📚 Year 8 OCR Statistics: Case Study Practical Exercise | Year 8 OCR 统计：案例分析实战演练
In this article, we will take you through a complete statistics case study based on a school canteen satisfaction survey. You will learn how to design a questionnaire, collect and organise data, create graphs, calculate averages and the range, and even touch on simple probability. This practical exercise mirrors the real-world skills required by the OCR Year 8 Statistics curriculum and will help you feel confident when tackling your own investigations.
在这篇文章中，我们将带你完成一个基于学校食堂满意度调查的完整统计案例。你将学习如何设计问卷、收集和整理数据、绘制图表、计算平均数和极差，甚至触及简单的概率。这个实战演练反映了 OCR Year 8 统计课程所需的真实技能，并将帮助你在面对自己的调查项目时充满信心。
1. The Case Study: Canteen Satisfaction Survey | 案例研究：食堂满意度调查
Imagine you are a Year 8 student asked to investigate how satisfied your peers are with the school canteen. The school wants to understand overall happiness levels and identify areas for improvement. You decide to give each participating student a simple rating scale from 1 to 5, where 1 means ‘Very Dissatisfied’ and 5 means ‘Very Satisfied’. After collecting 100 responses, you have a rich dataset to analyse. This case study will follow every step of the statistical inquiry cycle: posing a question, collecting data, processing and presenting data, and interpreting results.
想象你是一名 Year 8 学生，被要求调查同学们对学校食堂的满意度。学校希望了解整体满意程度并找出需要改进的地方。你决定给每位参与的学生一个简单的评分量表，从 1 到 5，其中 1 表示’非常不满意’，5 表示’非常满意’。在收集了 100 份反馈后，你拥有了一个可供分析的丰富数据集。本案例将遵循统计探究周期的每一个步骤：提出问题、收集数据、处理与呈现数据以及解释结果。
2. Designing the Questionnaire | 设计调查问卷
A well-designed questionnaire is the foundation of any reliable survey. We need to ask a clear, unbiased question. In our case, the question is: ‘On a scale of 1–5, how satisfied are you with the school canteen overall?’ We must also include clear labels for the scale points: 1 = Very Dissatisfied, 2 = Dissatisfied, 3 = Neutral, 4 = Satisfied, 5 = Very Satisfied. The questionnaire should be anonymous to encourage honest answers and we should avoid leading questions like ‘Don’t you think the canteen is great?’ because that would introduce bias.
精心设计的问卷是一切可靠调查的基础。我们需要提出一个清晰、无偏见的问题。在我们的案例中，问题是：’按 1–5 的评分，你对学校食堂的整体满意程度如何？’ 我们还必须为各个评分点注明清晰的标签：1 = 非常不满意，2 = 不满意，3 = 一般，4 = 满意，5 = 非常满意。问卷应匿名以鼓励诚实作答，我们还应避免像’难道你不觉得食堂很棒吗？’这样的诱导性问题，因为那会引入偏差。
- Keep questions simple and specific. / 问题应简单而具体。
- Use a consistent rating scale with clear descriptors. / 使用具有清晰描述的统一点评分量表。
- Make the survey anonymous to reduce social pressure. / 让调查匿名以减少社交压力。
- Avoid leading or double-barrelled questions. / 避免诱导性或双重问题。
3. Data Collection and Sampling | 数据收集与抽样
We need to decide who to ask and how many people to include. Our target population is all students in Year 8. Because interviewing everyone would take too long, we use a sample of 100 students. For the sample to be representative, we must select students fairly. A good method is stratified sampling: we might take a proportional number of students from each tutor group or from each lunch break slot. In this case study, we assume the sample was collected randomly at the canteen entrance across one week, ensuring a mix of days and times. We record each response as a number from 1 to 5 on a tally chart.
我们需要决定询问谁以及包含多少人。目标总体是所有 Year 8 学生。由于采访所有人耗时太长，我们使用一个包含 100 名学生的样本。为了使样本具有代表性，我们必须公平地选取学生。一种好方法是分层抽样：比如按辅导班或午餐时段的比例选取学生。在本案例中，我们假设在一周内的食堂入口处随机收集了样本，确保了日期和时间的混合。我们将每个回答记录为 1 到 5 的数字，登记在画记表上。
After collection, the raw data shows the following frequency distribution: Rating 1 → 8 students, Rating 2 → 12 students, Rating 3 → 30 students, Rating 4 → 35 students, Rating 5 → 15 students. The total is 100. This frequency table is our first step in organising the data.
收集完毕后，原始数据呈现如下频数分布：评分 1 → 8 名学生，评分 2 → 12 名学生，评分 3 → 30 名学生，评分 4 → 35 名学生，评分 5 → 15 名学生。总数为 100。这张频数表是我们整理数据的第一步。
4. Organising Data: Frequency Tables | 整理数据：频数表
A frequency table shows how many times each score appears. We can expand it to include a tally column and a frequency column. For our satisfaction ratings, the completed frequency table looks like this:
频数表显示了每个分值出现的次数。我们可以扩大表格以包含画记列和频数列。就我们的满意度评分而言，完整的频数表如下所示：
| Rating (Score) / 评分 (分值) | Tally / 画记 | Frequency / 频数 |
|---|---|---|
| 1 | IIII III | 8 |
| 2 | IIII IIII II | 12 |
| 3 | IIII IIII IIII IIII IIII IIII | 30 |
| 4 | IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII | 35 |
| 5 | IIII IIII IIII | 15 |
From this table we can quickly see that ratings 3 and 4 are the most common, while very few students gave the lowest rating. The frequency table makes it easier to calculate averages and draw graphs.
从这张表中我们可以快速看到评分 3 和 4 是最常见的，而给出最低评分的学生很少。频数表使得计算平均数和绘制图表更加容易。
5. Drawing Charts: Bar Charts and Pie Charts | 绘制图表：条形图和饼图
Visual representations help us communicate findings effectively. A bar chart is perfect for discrete numerical data like satisfaction ratings. The horizontal axis shows the rating (1 to 5) and the vertical axis shows frequency. Each bar’s height corresponds to the frequency. For our data, bar 4 is the tallest at 35, followed by bar 3 at 30. A pie chart shows the proportion of students who gave each rating. To draw a pie chart, we calculate the angle for each sector: (frequency ÷ total) × 360°. For example, rating 4: (35/100)×360° = 126°. All angles sum to 360°.
可视化呈现有助于我们有效地沟通发现。条形图非常适合像满意度评分这样的离散数值数据。横轴显示评分 (1 到 5)，纵轴显示频数。每根条形的高度对应频数。在我们的数据中，条形 4 最高 (35)，其次是条形 3 (30)。饼图显示了给出每个评分的学生比例。要绘制饼图，我们计算每个扇形的角度：(频数 ÷ 总数) × 360°。例如，评分 4：(35/100)×360° = 126°。所有角度相加等于 360°。
The angle calculations are: Rating 1: (8/100)×360° = 28.8°; Rating 2: 43.2°; Rating 3: 108°; Rating 4: 126°; Rating 5: 54°. When creating these charts by hand, we use a ruler and protractor, label axes clearly, and give the chart a title such as ‘Canteen Satisfaction Ratings’.
角度计算如下：评分 1：(8/100)×360° = 28.8°；评分 2：43.2°；评分 3：108°；评分 4：126°；评分 5：54°。手工绘制这些图表时，我们使用尺子和量角器，清晰地标记坐标轴，并为图表加上诸如’食堂满意度评分’的标题。
6. Calculating Averages: Mode, Median, Mean | 计算平均数：众数、中位数、均值
Averages summarise a data set with a single representative value. The mode is the most frequent score. Here, rating 4 appears 35 times, so the mode is 4. The mode is easy to spot but can be misleading if data is more spread out. The median is the middle value when data is ordered. With 100 responses, the median lies between the 50th and 51st values. Using cumulative frequency: Rating 1 (8), Rating 2 (8+12=20), Rating 3 (20+30=50). So the 50th value is 3 and the 51st is 4. Thus the median is (3+4)/2 = 3.5. The mean uses all values: sum all scores and divide by 100. Sum = (1×8)+(2×12)+(3×30)+(4×35)+(5×15) = 8+24+90+140+75 = 337. Mean = 337 ÷ 100 = 3.37.
平均数用一个代表性的值来概括数据集。众数是出现频率最高的分值。此处，评分 4 出现了 35 次，所以众数是 4。众数容易识别，但如果数据更分散则可能产生误导。中位数是将数据排序后的中间值。有 100 个回答，中位数位于第 50 和第 51 个值之间。使用累积频数：评分 1 (8)，评分 2 (8+12=20)，评分 3 (20+30=50)。所以第 50 个值是 3，第 51 个值是 4。因此中位数是 (3+4)/2 = 3.5。均值使用所有数值：将全部评分求和再除以 100。总和 = (1×8)+(2×12)+(3×30)+(4×35)+(5×15) = 8+24+90+140+75 = 337。均值 = 337 ÷ 100 = 3.37。
Our three averages tell a consistent story: the typical satisfaction level is around 3 to 4, indicating students are moderately satisfied with the canteen. The mean (3.37) is slightly below the median (3.5) because the distribution is slightly skewed by the lower ratings.
我们的三个平均数讲述了一致的情况：典型的满意度水平约在 3 到 4 之间，表明学生对食堂大体满意。均值 (3.37) 略低于中位数 (3.5)，因为分布因较低评分而略显偏斜。
7. Measuring Spread: Range | 计算离散程度：极差
While averages tell us about the centre of the data, the range describes how spread out the values are. The range is simply the difference between the highest and lowest values. In our data set, highest rating = 5, lowest = 1, so range = 5 − 1 = 4. A large range indicates diverse opinions; a small range suggests agreement. Here the range of 4 (on a 5-point scale) shows that students used the full breadth of the scale, so opinions vary considerably. Knowing both the average and the range gives a fuller picture: on average students are moderately satisfied, but there are both very happy and very unhappy students.
虽然平均数告诉我们数据的中心，极差则描述数值的分散程度。极差就是最高值与最低值之差。在我们的数据集中，最高评分 = 5，最低 = 1，所以极差 = 5 − 1 = 4。大的极差表示意见多样；小的极差表明意见一致。此处极差为 4 (在 5 分量表上) 显示学生用足了量表的整个范围，因此意见差异较大。同时了解平均数和极差能描绘出更全面的图景：平均而言学生基本满意，但既有非常开心的学生也有非常不满意的学生。
8. Interpreting Data and Drawing Conclusions | 数据解释与得出结论
Now we must answer our original question: ‘How satisfied are Year 8 students with the school canteen?’ Based on the data, the modal rating is 4 (Satisfied), and the median is 3.5, so the typical sentiment lies between Neutral and Satisfied. The mean (3.37) reinforces this. However, the range of 4 and the presence of 8 students giving a 1 suggest that a minority of students are very unhappy. As a school council representative, you might report: ‘Most students are reasonably satisfied, but around one in five students gave a rating of 1 or 2. The canteen could investigate ways to improve the experience for these students, perhaps by expanding menu options or reducing queue times.’ A strong conclusion always links back to the data and acknowledges any limitations, such as the small sample size or possible bias if only certain year groups were surveyed.
现在我们必须回答最初的问题：’Year 8 学生对学校食堂的满意程度如何？’ 根据数据，众数评分是 4 (满意)，中位数是 3.5，因此典型看法介于一般和满意之间。均值 (3.37) 强化了这一点。然而，极差为 4 且存在 8 名学生给出了 1 分，这表明少数学生非常不满。作为学生会代表，你可以这样报告：’大多数学生基本满意，但大约五分之一的学生给出了 1 或 2 分。食堂可以探索改善这些学生体验的方法，例如增加菜单选项或缩短排队时间。’ 一个强有力的结论总是要联系回数据，并承认任何局限性，例如样本量较小或如果仅调查某些年级组可能存在的偏差。
9. Introduction to Probability: Outcomes Based on Data | 概率入门：基于数据的可能结果
Statistics and probability are closely related. From our survey data, we can estimate the probability that a randomly chosen Year 8 student gives a particular rating. This is experimental probability, based on observed frequencies. The probability of a rating of 4 is 35/100 = 0.35 or 35%. The probability of a student being satisfied or very satisfied (rating 4 or 5) is (35+15)/100 = 50/100 = 0.5. Similarly, the probability of a low rating (1 or 2) is (8+12)/100 = 0.2. These probabilities can help the canteen predict how a new group of students might respond, but remember that experimental probabilities are estimates and can vary with different samples.
统计与概率紧密相关。从我们的调查数据中，我们可以估计随机选择一名 Year 8 学生给出某个特定评分的概率。这是基于观测频数的实验概率。评分为 4 的概率是 35/100 = 0.35 或 35%。一名学生感到满意或非常满意 (评分 4 或 5) 的概率是 (35+15)/100 = 50/100 = 0.5。同样，获得低评分 (1 或 2) 的概率是 (8+12)/100 = 0.2。这些概率可以帮助食堂预测新一批学生可能会如何回应，但要记住实验概率只是估计值，可能随不同样本而变化。
10. Case Study Summary and Practice | 案例总结与练习
This case study has walked you through the complete statistical process: defining a question, designing a fair survey, collecting a sample, organising data in a frequency table, drawing bar and pie charts, calculating the mode, median, mean, and range, interpreting findings, and linking to probability. The key skills tested in OCR Year 8 assessments are all here. To strengthen your understanding, try collecting your own set of data — perhaps on favourite school subjects, hours of sleep, or screen time — and repeat this analytical process. Ask yourself: what is the best average to use? Does the range tell you something important? Can you predict probabilities for different outcomes? Practising with real data turns these techniques into lifelong skills.
本案例带你走完了完整的统计过程：界定问题、设计公平的调查、收集样本、在频数表中整理数据、绘制条形图和饼图、计算众数、中位数、均值和极差、解释发现并联系概率。OCR Year 8 评估中考查的关键技能全都涵盖在内。为加深理解，尝试收集你自己的数据集——比如最喜欢的学校科目、睡眠时长或屏幕时间——并重复这一分析过程。问自己：使用哪种平均数最合适？极差是不是告诉了你一些重要信息？你能预测不同结果的概率吗？用真实数据进行练习，可以将这些技术转化为终身受用的技能。
Published by TutorHao | Statistics Revision Series | aleveler.com
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