📚 Case Study Practice: School Canteen Satisfaction Survey | 学校食堂满意度调查案例分析实战
In this case study, we will step into the role of a student researcher investigating satisfaction with the school canteen. Using real-world data collection, statistical analysis and probability, we will move from a simple question to evidence-based conclusions. All tasks are aligned with the Year 11 OCR Mathematics curriculum, giving you hands-on practice with data handling, representation and interpretation.
在这个案例分析中，我们将扮演学生研究员的角色，调查学校食堂的满意度。通过真实的数据收集、统计分析和概率计算，我们将从一个简单的问题出发，得出基于证据的结论。所有任务都紧扣 Year 11 OCR 数学大纲，帮助你实际操作数据处理、数据表示和解读。
1. Understanding the Problem | 理解问题
Our school leadership team wants to know whether students are happy with the canteen food. We have been asked to design a survey, collect responses and produce a report that summarises satisfaction levels. The key question is: ‘How satisfied are Year 11 students with the canteen, and how does this compare with Year 10?’ We will use a rating scale from 1 (very dissatisfied) to 5 (very satisfied).
学校领导团队想知道学生对食堂的饭菜是否满意。我们被要求设计一份问卷，收集回答并撰写一份总结满意度的报告。核心问题是：”Year 11 学生对食堂的满意度如何，与 Year 10 相比怎样？” 我们将使用从 1（非常不满意）到 5（非常满意）的评分量表。
2. Designing the Questionnaire | 设计调查问卷
A well-designed questionnaire is essential for reliable data. We include a clear rating question, avoid leading language and keep it short. Our question reads: ‘On a scale of 1 to 5, how satisfied are you with the school canteen? (1 = very dissatisfied, 5 = very satisfied).’ We also add a space for optional comments, but only the numerical ratings will be used for statistical analysis.
设计良好的问卷对于获得可靠数据至关重要。我们设计了一个明确的评分问题，避免诱导性语言，并保持简短。我们的问题是：”请用 1 到 5 分评价您对学校食堂的满意度（1 = 非常不满意，5 = 非常满意）。” 我们还添加了可选评论栏，但只有数字评分会用于统计分析。
3. Collecting Data | 收集数据
We survey a random sample of 50 Year 11 students and 50 Year 10 students during form time. Using a random sample helps reduce bias and ensures the results are representative of each year group. The raw data from Year 11 is shown below:
3,4,2,5,3,3,4,2,1,5,3,3,4,2,4,5,3,3,2,1,4,3,5,2,3,3,4,1,5,3,2,4,3,3,5,2,4,3,1,5,3,4,2,3,4,3,5,1,3,2
(These are the 50 ratings we will analyse.)
我们在班会时间随机抽样了 50 名 Year 11 学生和 50 名 Year 10 学生。随机抽样有助于减少偏差，确保结果能代表每个年级。Year 11 的原始数据如下：
3,4,2,5,3,3,4,2,1,5,3,3,4,2,4,5,3,3,2,1,4,3,5,2,3,3,4,1,5,3,2,4,3,3,5,2,4,3,1,5,3,4,2,3,4,3,5,1,3,2
（这是我们将要分析的 50 个评分。）
4. Organising Data and Frequency Tables | 数据整理与频数表
The first step is to tally the ratings into a frequency table. This helps us see the distribution at a glance. For Year 11, the table looks like this:
第一步是将评分制成频数表，这样我们就能一目了然地看到分布情况。Year 11 的频数表如下：
| Rating (评分) | Tally (计数) | Frequency (频数) |
|---|---|---|
| 1 | ~~||||~~ | 5 |
| 2 | ~~||||~~ ~~||||~~ | 10 |
| 3 | ~~||||~~ ~~||||~~ ~~||||~~ | 15 |
| 4 | ~~||||~~ ~~||||~~ ~~||~~ | 12 |
| 5 | ~~||||~~ ~~|||~~ | 8 |
We also add a cumulative frequency column to help find the median and quartiles later. The total number of responses is 50.
我们还添加了累积频数列，以便稍后寻找中位数和四分位数。回答总数为 50。
5. Data Visualisation: Bar Charts and Pie Charts | 数据可视化：条形图与饼图
Visual representations make patterns easier to spot. A bar chart displays the frequency for each rating using bars of equal width. The horizontal axis shows the rating 1 to 5, and the vertical axis shows frequency. Because the data is discrete, we leave gaps between bars. A pie chart shows the proportion of each rating. To construct a pie chart, we calculate the angle for each sector: angle = (frequency ÷ total) × 360°. For a rating of 3, angle = (15 ÷ 50) × 360° = 108°. A pie chart quickly reveals that the largest slice corresponds to rating 3.
可视化展示能让规律更容易被发现。条形图用等宽的长条显示每个评分的频数。横轴表示评分 1 到 5，纵轴表示频数。由于数据是离散的，我们在条形之间保留空隙。饼图则展示每个评分所占的比例。要绘制饼图，我们计算每个扇形的角度：角度 = (频数 ÷ 总数) × 360°。对于评分 3，角度 = (15 ÷ 50) × 360° = 108°。饼图可以迅速显示出评分 3 对应的扇形最大。
6. Measures of Central Tendency: Mean, Median, Mode | 集中趋势度量：平均数、中位数、众数
The mean is calculated by multiplying each rating by its frequency, summing these products and dividing by the total frequency.
平均数通过将每个评分乘以其频数，求和后除以总频数来计算。
Mean = (1×5 + 2×10 + 3×15 + 4×12 + 5×8) ÷ 50 = (5 + 20 + 45 + 48 + 40) ÷ 50 = 158 ÷ 50 = 3.16
The median is the middle value when data is ordered. With 50 responses, the median lies between the 25th and 26th values. Using cumulative frequency (5, 15, 30, …), both the 25th and 26th responses fall in the rating 3 group, so the median is 3. The mode is the most frequent rating, which is also 3. These statistics suggest a central tendency around rating 3, but the mean is slightly higher.
中位数是数据排序后的中间值。有 50 个回答，中位数位于第 25 和第 26 个值之间。利用累积频数（5, 15, 30, …），第 25 和第 26 个回答都落在评分 3 组，因此中位数为 3。众数是出现最频繁的评分，也是 3。这些统计量表明集中趋势在评分 3 附近，但平均数略高。
7. Spread: Range and Quartiles | 离散程度：范围与四分位数
The range is a simple measure of spread: maximum minus minimum. For Year 11, range = 5 − 1 = 4. Quartiles divide the ordered data into four equal parts. The lower quartile (Q₁) is at position (50+1)÷4 = 12.75, which falls in the rating 2 group (cumulative frequency reaches 15 by rating 2). So Q₁ = 2. The upper quartile (Q₃) is at position 3×12.75 = 38.25, which falls in the rating 4 group. Thus Q₃ = 4. The interquartile range (IQR) = Q₃ − Q₁ = 4 − 2 = 2. The IQR tells us that the middle 50% of responses lie within 2 points on the satisfaction scale.
范围是离散程度的简单度量：最大值减最小值。Year 11 的范围 = 5 − 1 = 4。四分位数将排序后的数据分为四等份。下四分位数 (Q₁) 位于 (50+1)÷4 = 12.75 的位置，落入评分 2 组（累积频数在评分 2 达到 15），因此 Q₁ = 2。上四分位数 (Q₃) 位于 3×12.75 = 38.25 的位置，落入评分 4 组，因此 Q₃ = 4。四分位距 (IQR) = Q₃ − Q₁ = 4 − 2 = 2。IQR 告诉我们中间 50% 的回答在满意度量表上分布在 2 分范围内。
8. Box Plots Comparing Year Groups | 箱线图比较不同年级
We also collected data from 50 Year 10 students. Their frequency distribution is: 1: 8, 2: 12, 3: 18, 4: 9, 5: 3. From this we find: minimum=1, Q₁=2, median=3, Q₃=4, maximum=5. However, because the frequencies at the extremes differ, the shape of the box plot changes slightly. Drawing both box plots on the same scale allows a visual comparison. Year 11 has a higher mean and a greater concentration of ratings 4 and 5, while Year 10 shows a lower median (still 3, but with a longer lower whisker) and fewer high ratings. This suggests Year 11 students are more satisfied.
我们还收集了 50 名 Year 10 学生的数据。他们的频数分布为：1: 8, 2: 12, 3: 18, 4: 9, 5: 3。据此我们得到：最小值=1, Q₁=2, 中位数=3, Q₃=4, 最大值=5。然而，由于两端频数不同，箱线图的形状略有变化。在同一尺度上绘制两个箱线图可以直观比较。Year 11 的平均数更高，评分 4 和 5 更集中，而 Year 10 中位数相同（仍为 3，但下方须线更长），高评分较少。这表明 Year 11 学生的满意度更高。
9. Probability Calculations: Random Selection | 概率计算：随机选择
Probability helps quantify uncertainty. If we select one Year 11 student at random, the probability they gave a rating of at least 4 is P(4 or 5) = (12+8)/50 = 20/50 = 0.4. The probability a randomly chosen Year 11 student is dissatisfied (rating 1 or 2) is (5+10)/50 = 15/50 = 0.3. We can also compare across year groups: for Year 10, P(at least 4) = (9+3)/50 = 12/50 = 0.24. The difference of 0.16 suggests a meaningful gap in satisfaction.
概率有助于量化不确定性。如果随机选取一名 Year 11 学生，其评分至少为 4 的概率为 P(4 或 5) = (12+8)/50 = 20/50 = 0.4。随机选取的 Year 11 学生不满意（评分 1 或 2）的概率为 (5+10)/50 = 15/50 = 0.3。我们还可以跨年级比较：Year 10 学生评分至少为 4 的概率为 (9+3)/50 = 12/50 = 0.24。0.16 的差距表明满意度存在显著差异。
10. Interpreting the Results and Making Decisions | 解读结果并做出决策
The analysis shows that Year 11 students are moderately satisfied, with a mean of 3.16 and a median of 3. The IQR of 2 indicates some variation, but opinions are not extremely spread out. Compared with Year 10, Year 11 has a noticeably higher proportion of top ratings. The school leadership can conclude that satisfaction is reasonable but could be improved, especially for Year 10. They might consider investigating why Year 10 students are less positive, perhaps through focus groups or menu changes.
分析显示，Year 11 学生满意度中等，平均数为 3.16，中位数为 3。IQR 为 2 表明存在一定差异，但意见并不极其分散。与 Year 10 相比，Year 11 获得高评分的比例明显更高。学校领导可以得出结论：满意度尚可，但仍有提升空间，尤其是 Year 10。他们可以考虑调查 Year 10 学生评分较低的原因，或许通过焦点小组或调整菜单。
11. Reflection and Evaluation of the Case Study | 案例反思与评估
Our investigation was systematic, but we should critically evaluate its limitations. The sample size of 50 per year group is fairly small; a larger sample would increase reliability. The rating scale is subjective – what one student calls a ‘4’ might be a ‘3’ for another. We also assumed the samples were truly random, but there may have been non-response bias. Despite these limitations, the case study illustrates how OCR Mathematics skills can be applied to real-life decision-making, from designing a data collection tool to drawing conclusions.
我们的调查具有系统性，但我们应该批判性地评估其局限性。每个年级 50 人的样本量较小；更大的样本会提高可靠性。评分量表带有主观性——一位学生眼中的”4″可能在另一位学生看来是”3″。我们还假设样本是真正随机的，但可能存在无应答偏差。尽管有这些局限，本案例展示了如何将 OCR 数学技能应用于现实决策，从设计数据收集工具到得出结论。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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