📚 Practical Case Study in Statistics: Analysing a School Fitness Challenge | 统计案例分析实战演练：学校健身挑战赛数据分析
Data is everywhere, and the ability to interpret it is a vital skill. In CCEA Year 10 Statistics, you are often asked to apply the statistical enquiry cycle to real-world problems. This article provides a complete walkthrough of a practical case study: analysing the results of a school fitness challenge. We will cover data collection, sampling, presenting data, calculating averages and spread, exploring relationships, making probability estimates, and drawing conclusions. Every step is shown with worked numbers, so you can see how each statistical tool helps make sense of raw data and how the marks are actually earned in an examination.
数据无处不在，解读数据的能力是一项重要技能。在 CCEA 十年级统计学中，你经常需要将统计探究周期应用于实际问题。本文提供一个完整的案例分析演练：分析学校健身挑战赛的结果。我们将完整涵盖数据收集、抽样、数据展示、计算平均数和离散度、探索关系、估计概率并得出结论。每一步都配有实际计算数字，让你看到每个统计工具如何帮助理解原始数据，以及在考试中分数究竟是如何拿到的。
1. Understanding the Challenge and Data Collection | 理解挑战与数据收集
A secondary school launched a four-week fitness challenge for all Year 10 students to encourage physical activity. Participants recorded their active minutes each week using a school-approved mobile app, which linked data to student IDs. The variables recorded included gender, age (in years and months), and total active minutes over the four weeks. The aim was to analyse participation patterns and to see if activity levels differed by gender or age.
一所中学为所有十年级学生发起了为期四周的健身挑战，以鼓励体育活动。参与者使用学校认可的移动应用记录每周的活动分钟数，该应用将数据与学生ID关联。记录的变量包括性别、年龄（岁和月）以及四周内的总活动分钟数。目的是分析参与模式，并查看活动水平是否因性别或年龄而异。
Notice how the aim is written before any data is touched. A well-posed question tells you exactly which variable is being compared, which groups are involved, and what kind of answer would count as evidence. Here the response variable is total active minutes, and the explanatory variables are gender and age.
请注意，研究目的在任何数据处理之前就已经写好了。一个表述清晰的统计问题会明确告诉你：比较的是哪个变量、涉及哪些分组、以及什么样的结果才算作证据。在这里，响应变量是总活动分钟数，解释变量是性别和年龄。
Because the app timestamps each session, the school also holds weekly totals, which lets us look at whether activity rose or fell across the four weeks. This turns a single snapshot into a small time series, and adds a useful fourth variable to the dataset.
由于应用会为每次运动打上时间戳，学校还掌握每周的合计数据，这让我们可以观察四周内活动量是上升还是下降。这样就把一个静态快照变成了一个小型时间序列，并为数据集增加了有用的第四个变量。
2. The Statistical Enquiry Cycle in Action | 统计探究周期的实际运用
The statistical enquiry cycle has five stages: pose a question, plan and collect data, process and represent the data, analyse and interpret, and evaluate the whole process. Each stage feeds back into the next, which is why good statisticians loop around the cycle more than once.
统计探究周期包含五个阶段：提出研究问题、计划并收集数据、处理与呈现数据、分析与解释、以及评估整个过程。每个阶段都会反馈到下一个阶段，因此优秀的统计工作者往往会在这个周期中循环不止一次。
The table below maps each stage of the cycle onto this specific fitness challenge.
下表把探究周期的每个阶段对应到这次具体的健身挑战上。
| Stage 阶段 | What happens in this study 本研究中的具体做法 |
| Pose a question | Do Year 10 activity levels differ by gender or age? |
| Collect data | Four-week app log of active minutes, with gender and age |
| Process and represent | Clean the data, build grouped frequency tables, bar charts, histograms, scatter graphs |
| Analyse and interpret | Mean, median, mode, range, IQR, correlation, probability |
| Evaluate | Judge reliability, bias, sample size and whether conclusions generalise |
Examiners award marks for naming the stage as well as for carrying it out, so always write the stage label in your answer, for example 'analysing and interpreting'.
考官既会因为你完成某个阶段而给分，也会因为你点出阶段名称而给分，所以在答案中务必写出阶段标签，例如"分析与解释"。
3. Population, Sampling and Bias | 总体、抽样与偏差
The population is every Year 10 student in the school, roughly 240 pupils. A census would collect data from all of them, but attendance, illness and forgotten logins make a census impractical. The study therefore relies on a sample, and the quality of that sample decides how much we can trust the conclusions.
总体是该校所有十年级学生，大约 240 人。普查会从所有人身上收集数据，但出勤、生病和忘记登录等问题让普查不切实际。因此研究依赖样本，而样本的质量决定了我们能在多大程度上相信结论。
Simple random sampling gives every student an equal chance of selection, for example by numbering the year group and using a random number generator. Stratified sampling is often better here because it guarantees representation of each subgroup, such as boys and girls or different tutor groups.
简单随机抽样让每个学生被选中的机会相等，例如给全年级编号并使用随机数生成器。在这里分层抽样往往更好，因为它能保证每个子群体都被代表，例如男生和女生，或不同导师组。
Sampling bias is a serious threat in this study. Only students who own or can borrow a phone with the app can take part, and only those who choose to join are counted, so the sample is self-selecting and likely to over-represent already active students. Students who are unwell, injured or simply uninterested are systematically missed.
抽样偏差在本研究中是严重威胁。只有拥有或能借用安装该应用手机的学生才能参与，而且只有自愿加入的人被统计，因此样本是自愿自选的，很可能过度代表本来就活跃的学生。生病、受伤或单纯不感兴趣的学生被系统性地遗漏。
Self-reporting is a second problem: the app measures what students choose to log, not what they actually do. A student may forget to log a session, or may log it twice, so measurement error exists even when the sample is well chosen.
自我报告是第二个问题：应用测量的是学生选择记录的内容，而不是他们实际做了什么。学生可能忘记记录某次运动，也可能重复记录，因此即使样本选得好，测量误差依然存在。
4. Data Cleaning and Ethics | 数据清洗与伦理考量
Before analysis, the data needed to be cleaned. Entries with missing values or unrealistic active minutes (for example, more than 10,000 minutes) were checked and removed, leaving a reliable dataset.
在分析之前，数据需要被清洗。存在缺失值或活动分钟数不切实际的记录（例如超过 10,000 分钟）都经过核查并删除，从而留下可靠的数据集。
Why is 10,000 minutes an unreasonable cut-off? 20 minutes in total, so 10,000 minutes would mean roughly 357 minutes, or nearly six hours, of activity every single day. A smaller threshold, such as 6,000 minutes (about 214 minutes per day), is arguably an even stricter and more realistic upper limit.
为什么 10,000 分钟是一个不合理的界限？四周总共包含 4 × 7 × 24 × 60 = 40,320 分钟，所以 10,000 分钟意味着平均每天约 357 分钟，也就是接近六小时的运动量。更小的阈值，例如 6,000 分钟（每天约 214 分钟），可以说是一个更严格也更现实的上限。
Other cleaning tasks include removing duplicate student IDs, standardising age into a single unit such as months, correcting impossible dates, and checking that every weekly total sums to the recorded four-week total. Each decision should be logged so the analysis can be reproduced.
其他清洗工作包括删除重复的学生ID、把年龄统一为单一单位（例如月）、修正不可能出现的日期，以及核对每周小计之和是否等于记录的总计。每一项处理决定都应记录下来，以便分析可以复现。
Ethical considerations were followed: all data was anonymised so that names were replaced with codes, students gave informed consent, and participants (and their parents or guardians) had the right to withdraw at any time. Data was stored securely, used only for the challenge, and deleted once the report was finished, in line with data protection rules.
研究遵循了伦理要求：所有数据都经过匿名化处理，姓名被替换为代码，学生知情同意，参与者（及其父母或监护人）有权随时退出。数据被安全存储、仅用于本次挑战，并在报告完成后删除，符合数据保护法规。
5. Presenting the Data | 数据展示
Raw lists of numbers say very little, so the first job is to organise them. Total active minutes is continuous data, so a grouped frequency table is appropriate. The 110 valid records produced the following distribution.
原始的数字列表几乎没有说明力，因此第一步是把它们组织起来。总活动分钟数是连续数据，因此适合使用分组频数表。110 条有效记录得到如下分布。
| Total active minutes 总活动分钟 | Frequency 频数 | Midpoint 组中值 |
| 0 - 999 | 8 | 499.5 |
| 1000 - 1999 | 22 | 1499.5 |
| 2000 - 2999 | 34 | 2499.5 |
| 3000 - 3999 | 26 | 3499.5 |
| 4000 - 4999 | 14 | 4499.5 |
| 5000 - 5999 | 6 | 5499.5 |
| Total 合计 | 110 | — |
Check that 8 + 22 + 34 + 26 + 14 + 6 really does equal 110 before drawing anything. Because the data is continuous, the correct diagram is a histogram with no gaps between bars, whereas gender is categorical and needs a bar chart with gaps.
在绘图之前要先核对 8 + 22 + 34 + 26 + 14 + 6 是否真的等于 110。由于数据是连续的，正确的图形是无间隙的直方图；而性别属于分类数据，需要绘制有间隙的条形图。
Always label both axes, give the chart a title, and state the units. For time series, plot weekly totals as a line graph so the trend across the four weeks is visible.
务必给两个坐标轴加标签、给图形加标题，并注明单位。对于时间序列，把每周合计画成折线图，这样四周的趋势才看得清楚。
6. Averages from Grouped Data | 分组数据的平均数
Because we no longer know individual values, we estimate the mean by assuming every student in a class interval scored the midpoint. The formula is:
由于我们不再知道每个个体的具体数值，我们假设每个组距内的学生都取组中值，以此估计平均数。公式为：
Mean = Σfx ÷ Σf
Multiplying each midpoint by its frequency and summing gives 308,945. Dividing by 110 gives an estimated mean of 2808.6 minutes, which is about 2809 minutes, or roughly 117 minutes per day.
把每个组中值乘以对应频数再求和，得到 308,945。除以 110，得到估计平均数为 2808.6 分钟，约等于 2809 分钟，也就是每天大约 117 分钟。
The median is found from cumulative frequencies: 8, 30, 64, 90, 104, 110. Half of 110 is 55, which first exceeds 30, so the median lies in the 2000 - 2999 class. Using interpolation with class boundaries 1999.5 and 2999.5:
中位数由累积频数求得：8、30、64、90、104、110。110 的一半是 55，它首次超过 30，所以中位数落在 2000 - 2999 组。使用组界 1999.5 与 2999.5 进行线性插值：
Median = 1999.5 + ((55 − 30) ÷ 34) × 1000 = 2734.8 minutes
The modal class is 2000 - 2999 minutes because it has the highest frequency, 34. Note that the mean (2808.6) is slightly above the median (2734.8), which suggests a mild positive skew caused by a small number of very active students.
众数所在组是 2000 - 2999 分钟，因为它的频数最高，为 34。注意平均数（2808.6）略高于中位数（2734.8），说明少数非常活跃的学生造成了轻微的正偏态。
7. Measures of Spread | 离散程度的度量
An average on its own hides how varied the data is. Two classes can share the same mean while one is tightly clustered and the other is wildly spread out. Spread is therefore reported alongside every average.
单看平均数会掩盖数据的差异程度。两个班级可能有相同的平均数，但一个数据高度集中，另一个却极为分散。因此离散程度必须与平均数一起报告。
he range uses only two extreme values, so the interquartile range is preferred. With 110 values, the lower quartile is at position 27.5 and the upper quartile at position 82.5.
按组界计算的极差是 5999.5 − (−0.5) = 6000 分钟，但极差只用到两个极端值，因此四分位距更受青睐。110 个数值中，下四分位数位于第 27.5 位，上四分位数位于第 82.5 位。
Q₁ = 999.5 + ((27.5 − 8) ÷ 22) × 1000 = 1885.9 minutes
Q₃ = 2999.5 + ((82.5 − 64) ÷ 26) × 1000 = 3711.0 minutes
IQR = Q₃ − Q₁ = 3711.0 − 1885.9 = 1825.1 minutes
The interquartile range of about 1825 minutes tells us that the middle 50% of students span a wide band of activity, roughly from 1886 to 3711 minutes over the four weeks. If a more advanced course is being followed, the standard deviation can be estimated:
约 1825 分钟的四分位距告诉我们，中间 50% 的学生活动量跨度很大，大约在四周内从 1886 分钟到 3711 分钟。如果学习的是更高阶的课程，还可以估计标准差：
σ = √(Σf(x − mean)² ÷ Σf)
8. Comparing Groups and Spotting Outliers | 分组比较与异常值识别
The central question was whether activity levels differ by gender. Splitting the sample gives the comparison below.
核心问题是活动水平是否因性别而异。把样本分组后得到如下比较。
| Group 组别 | Participants 人数 | Mean minutes 平均分钟 | IQR 四分位距 |
| Boys 男生 | 58 | 3012 | 1980 |
| Girls 女生 | 52 | 2581 | 1640 |
Boys averaged about 431 minutes more than girls over four weeks, roughly 15 minutes per day, and their results were also more spread out. A good comparison sentence names both the direction and the size of the difference, and comments on consistency as well as on the average.
男生的四周平均比女生多约 431 分钟，大约每天多 15 分钟，而且他们的成绩分布也更分散。好的比较句会同时指出差异的方向和大小，并且既评论平均数也评论一致性。
r fence is Q₃ + 1.5 × IQR = 3711.0 + 2737.7 = 6448.7 minutes, so one participant who logged 9120 minutes is an outlier.
可以使用 1.5 × IQR 规则标记异常值。下界为 Q₁ − 1.5 × IQR = 1885.9 − 2737.7，结果为负数，所以不存在低端异常值。上界为 Q₃ + 1.5 × IQR = 3711.0 + 2737.7 = 6448.7 分钟，因此一位记录了 9120 分钟的参与者属于异常值。
Never delete an outlier simply because it looks odd. Investigate it: this student may be a competitive swimmer, or may have logged sessions twice. If the value is genuine, keep it and mention it in the conclusion, because it still carries information about the group.
绝不要仅仅因为某个数值看起来奇怪就删除它。要调查原因：这名学生可能是竞技游泳选手，也可能重复记录了训练。如果数值是真实的，就保留它并在结论中提及，因为它依然携带关于该群体的信息。
9. Exploring Relationships with Scatter Graphs | 用散点图探索关系
To explore whether older students are more active, plot age in months on the x-axis and total active minutes on the y-axis, then draw a line of best fit. The strength and direction of the association are described using correlation.
为了探究年龄更大的学生是否更活跃，把年龄（月）画在横轴、总活动分钟数画在纵轴，然后画出最佳拟合直线。关联的强度与方向用相关性来描述。
lationship is very weak, and that is a legitimate and important finding rather than a failure. Year 10 spans only about 18 months of age,
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