📚 Case Study in Statistics: Practical Exercise | 统计案例分析实战演练
Statistics comes to life when we apply it to real-world situations. In this practical exercise, we will work through a complete case study: a school canteen survey. By following the statistical enquiry cycle, you will learn how to plan, collect, process and interpret data, just like a real data analyst.
当我们将统计学应用到现实情境中时，它就变得生动起来。在这次实践练习中，我们将完成一个完整的案例分析：学校食堂调查。通过遵循统计探究循环，你将学习如何像真正的数据分析师一样计划、收集、处理并解读数据。
1. Introduction to the Case Study | 案例导入
Imagine you are a student researcher at a secondary school. The school leadership wants to improve the canteen menu and has asked your team to investigate students' lunch preferences and satisfaction. You decide to carry out a survey to gather the necessary information.
假设你是一所中学的学生研究员。学校领导层希望改善食堂菜单，因此要求你的团队调查学生的午餐偏好和满意度。你们决定开展一项调查来收集必要的信息。
Your study will focus on two main questions: 'Which lunch option is the most popular?' and 'How satisfied are students with the current meals?' You will collect two types of data — categorical data (favourite lunch type) and numerical data (satisfaction rating on a scale from 1 to 5). This case study will walk you through the entire statistical process.
你的研究将聚焦两个主要问题：“哪种午餐最受欢迎？”以及“学生对当前餐食的满意度如何？”你将收集两类数据 — 分类数据（最喜爱的午餐类型）和数值数据（1 到 5 分的满意度评分）。本案例将带你走完整个统计流程。
2. Defining the Problem and Objectives | 定义问题与目标
Before collecting any data, it is essential to define clear research questions. In our case, the main problems are: determining the most preferred lunch option among students and assessing their overall satisfaction level. From these, we derive specific objectives: to summarise preferences using frequency tables and charts, to calculate averages and spread for ratings, and to use the data to make simple probability statements.
在收集任何数据之前，必须明确研究问题。我们的案例中，主要问题是：确定学生最偏爱的午餐选择，以及评估他们的总体满意度。由此我们制定具体目标：利用频数表和图表汇总偏好，计算评分的平均数和离散程度，并利用数据做出简单的概率陈述。
By setting these objectives, we ensure that the survey is focused and that the analysis will directly answer the questions posed by the school. This step aligns with the 'Plan' phase of the statistical enquiry cycle.
通过设定这些目标，我们确保调查有针对性，且分析将直接回答学校提出的问题。这一步符合统计探究循环中的“计划”阶段。
3. Collecting Data | 数据收集
We designed a short questionnaire with two questions. The first question asked: 'What is your favourite lunch type? (Pizza, Burger, Salad, Noodles)'. The second asked: 'On a scale of 1 to 5, how would you rate your overall satisfaction with the school lunches?' (1 = very dissatisfied, 5 = very satisfied).
我们设计了一份简短问卷，包含两个问题。第一个问题询问：“你最喜欢的午餐类型是什么？（披萨、汉堡、沙拉、面条）”。第二个问题询问：“以 1 到 5 分打分，你对学校午餐的整体满意度如何？”（1 = 非常不满意，5 = 非常满意）。
To obtain a representative sample, we used simple random sampling. We assigned a number to every student in the school and then used a random number generator to select 50 participants. This method helps avoid bias and gives every student an equal chance of being chosen. All responses were recorded anonymously to encourage honest answers.
为了获得代表性样本，我们使用了简单随机抽样。我们给学校每位学生分配了编号，然后用随机数生成器选出了 50 名参与者。该方法有助于避免偏见，并让每位学生都有均等的机会被选中。所有回答均匿名记录，以鼓励真实作答。
4. Organising Data: Frequency Tables | 数据整理：频数表
After collecting the questionnaires, the raw data were organised into frequency tables. The first table summarises the favourite lunch types.
收集完问卷后，我们将原始数据整理成频数表。第一张表汇总了最喜爱的午餐类型。
| Lunch Type | Frequency |
|---|---|
| Pizza | 18 |
| Burger | 12 |
| Salad | 10 |
| Noodles | 10 |
From this table we can see that Pizza was the most common choice with 18 votes, while Salad and Noodles each received 10 votes. The second table shows the satisfaction ratings along with cumulative frequencies, which will be useful for finding the median later.
从该表可以看出，披萨是最普遍的选择，获得 18 票，沙拉和面条各得到 10 票。第二张表展示了满意度评分以及累计频数，这对稍后求中位数很有用。
| Rating | Frequency | Cumulative Frequency |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 8 | 11 |
| 3 | 15 | 26 |
| 4 | 16 | 42 |
| 5 | 8 | 50 |
Notice that the cumulative frequency for rating 3 is 26, meaning 26 students gave a rating of 3 or below. The total number of responses is 50, confirming all students answered both questions.
请注意，评分 3 的累计频数为 26，意味着 26 名学生给出了 3 分或更低。回答总数为 50，确认所有学生都回答了两个问题。
5. Representing Data: Charts | 数据表示：图表
To visualise the lunch preferences, we would draw a bar chart. On the horizontal axis we place the lunch types, and on the vertical axis the frequency. The bar for Pizza would be the tallest, reaching 18. Burger would be shorter at 12, while Salad and Noodles would have bars of equal height at 10. A bar chart makes it easy to compare categories at a glance.
为了可视化午餐偏好，我们会绘制条形图。横轴表示午餐类型，纵轴表示频数。披萨的条形最高，达到 18。汉堡较矮，为 12，而沙拉和面条的条形高度相等，均为 10。条形图便于一眼比较各个类别。
r the lunch type data to show proportions. In a pie chart, each category is represented as a slice; the angle for Pizza would be (18/50) × 360° = 129.6°, Burger 86.4°, Salad 72°, and Noodles 72°. This helps viewers understand the relative popularity of each option.
对于满意度评分，也可以使用条形图，横轴标出 1 到 5 的评分。此外，我们还可以为午餐类型数据绘制饼图以展示比例。在饼图中，每个类别用一个扇形表示；披萨的角度为 (18/50) × 360° = 129.6°，汉堡 86.4°，沙拉 72°，面条 72°。这有助于观察者理解各项选择的相对受欢迎程度。
6. Measures of Central Tendency | 集中趋势度量
Now we calculate summary statistics for the satisfaction ratings. First, the mean. We multiply each rating by its frequency and sum the products.
现在我们计算满意度评分的汇总统计量。首先是平均数。我们将每个评分乘以其频数，然后求积之和。
Mean = (Sum of all ratings) ÷ Number of students
8. Therefore, Mean = 168 ÷ 50 = 3.36. So the average satisfaction rating is 3.36 out of 5.
求和 = (1 × 3) + (2 × 8) + (3 × 15) + (4 × 16) + (5 × 8) = 3 + 16 + 45 + 64 + 40 = 168。因此，平均数 = 168 ÷ 50 = 3.36。故平均满意度评分为 3.36 分（满分 5 分）。
Next, the median. With 50 data points, the median lies between the 25th and 26th values. Looking at the cumulative frequency, up to rating 3 we have 26 students. That means the 25th and 26th students both fall in rating 3. Hence, the median is 3.
接下来是中位数。数据集有 50 个点，中位数位于第 25 和第 26 个值之间。查看累计频数，评分达到 3 时已有 26 名学生。这表示第 25 和第 26 名学生都落在评分 3。因此中位数为 3。
Finally, the mode — the most frequent rating. Rating 4 appears 16 times, more than any other rating, so the mode is 4. This tells us that the most common satisfaction score was 4, which indicates a generally positive feeling.
最后是众数——出现频率最高的评分。评分 4 出现了 16 次，多于任何其他评分，因此众数为 4。这表明最常见的满意度得分是 4 分，反映出总体积极的感受。
7. Measures of Spread: Range | 离散度量：极差
To measure how spread out the satisfaction ratings are, we calculate the range. The range is the difference between the highest and lowest values.
为了度量满意度评分的离散程度，我们需要计算极差。极差是最高值与最低值之差。
Range = Highest rating − Lowest rating
s tells us that all ratings lie within a span of 4 points. Combined with the mean and median, we now have a good sense of the distribution — are clustered around 3 and 4, but there are still some students who are very dissatisfied.
在我们的数据中，最高评分为 5，最低为 1。因此极差 = 5 − 1 = 4。这告诉我们所有评分都落在一个 4 分的跨度内。结合平均数和中位数，我们现在对分布有了较好的了解——大多数评分集中在 3 和 4 附近，但仍有部分学生非常不满意。
8. Interpreting Results | 解读结果
Having performed the analysis, we can now answer the original questions. The most popular lunch type is clearly Pizza, chosen by 18 out of 50 students. Burger comes second with 12, while Salad and Noodles share third place with 10 each. The bar chart and pie chart would visually reinforce this.
完成分析后，我们现在可以回答最初的问题了。最受欢迎的午餐类型显然是披萨，50 名学生中有 18 人选择。汉堡以 12 票位列第二，沙拉和面条各以 10 票并列第三。条形图和饼图将在视觉上强化这一结论。
Regarding satisfaction, the average rating of 3.36 suggests a moderate level of contentment. The median of 3 indicates that half the students gave a rating of 3 or less, while the other half gave 4 or 5. The mode of 4 highlights that the most frequent response was fairly high. However, the range of 4 and the presence of ratings 1 and 2 show that a noticeable minority is unhappy. The school might use this insight to explore what those students dislike and consider menu adjustments.
关于满意度，
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