📚 Case Study: A Practical Walkthrough | 案例分析实战演练
In Year 11 OCR Statistics, applying statistical techniques to real-world scenarios is essential. This article walks you through a complete case study investigating the link between students’ sleep duration and their exam performance. You’ll see how to design a study, collect data, draw graphs, calculate statistics, and form conclusions—exactly as required in the exam.
在Year 11 OCR统计学中，将统计技术应用于实际情境至关重要。本文将通过一个完整的案例研究，探讨学生睡眠时长与考试成绩之间的关系，引导你逐步完成研究设计、数据收集、图表绘制、统计量计算以及结论形成，完全符合考试要求。
1. The Scenario | 案例背景
A school wishes to investigate whether the amount of sleep a student gets the night before an exam is related to their exam score. We will apply the statistical enquiry cycle: plan, collect, process, discuss, and conclude.
一所学校希望调查学生考前一天的睡眠时长是否与考试成绩有关。我们将应用统计调查循环：计划、收集、处理、讨论并得出结论。
We have gathered data from a random sample of 12 Year 11 students, recording sleep hours and the corresponding exam score (out of 100).
我们从12名Year 11学生中随机抽样，记录了睡眠时长和相应的百分制考试成绩。
| Student | Sleep (hours) | Exam score (%) |
|---|---|---|
| A | 7.5 | 65 |
| B | 6.0 | 52 |
| C | 8.5 | 78 |
| D | 7.0 | 60 |
| E | 5.5 | 45 |
| F | 9.0 | 88 |
| G | 6.5 | 55 |
| H | 8.0 | 72 |
| I | 7.0 | 68 |
| J | 6.0 | 50 |
| K | 8.5 | 80 |
| L | 7.5 | 70 |
2. Defining the Problem and Hypothesis | 定义问题与假设
We begin by framing the investigation as a testable question: “Is there a relationship between pre-exam sleep and exam achievement for Year 11 students?”
我们首先将调查转化为可检验的问题：“Year 11学生考前睡眠与考试成绩之间是否存在关系？”
The null hypothesis H₀ states that there is no correlation between the two variables. The alternative hypothesis H₁ suggests a positive correlation: more sleep tends to be associated with higher scores. We will use a 5% significance level.
零假设 H₀ 声称两个变量之间没有相关性。备择假设 H₁ 表明存在正相关：睡眠时间越长，分数往往越高。我们将使用5%的显著性水平。
3. Data Collection and Sampling Methods | 数据收集与抽样方法
Data were collected using a questionnaire in which students reported their sleep hours the night before the exam. Their exam scores were then obtained from school records. The sample was selected by simple random sampling from the Year 11 cohort to reduce bias.
数据通过问卷收集：学生报告考试前一天的睡眠时长，同时从学校记录中获取考试成绩。样本通过从Year 11群体中简单随机抽样获得，以减少偏差。
However, limitations exist: self-reported sleep may be inaccurate, the sample is small (n = 12), and only one year group is represented. These will be addressed in the evaluation.
然而，存在局限性：自报的睡眠时间可能不准确，样本量较小（n = 12），且仅代表一个年级。这些问题将在评估部分讨论。
4. Presenting the Data: Tables and Diagrams | 数据表示：表格与图表
The data table above gives the raw values. For visual exploration, a scatter diagram is most appropriate because both variables are continuous. The scatter graph plots sleep hours on the x-axis and exam score on the y-axis.
上方的数据表给出了原始数值。为了可视化探索，散点图最为合适，因为两个变量都是连续的。散点图将睡眠时长标在x轴，考试成绩标在y轴。
Each point represents one student. The plot reveals a clear upward trend, suggesting a strong positive association. No obvious outliers are present.
每个点代表一名学生。图形显示出明显的上升趋势，表明强烈的正相关关系。没有明显的异常值。
5. Measures of Central Tendency and Spread | 中心趋势与离散程度的度量
For sleep hours, the ordered data are: 5.5, 6.0, 6.0, 6.5, 7.0, 7.0, 7.5, 7.5, 8.0, 8.5, 8.5, 9.0. The mean is (5.5+6.0+6.0+6.5+7.0+7.0+7.5+7.5+8.0+8.5+8.5+9.0)/12 = 87/12 = 7.25 hours. The median is the average of the 6th and 7th values (7.0 + 7.5)/2 = 7.25 hours. The range is 9.0 − 5.5 = 3.5 hours. The lower quartile Q₁ is the median of the first six values: (6.0+6.5)/2 = 6.25 hours. The upper quartile Q₃ is the median of the last six: (8.0+8.5)/2 = 8.25 hours. The interquartile range (IQR) = 8.25 − 6.25 = 2.0 hours.
对于睡眠时长，排序后的数据为：5.5, 6.0, 6.0, 6.5, 7.0, 7.0, 7.5, 7.5, 8.0, 8.5, 8.5, 9.0。均值为 (5.5+6.0+6.0+6.5+7.0+7.0+7.5+7.5+8.0+8.5+8.5+9.0)/12 = 87/12 = 7.25 小时。中位数为第6和第7个数值的平均 (7.0+7.5)/2 = 7.25 小时。极差为 9.0 − 5.5 = 3.5 小时。下四分位数 Q₁ 是前6个值的中位数 (6.0+6.5)/2 = 6.25 小时。上四分位数 Q₃ 是后6个值的中位数 (8.0+8.5)/2 = 8.25 小时。四分位距 IQR = 8.25 − 6.25 = 2.0 小时。
For exam scores (ordered: 45, 50, 52, 55, 60, 65, 68, 70, 72, 78, 80, 88), mean = 783/12 = 65.25%, median = (65+68)/2 = 66.5%, range = 88 − 45 = 43%, Q₁ = (52+55)/2 = 53.5%, Q₃ = (72+78)/2 = 75%, IQR = 21.5%.
对于考试成绩（排序：45, 50, 52, 55, 60, 65, 68, 70, 72, 78, 80, 88），均值 = 783/12 = 65.25%，中位数 = (65+68)/2 = 66.5%，极差 = 88 − 45 = 43%，Q₁ = (52+55)/2 = 53.5%，Q₃ = (72+78)/2 = 75%，IQR = 21.5%。
6. Constructing a Scatter Diagram | 绘制散点图
When we draw the scatter diagram with sleep hours (x) and exam score (y), we observe a strong positive linear pattern. As sleep increases, exam scores tend to rise consistently.
当我们以睡眠时长（x）和考试成绩（y）绘制散点图时，观察到强正线性模式。随着睡眠时间增加，考试成绩趋势一致地上升。
We can add a line of best fit by eye, and the points cluster tightly around it. The form, direction, and strength of this relationship support further analysis using a correlation coefficient.
我们可以目测添加一条最佳拟合线，各点紧密分布在直线周围。这种关系的形状、方向和强度支持使用相关系数进行进一步分析。
7. Calculating Spearman’s Rank Correlation Coefficient | 计算斯皮尔曼等级相关系数
Since we want a numerical measure of the association, we use Spearman’s rank correlation coefficient rₛ. This test does not require the data to be normally distributed and is ideal for small samples.
由于我们需要关联度的数值度量，我们使用斯皮尔曼等级相关系数 rₛ。该检验不要求数据呈正态分布，非常适合小样本。
We rank sleep hours and exam scores separately. Tied values receive the mean rank. Below is the ranking table:
我们分别对睡眠时长和考试成绩排序。相同值取平均等级。下面为排序表：
| Student | Sleep rank (R₁) | Score rank (R₂) | d = R₁ − R₂ | d² |
|---|---|---|---|---|
| E | 1 | 1 | 0 | 0 |
| B | 2.5 | 3 | -0.5 | 0.25 |
| J | 2.5 | 2 | 0.5 | 0.25 |
| G | 4 | 4 | 0 | 0 |
| D | 5.5 | 5 | 0.5 | 0.25 |
| I | 5.5 | 7 | -1.5 | 2.25 |
| A | 7.5 | 6 | 1.5 | 2.25 |
| L | 7.5 | 8 | -0.5 | 0.25 |
| H | 9 | 9 | 0 | 0 |
| C | 10.5 | 10 | 0.5 | 0.25 |
| K | 10.5 | 11 | -0.5 | 0.25 |
| F | 12 | 12 | 0 | 0 |
Sum of d² = 0 + 0.25 + 0.25 + 0 + 0.25 + 2.25 + 2.25 + 0.25 + 0 + 0.25 + 0.25 + 0 = 6.0.
d² 的总和 = 0 + 0.25 + 0.25 + 0 + 0.25 + 2.25 + 2.25 + 0.25 + 0 + 0.25 + 0.25 + 0 = 6.0。
Now apply the formula:
现在应用公式：
rₛ = 1 −
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