📚 IGCSE WJEC Statistics: Case Study Practical Drill | IGCSE WJEC 统计：案例分析实战演练
This article walks through a full statistics case study based on the IGCSE WJEC specification. We explore a realistic investigation into the weekly exercise habits and sleep duration of secondary school students, following the complete statistical enquiry cycle from problem definition to final conclusions.
本文基于 IGCSE WJEC 考纲，完整演示一个统计案例分析。我们以中学生每周锻炼时长与睡眠时间的真实调查为例，遵循从问题定义到最终结论的完整统计探究流程。
1. Introduction to the Case Study | 案例研究简介
The case study investigates the relationship between weekly exercise hours and nightly sleep duration among Year 11 students at Oakwood School. The school’s health committee suspects that students who exercise more may sleep better, but they also worry about possible gender differences and whether a balanced lifestyle is common. A statistical investigation is designed to answer these questions.
本案例研究调查奥克伍德学校 11 年级学生每周锻炼时长与夜间睡眠时间的关系。学校健康委员会怀疑锻炼较多的学生睡眠更好，但也担心可能存在性别差异，以及学生是否普遍拥有平衡的生活方式。为此我们设计了一项统计调查来回答这些问题。
2. Defining the Problem and Planning | 定义问题与计划
The primary objective is to determine whether there is a positive correlation between hours of exercise per week and hours of sleep per night. Secondary objectives include comparing the distributions for male and female students and estimating the proportion of students meeting the recommended minimum of 8 hours of sleep. The target population is all 180 Year 11 students. A stratified sample of 60 students (30 male, 30 female) will be used to ensure fair gender representation.
主要目标是确定每周锻炼时长与每晚睡眠时长之间是否存在正相关。次要目标包括比较男女学生的分布，并估计达到建议最低 8 小时睡眠的学生比例。目标总体是全部 180 名 11 年级学生。我们将使用一个 60 人的分层样本（30 名男生、30 名女生），以确保性别代表性公平。
3. Data Collection Methods | 数据收集方法
Data is collected via a structured questionnaire. Each selected student reports their average weekly exercise (in hours, rounded to the nearest 0.5 hour) and their average nightly sleep (in hours, rounded to the nearest 0.5 hour) over the past month. To reduce response bias, the questionnaire is anonymous and administered during form time. A pilot study with 5 students checked that questions were clear. Any student with an extreme or illogical response (e.g., 20 hours of sleep) was excluded and replaced from the same stratum.
数据通过结构化问卷收集。每位被选中的学生报告过去一个月中平均每周锻炼时长（小时，四舍五入到 0.5 小时）和平均每晚睡眠时长（小时，四舍五入到 0.5 小时）。为减少回答偏差，问卷采用匿名方式，在班会时间填写。我们先对 5 名学生进行了预调查，以确保问题清晰。若出现极端或不合理的回答（如睡眠 20 小时），则排除该学生并从同一层中替换。
4. Organizing the Data | 数据整理
The raw data for 60 students is entered into a spreadsheet. For analysis, exercise hours are grouped into intervals: 0–1.5, 2–3.5, 4–5.5, 6–7.5, 8–9.5. Sleep hours are grouped into: 6.0–6.5, 7.0–7.5, 8.0–8.5, 9.0–9.5. Two frequency tables are produced, one for each variable, separately for males and females. The table below shows the combined frequency distribution for sleep hours.
60 名学生的原始数据被录入电子表格。为便于分析，每周锻炼时长分为以下组距：0–1.5、2–3.5、4–5.5、6–7.5、8–9.5 小时。睡眠时长分为：6.0–6.5、7.0–7.5、8.0–8.5、9.0–9.5 小时。我们制作了男女分别的频率表，下面这张表展示了睡眠时长的合并频率分布。
| Sleep (hours) | Frequency |
|---|---|
| 6.0 – 6.5 | 10 |
| 7.0 – 7.5 | 20 |
| 8.0 – 8.5 | 22 |
| 9.0 – 9.5 | 8 |
5. Graphical Representation | 图形表示
A compound bar chart is drawn to compare the exercise distributions of males and females. The horizontal axis shows exercise groups, and separate coloured bars represent each gender. A pie chart is used to display the proportion of all students in each sleep category. Additionally, a scatter diagram plots each student’s exercise hours against sleep hours, with different symbols for males and females, to visually check for correlation.
我们绘制了复合条形图来比较男女生的锻炼分布。横轴为锻炼时长组，不同颜色的条形代表不同性别。饼图用于展示全体学生在每个睡眠组的比例。此外，散点图将每位学生的锻炼时长与睡眠时长相配对，男生和女生使用不同符号，以便直观查看相关性。
6. Measures of Central Tendency | 集中趋势的度量
For exercise hours (all students), the mean is calculated as 4.6 hours, the median class is 4–5.5 hours, and the modal class is also 4–5.5 hours. For sleep hours, the mean is 7.8 hours. The mean sleep for males is 7.5 hours, while for females it is 8.1 hours. This suggests that, on average, female students in this sample sleep slightly longer.
对于锻炼时长（全体学生），平均值为 4.6 小时，中位数所在组为 4–5.5 小时，众数组也是 4–5.5 小时。睡眠时长的平均值为 7.8 小时。男生的平均睡眠为 7.5 小时，女生为 8.1 小时。这表明在该样本中，女生平均睡眠时间略长。
7. Measures of Spread | 离散程度的度量
The range of sleep hours is 3.5 hours (from 6.0 to 9.5). The interquartile range (IQR) is found by locating the lower quartile (7.0 hours) and upper quartile (8.5 hours), giving an IQR of 1.5 hours. The standard deviation for sleep is calculated as 0.94 hours, indicating moderate variability. For exercise, the range is 9 hours (0 to 9), the IQR is 3 hours, and the standard deviation is 2.1 hours, showing greater spread.
睡眠时长的极差为 3.5 小时（从 6.0 到 9.5 小时）。通过定位下四分位数（7.0 小时）和上四分位数（8.5 小时），得到四分位距（IQR）为 1.5 小时。睡眠时长的标准差为 0.94 小时，显示中等变异程度。锻炼时长的极差为 9 小时（0 至 9），四分位距为 3 小时，标准差为 2.1 小时，表明离散程度更大。
8. Probability and Expected Frequency | 概率与期望频率
If a student is selected at random from the sample, the probability that they sleep at least 8 hours is P(sleep ≥ 8) = (22+8)/60 = 30/60 = 0.5. The probability that a randomly chosen male sleeps less than 8 hours is 18/30 = 0.6. Using expected frequency, if the whole year group of 180 students followed the same sleeping pattern, we would expect 180 × 0.5 = 90 students to sleep at least 8 hours.
如果从样本中随机选取一名学生，其睡眠至少 8 小时的概率为 P(睡眠 ≥ 8) = (22+8)/60 = 30/60 = 0.5。随机选取一名男生其睡眠少于 8 小时的概率为 18/30 = 0.6。利用期望频率，若整个年级 180 名学生遵循相同的睡眠模式，我们预期有 180 × 0.5 = 90 名学生睡眠时间至少为 8 小时。
9. Bivariate Data and Correlation | 双变量数据与相关性
The scatter graph shows a weak positive correlation between exercise hours and sleep hours. The calculated product-moment correlation coefficient (r) is 0.32, confirming a weak positive linear relationship. However, the data points are widely scattered, and several outliers exist, such as a student with 0 exercise hours who sleeps 9 hours. This suggests that while a slight trend may exist, exercise is not a strong predictor of sleep duration on its own.
散点图显示锻炼时长与睡眠时长之间存在弱正相关。计算得出的积矩相关系数（r）为 0.32，证实了这种弱的正线性关系。然而数据点非常分散，并且存在若干异常值，例如一名完全不锻炼的学生却睡 9 小时。这表明虽然可能存在轻微趋势，但锻炼本身并不是睡眠时长的强预测因子。
10. Drawing Conclusions and Limitations | 得出结论与局限性
The investigation found that the average sleep duration meets the recommended 8 hours only for females; males are slightly below. The weak positive correlation indicates that encouraging exercise might slightly improve sleep, but other factors (screen time, stress, diet) likely play a larger role. The main limitation is the self-reported data, which may be inaccurate. The sample, although stratified, comes from a single school, so results cannot be generalised to all teenagers. The grouping of data also loses some detail.
调查发现，仅女生的平均睡眠时长达到了推荐的 8 小时；男生略低于此标准。弱正相关表明，鼓励锻炼或许能轻微改善睡眠，但其他因素（屏幕使用时间、压力、饮食）可能发挥着更重要的作用。主要局限在于数据为自报数据，可能不准确。虽然样本进行了分层，但仅来自一所学校，因此结果不能推广到所有青少年。数据分组也造成了一些细节的丢失。
11. Communication of Findings | 调查结果沟通
A clear statistical report is prepared for the school health committee. Key graphs, summary statistics, and the correlation coefficient are included. The report highlights the gender difference in sleep and recommends a larger, multi-school study to verify findings. The use of straightforward language and well-labelled visuals ensures the committee, who may not be statisticians, can understand and act on the information.
为学校健康委员会准备了一份清晰的统计报告。报告包含了关键图表、汇总统计量和相关系数。报告特别指出了睡眠方面的性别差异，并建议进行更大规模的多校研究以验证结果。报告采用易懂的语言和标注清晰的图像，确保非统计专业的委员会成员能够理解并据此采取行动。
12. Ethical Considerations and Data Protection | 伦理考量与数据保护
All participants gave informed consent, and their anonymity was preserved by using ID codes instead of names. Data was stored securely on a password-protected school server and will be deleted after the investigation. No individual student could be identified in any published output, in line with data protection guidelines.
所有参与学生均已知情同意，并通过使用编号代替姓名来保护其匿名性。数据被安全地存储在设有密码保护的学校服务器上，调查结束后将被删除。遵循数据保护指南，任何公开结果均无法识别出单个学生。
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