps between them. The heights are: 2 (score 1), 6 (score 2), 8 (score 3), 9 (score 4), 5 (score 5). This shows that score 4 is the most common and score 1 the least common.
要构建满意度条形图，画一条水平轴表示分数（1 到 5），一条垂直轴表示频数（0 到 10）。绘制等宽的条形，条间留有小间隙。条形高度为：2（分数 1），6（分数 2），8（分数 3），9（分数 4），5（分数 5）。这表明分数 4 最常见，分数 1 最不常见。
k sum: 96+60+84+72+48 = 360°). Colour each sector differently and include a legend. The pie chart instantly shows that Apple is the favourite and Strawberry the least popular.
对于饼图，使用以下角度将圆分成五个扇区：苹果 96°，香蕉 60°，橙子 84°，葡萄 72°，草莓 48°。（验证总和：96+60+84+72+48 = 360°）。将每个扇区涂上不同颜色并添加图例。饼图即时显示苹果最受欢迎，草莓最不受欢迎。
Drawing accurate diagrams is a key exam skill. Use a ruler, a pencil, and a protractor for pie charts. In digital work, you can use spreadsheet software, but understanding the manual method is essential for CAIE assessments.
绘制精确的图表是一项关键的考试技能。画饼图时使用直尺、铅笔和量角器。在数字化作业中，你可以使用电子表格软件，但理解手工方法对 CAIE 评估至关重要。
6. Calculating the Mean | 计算平均数
The mean (average) satisfaction score gives a single value that summarises the typical response. To calculate it, multiply each score by its frequency, sum these products, and divide by the total number of responses (30).
平均满意度分数提供了一个单一数值，概括了典型的回答。要计算它，将每个分数乘以其频数，将这些乘积相加，然后除以回答总数（30）。
Mean = (1×2 + 2×6 + 3×8 + 4×9 + 5×5) ÷ 30
Work step by step:
1×2 = 2
2×6 = 12
3×8 = 24
4×9 = 36
5×5 = 25
Sum = 2+12+24+36+25 = 99
逐步计算：
1×2 = 2
2×6 = 12
3×8 = 24
4×9 = 36
5×5 = 25
总和 = 2+12+24+36+25 = 99
Mean = 99 ÷ 30 = 3.3
The mean satisfaction score is 3.3 (out of 5). This suggests that overall, students are between 'neutral' and 'satisfied', leaning slightly towards satisfied.
平均满意度分数为 3.3（满分 5）。这表明总体而言，学生介于“一般”和“满意”之间，略微偏向满意。
7. Finding Median and Mode | 求中位数与众数
he frequency table: 1,1, 2,2,2,2,2,2, 3,3,3,3,3,3,3,3, 4,4,4,4,4,4,4,4,4, 5,5,5,5,5. Count to the 15th value — it is 3, and the 16th is also 3. So the median is (3+3)÷2 = 3.
中位数是数据排序后的中间值。有 30 个评分，中位数位于第 15 和第 16 个值之间。利用频数表将分数按顺序排列：1,1, 2,2,2,2,2,2, 3,3,3,3,3,3,3,3, 4,4,4,4,4,4,4,4,4, 5,5,5,5,5。数到第 15 个值——它是 3，第 16 个也是 3。因此中位数为 (3+3)÷2 = 3。
The mode is the value with the highest frequency. From the table, score 4 appears 9 times, which is the largest frequency. Hence the mode is 4. This tells us the most common satisfaction level is 'satisfied'.
众数是出现频数最高的值。从表中可知，分数 4 出现了 9 次，这是最大的频数。因此众数是 4。这告诉我们最常见的满意度是“满意”。
Having three averages gives a richer picture: the mean is 3.3, median 3, mode 4. The mean is pulled slightly higher by the few scores of 5, while the median is robust. The mode highlights the most typical feeling.
拥有三种平均数能提供更丰富的图景：平均数为 3.3，中位数为 3，众数为 4。平均数被少数几个 5 分略微拉高了，而中位数是稳健的。众数则凸显了最典型的感受。
8. Understanding the Range | 理解极差
satisfaction scores, the highest score is 5 and the lowest is 1, so the range = 5 − 1 = 4.
极差衡量数据的分散程度。它是最大值与最小值之差。对于满意度分数，最高分是 5，最低分是 1，因此极差 = 5 − 1 = 4。
A range of 4 on a 1-to-5 scale tells us that opinions are spread across the whole scale, but a small range doesn't always mean agreement—it just tells you the distance between extremes. Always combine range with an average to describe data properly.
在 1 到 5 的尺度上，极差为 4 告诉我们意见遍布于整个尺度，但极差小并不总意味着意见一致——它只告诉你极端值之间的距离。始终将极差与某种平均数结合起来才能恰当地描述数据。
If we compare with the fruit data, range is not meaningful for categorical data. For numerical data, it helps us see whether satisfaction is concentrated or widely varied. Here, scores range from very dissatisfied to very satisfied, which may prompt the canteen to investigate why.
若与水果数据相比，极差对于类别型数据没有意义。对于数值型数据，它帮助我们了解满意度是集中还是差异很大。此处，评分从很不满意到很满意都有，这可能促使食堂调查原因。
9. Interpreting Results and Drawing Conclusions | 解释结果并得出结论
Now we tell the story behind the numbers. The most popular fruit is Apple, chosen by about 27% of students. The canteen should consider offering more apple-based options. Meanwhile, Strawberry is the least favourite, so strawberry products might be reduced or paired with popular fruits.
现在我们讲述数字背后的故事。最受欢迎的水果是苹果，约 27% 的学生选择了它。食堂应考虑提供更多以苹果为基础的选择。同时，草莓最不受欢迎，因此草莓产品可以减少或与受欢迎的水果搭配。
For satisfaction, the average score is just above neutral at 3.3 and the mode is 4, suggesting many students are satisfied but some are quite unhappy (two scores of 1). The range of 4 confirms that there is a spread of views. The canteen could do a follow-up survey asking for reasons why students gave scores of 1 or 2.
在满意度方面，平均分刚好高于中性 3.3，且众数为 4，表明许多学生是满意的，但有些人相当不满（两个 1 分）。极差为 4 证实了观点的分散。食堂可以跟进调查，询问学生给出 1 分或 2 分的原因。
Always support conclusions with numerical evidence. Phrases like 'the data shows...' or 'with a mean of 3.3 and a mode of 4, we conclude...' make your report stronger and meet CAIE communication standards.
始终用数字证据支持结论。像“数据显示……”或“根据 3.3 的平均数和 4 的众数，我们推断……”这样的表述能让你的报告更有力，并符合 CAIE 的交流标准。
10. Common Pitfalls in Case Studies | 案例分析中的常见陷阱
Many students lose marks by misreading the data or using the wrong graph. Avoid these frequent mistakes:
许多学生因误读数据或使用错误图表而失分。避免这些常见错误：
- Using a pie chart for numerical scores — pie charts are for categorical data, not ordered numerical data.
- Forgetting to check that mean calculation includes all frequencies correctly.
- osition: for n=30, median is the average of the 15th and 16th values, not simply the 15th.
- Omitting units or axis labels on graphs.
- 对数值分数使用饼图——饼图用于类别数据，而非有序的数值数据。
- 忘记检查平均数计算是否正确包含所有频数。
- 混淆中位数的位置：当 n=30 时，中位数是第 15 和第 16 个值的平均数，而不仅仅是第 15 个。
- 在图表中遗漏单位或坐标轴标签。
Another trap is forgetting to interpret the range in context: just stating 'range = 4' is not enough; say what that implies about the spread of satisfaction.
另一个陷阱是忘记在上下文中解释极差：仅仅陈述“极差 = 4”是不够的；要说明这对满意度的分散程度意味着什么。
11. Practice Exercise: Your Turn | 练习：轮到你了
Now test your skills with a new mini dataset. A different Year 8 class surveyed 25 students about the number of books read last month. The results were:
3, 5, 2, 1, 4, 3, 2, 5, 6, 1, 3, 4, 2, 5, 3, 1, 4, 3, 2, 6, 3, 4, 2, 1, 3
现在用一个新的小数据集测试你的技能。另一个八年级班级调查了 25 名学生上个月阅读的书籍数量。结果如下：
3, 5, 2, 1, 4, 3, 2, 5, 6, 1, 3, 4, 2, 5, 3, 1, 4, 3, 2, 6, 3, 4, 2, 1, 3
Complete these tasks:
a) Create a frequency table for the data.
b) Calculate the mean number of books read (show your working).
c) Find the median and the mode.
d) Determine the range and explain what it tells you.
e) Draw a suitable bar chart to display the data.
完成以下任务：
a) 为数据创建一个频数表。
b) 计算阅读的书籍平均数（写出计算过程）。
c) 求中位数和众数。
d) 确定极差并解释它说明了什么。
e) 绘制一个合适的条形图来展示数据。
. Median = 3 (the 13th value); Mode = 3. Range = 6−1 = 5; a wide spread of reading habits.
答案：频数表（书籍数量：1→4, 2→6, 3→8, 4→4, 5→3, 6→2）。平均数 = (1×4+2×6+3×8+4×4+5×3+6×2)÷25 = 77÷25 = 3.08。中位数 = 3（第 13 个值）；众数 = 3。极差 = 6−1 = 5；阅读习惯差异很大。
12. Summary | 总结
Case studies in statistics bridge the gap between textbook theory and real-world application. By methodically organising data, choosing correct visual displays, and calculating the mean, median, mode and range, you can extract meaningful insights. Always end with a clear interpretation backed by numbers.
统计中的案例研究弥合了教科书理论与现实应用之间的鸿沟。通过有条理地整理数据、选择正确的可视展示，并计算平均数、中位数、众数和极差，你可以提取有意义的见解。始终以数字为支撑给出清晰的解释作为结尾。
Remember that CAIE questions often present a scenario and ask you to 'write a short report'. Practising with data like the canteen survey builds exactly that skill. The more you practise organising raw data and telling its story, the more confident you become.
请记住，CAIE 的题目常常给出一个场景并要求你“写一份简短的报告”。用像食堂调查这样的数据练习正好能培养这项技能。你越多地练习整理原始数据并讲述其故事，你就会变得越自信。
Published by TutorHao | Statistics Revision Series | aleveler.com
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