📚 Case Study Practical Exercises | 案例分析实战演练
In Year 10 OCR Statistics, applying theory to real-world scenarios is the surest way to master the subject. This article walks you through a series of practical case studies, each designed to strengthen your skills in data collection, representation, measures of centre and spread, probability, and bivariate analysis. Use these exercises to test your understanding, identify weaknesses, and build exam confidence.
在10年级OCR统计学中，将理论应用于真实场景是掌握这门学科最可靠的方法。本文将通过一系列实用案例研究，带你逐一练习数据收集、图表展示、集中量与离散量、概率以及双变量分析等技能。运用这些练习来检验理解、发现薄弱点，并建立考试信心。
1. Understanding the Role of Case Studies in Statistics | 理解案例研究在统计中的作用
Case studies are extended problems that mimic the statistical enquiry cycle: stating a hypothesis, collecting data, analysing it, and drawing conclusions. In the OCR exam, you may be given a description of a survey or experiment and asked to critique the method, suggest improvements, or perform calculations on the data provided. Getting comfortable with case studies early will help you tackle the longer, more demanding questions.
案例研究是模拟统计探究循环的综合问题：提出假设、收集数据、分析数据并得出结论。在OCR考试中，你可能会遇到对某项调查或实验的描述，并被要求评价方法、提出改进建议，或根据所提供的数据进行计算。尽早熟悉案例研究，有助于你应对较长的、要求更高的问题。
2. Case Study 1: Survey on Daily Screen Time | 案例一：每日屏幕时间调查
A group of Year 10 students wants to find out whether pupils at their school spend more than 3 hours per day on screens (phones, tablets, computers) outside of schoolwork. They design a questionnaire and collect responses from 60 students. The raw data (in hours) is shown below:
2.5, 4.0, 3.2, 5.5, 1.8, 3.0, 4.5, 6.0, 3.8, 2.0, ... (60 values).
Before diving into analysis, you must consider how the data was obtained. Did they use random sampling? Was the question clear? Could there be response bias?
一组10年级学生想要了解本校学生每天在学校作业之外使用电子屏幕（手机、平板、电脑）的时间是否超过3小时。他们设计了一份问卷，收集了60名学生的回答。原始数据（单位：小时）如下：
2.5, 4.0, 3.2, 5.5, 1.8, 3.0, 4.5, 6.0, 3.8, 2.0, ... （60个数值）。
在深入分析之前，你必须思考数据是如何获取的。他们是否用了随机抽样？问题是否清晰？是否存在回答偏差？
For this case study, assume the students used stratified sampling by year group to ensure representation. The questionnaire asked: 'On a typical weekday, how many hours do you spend on screen-based leisure activities?' This is a well-defined question, reducing misinterpretation. However, self-reported data may still be inaccurate because pupils might underestimate their screen time.
针对本案例，假设学生按年级进行了分层抽样以确保代表性。问卷问题是：“在一个典型的工作日，你花在屏幕休闲活动上的时间是多少小时？”这是一个清晰明确的问题，能减少误解。但自我报告的数据仍可能不准确，因为学生可能会低估自己的屏幕时间。
3. Data Representation: Bar Charts and Pie Charts | 数据展示：条形图和饼图
After collecting the data, the next step is to present it visually. For discrete or categorical summaries, bar charts and pie charts are common choices. Suppose we categorise screen time into intervals: 0–2 hours, 2–4 hours, 4–6 hours, and 6–8 hours. Frequencies: 12, 25, 18, 5. A bar chart will compare these frequencies directly, while a pie chart will show proportions of the total. Remember that in a bar chart the vertical axis must start at zero, and bars should be of equal width with gaps between them for discrete categories.
收集数据后，下一步是将其可视化。对于离散或分类汇总数据，条形图和饼图是常见的选择。假设我们将屏幕时间分为几个区间：0–2小时、2–4小时、4–6小时和6–8小时。频数分别为：12、25、18、5。条形图直接比较这些频数，而饼图则显示各部分占总体的比例。请记住，条形图的纵轴必须从零开始，条宽应相等，并且离散分类的条形之间应有间隙。
ng a pie chart, calculate the angle for each sector: (frequency ÷ total) × 360°. For the 2– ors clearly or provide a key. In OCR exams, you may be asked to critique an existing chart— ffects that distort proportions, or inappropriate use of line graphs for categorical data.
制作饼图时，要计算每个扇形的角度：（频数 ÷ 总数）× 360°。对于2–4小时区间：(25 ÷ 60) × 360° = 150°。务必清晰地标注各扇形或提供图例。在OCR考试中，你可能需要评价一幅已有图表——注意缺失标签、扭曲比例的三维效果，或对分类数据误用折线图等问题。
4. Measures of Central Tendency and Spread | 中心趋势和离散程度测量
ical and roughly symmetric. Suppose the sum of all 60 values is 210 hours. Then the mean x̄ = 210 ÷ 60 = 3.5 hours. The median is the middle value when data is ordered— out to be 3.4 hours. The mode is the most frequent value, which might be around 3.0 hours.
利用原始的屏幕时间数据，你可以计算均值、中位数和众数。均值在这里是合适的，因为数据是数值型且大致对称。假设60个数值的总和为210小时，则均值 x̄ = 210 ÷ 60 = 3.5小时。中位数是将数据排序后的中间值——若将所有60个值排序，中位数就是第30和第31个数值的平均值。在该数据集中，中位数为3.4小时。众数是最常出现的数值，大约在3.0小时。
For spread, the range is the simplest measure: maximum – sensitive to outliers. A more robust measure is the interquartile range (IQR), which we will examine in a later case study. When reporting, always pair a measure of centre with a measure of spread—for instance, 'The median screen time is 3.4 hours with a range of 5.5 hours.'
对于离散程度，极差是最简单的度量：最大值 – 最小值。这里，最大值 = 6.5 h，最小值 = 1.0 h，极差 = 5.5 h。然而，极差容易受异常值影响。更稳健的度量是四分位距（IQR），我们将在后续案例中研究。报告时，务必将中心度量与离散程度配对——例如，“屏幕时间中位数为3.4小时，极差为5.5小时。”
5. Case Study 2: Comparing Test Scores Between Two Groups | 案例二：比较两组成绩
Another common case study involves comparing two sets of data. Imagine two Year 10 classes, Class A and Class B, take the same statistics test. The scores out of 50 are as follows:
Class A: 28, 35, 42, 38, 45, 30, 33, 40, 37, 41
Class B: 20, 48, 25, 32, 50, 22, 28, 45, 35, 49
You are asked to compare the performance of the two classes. Which class did better? Which was more consistent?
另一种常见的案例研究涉及两组数据的比较。假设两个10年级班级，A班和B班，参加相同的统计测验。满分50分的成绩如下：
A班：28, 35, 42, 38, 45, 30, 33, 40, 37, 41
B班：20, 48, 25, 32, 50, 22, 28, 45, 35, 49
要求你比较两个班级的表现。哪个班成绩更好？哪个班更稳定？
an: order the scores: 28,30,33,35,37,38,40,41,42,45; median = (37+38)/2 = 37.5. Range = 45 – mean = 35.4. Ordered: 20,22,25,28,32,35,45,48,49,50; median = (32+35)/2 = 33.5. Range = 50 – 20 = 30.
首先，计算每组的汇总统计量。A班：均值 = (28+35+...+41) ÷ 10 = 369 ÷ 10 = 36.9。中位数：将成绩排序：28,30,33,35,37,38,40,41,42,45；中位数 = (37+38)/2 = 37.5。极差 = 45 – 28 = 17。B班：总分 = 354，均值 = 35.4。排序：20,22,25,28,32,35,45,48,49,50；中位数 = (32+35)/2 = 33.5。极差 = 50 – 20 = 30。
From the means, Class A performed slightly better on average, but Class B's range is much larger, indicating less consistency. To make a fairer comparison, we should look at quartiles and box plots, which resist the influence of the extreme scores in Class B.
从均值来看，A班平均成绩略好，但B班的极差大得多，说明成绩更不稳定。为了进行更公平的比较，我们应当查看四分位数和盒式图，它们能抵御B班极端分数的影响。
6. Using Box Plots and Quartiles | 盒式图与四分位数
For each class, find the lower quartile (Q1), median (Q2), and upper quartile (Q3).
alues: 28,30,33,35,37) = 33. Q3 is median of upper half (38,40,41,42,45) = 41. So IQR = Q3 – Q1 = 8.
ss B: lower half: 20,22,25,28,32 → Q1 = 25. Upper half: 35,45,48,49,50 → Q3 = 48. IQR = 48 – 25 = 23.
对每个班，找出下四分位数（Q1）、中位数（Q2）和上四分位数（Q3）。
A班，n=10：Q1是下半部分（前5个值：28,30,33,35,37）的中位数 = 33。Q3是上半部分（38,40,41,42,45）的中位数 = 41。因此 IQR = Q3 – Q1 = 8。
B班：下半部分：20,22,25,28,32 → Q1 = 25。上半部分：35,45,48,49,50 → Q3 = 48。IQR = 48 – 25 = 23。
Now draw comparative box plots. Class A's box is much shorter, showing that the middle 50% of scores are tightly packed between 33 and 41. Class B's box stretches from 25 to 48, indicating wide variation. The median lines show Class A's median (37.5) inside a compact box, while Class B's median (33.5) sits lower. This visual evidence supports the conclusion that Class A performed more consistently, and on average slightly higher, despite Class B having the highest individual score (50).
现在绘制比较盒式图。A班的箱子短得多，表明中间50%的成绩集中在33到41之间。B班的箱子从25延伸到48，说明差异很大。中位数线显示A班的中位数（37.5）位于紧凑的箱体内，而B班的中位数（33.5）偏低。这样的直观证据支持以下结论：A班成绩更稳定，且平均成绩略高，尽管B班有最高分（50）。
7. Case Study 3: Coin Flipping Experiment | 案例三：抛硬币试验
Probability case studies often involve experiments with equally likely outcomes. A student flips a fair coin 100 times and records 43 heads. They claim the coin must be biased because the result is not exactly 50 heads. Is this a valid conclusion? Use relative frequency and theoretical probability to analyse the situation.
概率案例研究常涉及等可能结果的实验。一名学生抛掷一枚均匀硬币100次，记录到43次正面。他声称这枚硬币一定不均匀，因为结果并非正好50次正面。这个结论合理吗？请用相对频率和理论概率来分析这一情况。
3 differs from 0.5, we must consider variation due to chance. In 100 flips, it is not unusual to observe 43 heads; the expected number is 50, but the standard deviation of the number of heads is √100×0.5×0.5 So values between 40 and 60 would not be surprising. This is an example of the concept of 'sample variability'.
正面的理论概率为 P(正面) = 0.5。实验得到的相对频率为 43/100 = 0.43。虽然0.43与0.5不同，但我们必须考虑随机变异。在100次抛掷中，观察到43次正面并不罕见；期望次数是50，但正面次数的标准差为 √100×0.5×0.5 = 5。因此，出现40至60次正面都不会令人意外。这就是“样本变异性”概念的一个例子。
8. Probability and Relative Frequency | 概率与相对频率
r of trials increases, relative frequency tends to get closer to the theoretical probability. This is known as the law of large numbers. Write down the probability estimate: P(Heads) ≈ 0.48. You can also calculate the probability of getting two heads in a row: (0.5)² = 0.25. In OCR, be prepared to construct sample space diagrams or tree diagrams for combined events.
继续硬币实验：该学生决定抛掷硬币500次，得到240次正面。相对频率 = 240/500 = 0.48。随着试验次数的增加，相对频率趋向于接近理论概率。这就是大数定律。你可以写下概率估计：P(正面) ≈ 0.48。你还可以计算连续两次抛出正面的概率：(0.5)² = 0.25。在OCR考试中，要准备好为复合事件构建样本空间图或树状图。
A practical case study question might ask: 'Design an experiment to estimate the probability that a drawing pin lands point up.' You would describe how to perform a large number of trials, record the outcomes, and compute the relative frequency. Also comment on how to make the estimate more reliable—increasing the number of trials and ensuring consistent conditions.
一个实际案例研究问题可能会问：“设计一个实验来估计图钉针尖朝上落地的概率。”你需要描述如何进行大量试验、记录结果并计算相对频率。同时还要说明如何使估计值更可靠——增加试验次数，并确保条件一致。
9. Case Study 4: Bivariate Data – Height vs. Arm Span | 案例四：双变量数据 – 身高与臂展
Many statistical investigations look for relationships between two variables. A student wonders: 'Are people with longer arms also taller?' She measures the height (cm) and arm span (cm) of 15 classmates. A table is constructed:
| Student | Height (cm) | Arm Span (cm) |
|---|---|---|
| 1 | 155 | 152 |
| 2 | 160 | 158 |
| 3 | 168 | 170 |
| ... | ... | ... |
This is bivariate data. The next step is to plot a scatter graph with height on the horizontal axis and arm span on the vertical axis.
许多统计调查都在寻找两个变量之间的关系。一名学生想：“手臂更长的人是否个子也更高？”她测量了15名同学的身高（cm）和臂展（cm）。构建表格如下：（表格略）。这是双变量数据。下一步是绘制散点图，以身高为横轴，臂展为纵轴。
10. Scatter Graphs and Correlation | 散点图与相关性
Plot the points. In this case, the pattern goes upward from left to right, indicating a positive correlation. Describe the relationship: 'There is a strong positive correlation between height and arm span.' You can add a line of best fit drawn by eye, passing through the mean point (x̄, ȳ). The line of best fit can be used to estimate one variable given the other. For example, if a student is 165 cm tall, the estimated arm span from the line might be approximately 163 cm.
描点。在本例中，点阵从左下向右上延伸，表明存在正相关。描述关系：“身高与臂展之间存在强正相关。”你可以用目测法添加一条最佳拟合线，使其穿过均值点（x̄, ȳ）。最佳拟合线可用于通过一个变量估计另一个变量。例如，若某学生身高165 cm，从拟合线可估计其臂展约为163 cm。
outside the range of the data (extrapolation). Also, correlation does not imply causation—there is an association, but it would be wrong to say 'increasing your height will increase your arm span' n coefficient, which here might be r ≈ 0.92, suggesting a very strong linear relationship.
需注意：最佳拟合线不应用于预测数据范围之外很远的值（外推）。同时，相关性并不意味着因果关系——确实存在关联，但若未考虑生物学成长机制，就说“增高会使臂展增长”则是错误的。在OCR案例研究中，你可能需要解读相关系数，本例中 r ≈ 0.92，表明线性关系极强。
11. Common Pitfalls in Statistical Case Studies | 统计案例研究中的常见陷阱
Many students lose marks by confusing types of data, using the wrong chart, or misinterpreting correlation. Always check whether data is discrete or continuous before choosing a diagram. Avoid drawing a line graph for discrete categories. When calculating averages, ensure you are using the correct formula—for grouped data, the mean involves midpoints. Also, do not forget to label axes and provide a title for any chart you draw in an exam.
许多学生因混淆数据类型、选错图表或误解相关性而失分。在选择图表之前，务必先检查数据是离散还是连续的。避免为离散类别绘制折线图。计算均值时，确保使用正确的公式——对于分组数据，均值需用到组中点。此外，不要忘记在考试中为任何图表标注坐标轴并提供标题。
Sampling bias is another major pitfall. If a case study describes a sample that is not representative—such as surveying only the school's netball team for fitness levels—the conclusion cannot be generalised. Always suggest using random or stratified sampling to minimise bias. Finally, when commenting on a case study, structure your answer: identify the problem, explain why it matters, and propose a realistic improvement.
抽样偏差是另一个重大陷阱。若案例研究描述了一个不具有代表性的样本——例如只调查学校篮网球队的健康水平——结论就无法推广。始终建议采用随机抽样或分层抽样以最大程度地减小偏差。最后，在对案例研究进行评述时，要组织好答案结构：指出问题，解释其重要性，并提出切合实际的改进措施。
12. Conclusion and Exam Tips | 结论与考试技巧
Working through case studies sharpens every skill in the OCR Statistics syllabus. Always start by identifying the type of data and the statistical enquiry cycle. Use suitable diagrams, calculate both a measure of centre and spread, and support your conclusions with clear reasoning. Practice computing quartiles, IQR, and drawing box plots until it becomes second nature, because these appear frequently in comparison problems.
通过案例研究可以打磨OCR统计学教学大纲中的每一项技能。始终要先识别数据类型和统计探究循环。使用合适的图表，同时计算集中量和离散量，并用清晰的推理支持你的结论。练习计算四分位数、IQR和绘制盒式图，直到熟能生巧，因为这些在比较问题中频频出现。
For bivariate data, always plot the points before making claims. Check for outliers that may distort the line of best fit. When asked about probability, ground your answer in relative frequency or theoretical reasoning. In the exam, read case study descriptions carefully, noting the sampling method, sample size, and possible sources of bias. By methodically applying these techniques, you can turn a complex case study into a set of manageable calculations and written arguments.
对于双变量数据，要先绘制点阵再下结论。检查可能扭曲最佳拟合线的异常值。遇到概率问题时，要将答案建立在相对频率或理论推理的基础上。在考试中，仔细阅读案例描述，注意抽样方法、样本量和可能的偏差来源。通过有条不紊地运用这些技巧，你就能将一个复杂的案例研究转化为一系列易于操作的计算和书面论证。
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