📚 Year 8 CIE Statistics: Case Study in Action | Year 8 CIE 统计：案例分析实战演练
Welcome to a practical case study designed to strengthen your statistical skills at Year 8 CIE level. Throughout this article, we will work step by step through a real-world investigation — a survey on weekly after‑school activity hours — applying data collection, presentation, analysis and probability. By the end, you will see how statistics turn raw numbers into meaningful conclusions.
欢迎来到这个专为 Year 8 CIE 阶段设计的实战案例研究。在本文中，我们将一步步地完成一项真实调查——关于每周课外活动小时数的调查——应用数据收集、展示、分析和概率等知识。到最后，你将看到统计学如何将原始数字转化为有意义的结论。
1. Introduction to Our Case Study | 案例介绍
Imagine you are a student researcher investigating how many hours per week your classmates spend on after‑school activities such as sports, music or clubs. The school wants to know if more activities should be offered, so you decide to conduct a statistical study. Your goal is to collect, display and interpret the data to make a recommendation.
想象你是一名学生研究员，正在调查同学们每周花在体育、音乐或社团等课外活动上的小时数。学校想知道是否需要提供更多活动，因此你决定开展一项统计研究。你的目标是收集、展示和解读数据，以提出建议。
You will follow the complete statistical cycle: pose a question, design a data collection tool, gather raw data, organise it into tables and graphs, calculate averages and spread, explore relationships and even touch on probability. This case study mirrors the CIE assessment objectives for Year 8.
你将遵循完整的统计循环：提出问题、设计数据收集工具、收集原始数据、将其整理成表格和图表、计算平均数和离散程度、探索关系，甚至接触概率。这个案例研究反映了 CIE Year 8 的评估目标。
2. Designing a Questionnaire | 设计问卷
Before collecting data, you need a clear question and a well‑structured questionnaire. The main question is: ‘How many hours do you spend on after‑school activities in a typical week?’ To make the data reliable, you also decide to ask a second question about daily screen time, which might be related. You keep the questions simple and closed, so answers are numbers.
在收集数据之前，你需要一个明确的问题和结构良好的问卷。主要问题是：“你通常每周花多少小时在课外活动上？”为了使数据可靠，你还决定问第二个关于每日屏幕时间的问题，这可能存在关联。你让问题简单且封闭，这样答案都是数字。
Your questionnaire includes: (1) Student number (to keep it anonymous), (2) ‘What is your total weekly after‑school activity time (in hours)?’, (3) ‘How many hours of recreational screen time do you have per day?’. A pilot test with five students confirmed the questions are clear.
你的问卷包括：（1）学生编号（保持匿名），（2）“你每周的课外活动总时间是多少小时？”（3）“你每天的娱乐屏幕时间是多少小时？”对五名学生进行的试点测试确认问题清晰明了。
3. Collecting Raw Data | 收集原始数据
You surveyed 20 students from your Year 8 class. The raw data for weekly activity hours is: 2, 3, 1, 4, 2, 5, 3, 2, 4, 6, 1, 2, 3, 5, 2, 4, 3, 1, 2, 5. For daily screen time (in hours), you recorded: 2.0, 3.5, 1.5, 2.0, 2.5, 4.0, 3.0, 2.0, 2.5, 3.0, 1.0, 2.5, 3.5, 4.0, 2.0, 2.5, 3.0, 1.5, 2.5, 3.5. These two sets are paired by student.
你调查了班上 20 名学生。每周活动时间的原始数据为：2, 3, 1, 4, 2, 5, 3, 2, 4, 6, 1, 2, 3, 5, 2, 4, 3, 1, 2, 5。每日屏幕时间（小时）记录为：2.0, 3.5, 1.5, 2.0, 2.5, 4.0, 3.0, 2.0, 2.5, 3.0, 1.0, 2.5, 3.5, 4.0, 2.0, 2.5, 3.0, 1.5, 2.5, 3.5。这两组数据是按学生配对的。
Raw data alone is messy and hard to interpret. The next step is to organise it into a frequency table to spot patterns.
仅有原始数据是杂乱且难以解读的。下一步是将其整理成频数表，以便发现规律。
4. Organising Data into Frequency Tables | 整理数据成频数表
For activity hours, we list each distinct value and count how many students reported it. This gives a frequency table. Because the data are discrete integer values, no grouping is needed.
对于活动时间，我们列出每个不同的取值，并统计有多少名学生报告了该值。这就得到了一个频数表。由于数据是离散的整数值，不需要分组。
| Activity hours | Tally | Frequency |
|---|---|---|
| 1 | ||| | 3 |
| 2 | |||| || | 7 |
| 3 | |||| | 5 |
| 4 | ||| | 3 |
| 5 | ||| | 3 |
| 6 | | | 1 |
Notice that 2 hours is the most common value, while 6 hours is an outlier with only one student. A frequency table instantly makes the distribution clearer.
注意到 2 小时是最常见的值，而 6 小时是离群值，只有一名学生。频数表立即使分布变得更清楚。
For screen time, the data are continuous, so we group them into intervals. A sensible class width is 0.5 hours. The grouped frequency table is shown below.
对于屏幕时间，数据是连续的，因此我们将其分组成区间。合理的组距是 0.5 小时。以下显示分组频数表。
| Screen time, t (hours) | Frequency |
|---|---|
| 1.0 ≤ t < 1.5 | 1 |
| 1.5 ≤ t < 2.0 | 3 |
| 2.0 ≤ t < 2.5 | 6 |
| 2.5 ≤ t < 3.0 | 4 |
| 3.0 ≤ t < 3.5 | 3 |
| 3.5 ≤ t < 4.0 | 2 |
| 4.0 ≤ t < 4.5 | 1 |
5. Visualising Data: Bar Charts | 数据可视化：条形图
A bar chart is perfect for discrete data like activity hours. The horizontal axis shows the number of hours, and the vertical axis shows frequency. Each bar’s height represents the count for that value. We leave equal gaps between bars to show the data are categorical in nature.
条形图非常适合像活动时间这样的离散数据。横轴显示小时数，纵轴显示频数。每个条形的高度代表该值的计数。我们在条形之间留出相等的间隙，以表明数据本质上是分类的。
From the bar chart (imagine one drawn from our frequency table), we instantly see the mode is 2 hours, and the distribution is skewed to the right, with a tail towards higher hours. This visual makes initial interpretation quick and intuitive.
从条形图中（想象一下根据我们的频数表绘制出的图形），我们立即看到众数是 2 小时，分布向右偏斜，尾部指向更高的时数。这种可视化使初步解读变得快速且直观。
Always label your axes clearly and give the chart a title. For instance: ‘Bar chart showing weekly after‑school activity hours for 20 Year 8 students’.
务必清楚地标注坐标轴并给图表加上标题。例如：“显示 20 名 Year 8 学生每周课外活动时间的条形图”。
6. Visualising Data: Pie Charts | 数据可视化：饼图
A pie chart is useful for showing proportions. To construct one, we calculate the angle for each sector: frequency ÷ total × 360°. With total frequency = 20, the angle for 2 hours is (7/20) × 360° = 126°. The full set of angles is calculated below.
饼图对于显示比例非常有用。要构建饼图，我们计算每个扇形的角度：频数 ÷ 总数 × 360°。总频数为 20，2 小时对应的角度为 (7/20) × 360° = 126°。全部角度计算如下。
Angles: 1 hour → 54°, 2 hours → 126°, 3 hours → 90°, 4 hours → 54°, 5 hours → 54°, 6 hours → 18°. These sum to 360°. The pie chart reveals that the ‘2 hours’ slice takes up more than a third of the whole, making it visually dominant.
角度：1 小时 → 54°，2 小时 → 126°，3 小时 → 90°，4 小时 → 54°，5 小时 → 54°，6 小时 → 18°。这些加起来等于 360°。饼图显示“2 小时”的扇形占据了整个图形的三分之一以上，在视觉上最为突出。
When comparing categories, a pie chart works well if you want to emphasise relative sizes. However, it can be harder to read exact frequencies than a bar chart.
在比较类别时，如果你想突出相对大小，饼图的效果很好。但相比条形图，它较难读取出精确的频数。
7. Measures of Central Tendency | 集中趋势的度量
We calculate three averages for activity hours: mean, median and mode. The mode is simply 2 hours, since it appears most frequently (7 times).
我们计算活动时间的三个平均数：均值、中位数和众数。众数就是 2 小时，因为它出现得最频繁（7 次）。
To find the median, first order the data: 1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,6. There are 20 values (even), so the median is the average of the 10th and 11th values. The 10th value is 2 and the 11th is 3, so median = (2+3)/2 = 2.5 hours.
要找到中位数，首先将数据排序：1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,6。共有 20 个数值（偶数个），因此中位数是第 10 和第 11 个数值的平均值。第 10 个数值是 2，第 11 个是 3，所以中位数 = (2+3)/2 = 2.5 小时。
The mean is the sum of all values divided by 20. Sum = (1×3)+(2×7)+(3×5)+(4×3)+(5×3)+(6×1) = 3+14+15+12+15+6 = 65. Mean = 65 ÷ 20 = 3.25 hours.
均值是所有数值的总和除以 20。总和 = (1×3)+(2×7)+(3×5)+(4×3)+(5×3)+(6×1) = 3+14+15+12+15+6 = 65。均值 = 65 ÷ 20 = 3.25 小时。
The mean is pulled upwards by the two high values (5 and 6), so it is greater than the median. This is a sign of positive skew. The mode remains the typical value.
均值被两个高值（5 和 6）拉高，因此它大于中位数。这是正偏态的一个标志。众数仍然是典型值。
8. Understanding Spread: Range | 理解分散度：极差
Range is the simplest measure of spread. For activity hours, maximum = 6, minimum = 1, so range = 6 − 1 = 5 hours. This tells us the data span 5 hours, but it is heavily influenced by the single 6‑hour student.
极差是最简单的离散程度度量。对于活动时间，最大值 = 6，最小值 = 1，所以极差 = 6 − 1 = 5 小时。这告诉我们数据跨越了 5 小时，但它严重受到唯一一名 6 小时学生的影响。
A large range relative to the typical values suggests variability. However, the range does not show whether most data cluster around a central value or are evenly spread. Combining the range with the median and interquartile range (IQR) would give a better picture, but for Year 8, the range is a good starting point.
相对于典型值而言较大的极差意味着存在变异性。然而，极差不能显示大多数数据是聚集在中心值附近还是均匀分布。将极差与中位数和四分位距（IQR）结合起来可以提供更好的图景，但对于 Year 8 而言，极差是一个很好的起点。
For screen time, the range is 4.0 − 1.0 = 3.0 hours. Comparing ranges helps you see which variable is more spread out in your sample.
对于屏幕时间，极差是 4.0 − 1.0 = 3.0 小时。比较极差有助于你了解样本中哪个变量的分布更广。
9. Scatter Graphs and Correlation | 散点图与相关性
We now investigate the relationship between screen time (horizontal axis) and activity hours (vertical axis). Plotting the 20 paired points yields a scatter graph. The cloud of points seems to slope downwards, suggesting a negative correlation: students who spend more time on screens tend to do fewer after‑school activities.
现在我们来研究屏幕时间（横轴）与活动时间（纵轴）之间的关系。将 20 对数据点绘制出来就得到了一幅散点图。点云的走向似乎向下倾斜，表明存在负相关：屏幕时间越长的学生，参加的课外活动往往越少。
To describe correlation, we use terms like ‘strong negative’, ‘weak positive’ or ‘no correlation’. In our case, the correlation appears moderate and negative. We can check by looking at approximate fitness of a straight line. A line of best fit could be drawn by eye, passing through (2.5, 3.0) roughly.
为了描述相关性，我们使用如“强负相关”“弱正相关”或“无相关”等词语。在我们的案例中，相关性看起来是中等程度的负相关。我们可以通过观察一条直线的拟合程度来检验。可以凭目测画出一条最佳拟合线，大致通过 (2.5, 3.0)。
Remember, correlation does not imply causation. More screen time might be linked to fewer activities, but we cannot say it is the cause without further evidence. This is a critical statistical thinking skill.
请记住，相关性并不意味着因果关系。屏幕时间较长可能与活动较少有关，但如果没有进一步的证据，我们不能说它就是原因。这是一项关键的统计思维技能。
10. Probability in Practice | 实际中的概率
Basic probability can be linked to our frequency table. If you pick one student at random from the 20, what is the probability that their weekly activity hours are more than 3? Values greater than 3 are 4, 5 and 6. Their frequencies sum to 3+3+1 = 7. So P(>3) = 7/20 = 0.35.
基本的概率可以与我们的频数表联系起来。如果你从 20 名学生中随机抽取一名，其每周活动时间超过 3 小时的概率是多少？大于 3 的值有 4、5 和 6。它们的频数之和为 3+3+1 = 7。因此 P(>3) = 7/20 = 0.35。
Similarly, the probability of doing exactly 2 hours is 7/20. The probability of doing between 2 and 4 hours inclusive is (7+5+3)/20 = 15/20 = 3/4. These simple calculations use the idea of relative frequency as an estimate of probability.
类似地，恰好参加 2 小时活动的概率是 7/20。参加 2 至 4 小时（含）活动的概率为 (7+5+3)/20 = 15/20 = 3/4。这些简单的计算运用了相对频数作为概率估计的概念。
You can also express probability as a fraction, decimal or percentage. This prepares you for more advanced probability topics where sample spaces and outcomes are considered.
你还可以将概率表示为分数、小数或百分比。这为你学习更高级的概率主题（考虑样本空间和结果）做好了准备。
11. Drawing Conclusions | 得出结论与反思
Based on the analysis, most Year 8 students do 2–3 hours of after‑school activities per week, with an average of 3.25 hours. There is a wide spread (range 5 hours), showing that while some are highly active, others do very little. The negative correlation with screen time might suggest a trade‑off, but it needs a larger sample to be convincing.
根据分析，大多数 Year 8 学生每周进行 2–3 小时的课外活动，平均为 3.25 小时。分布较广（极差为 5 小时），表明虽然有些学生非常活跃，但另一些学生活动很少。与屏幕时间的负相关可能暗示一种权衡，但需要更大的样本才具有说服力。
A good statistical conclusion relates back to the original question. The school might consider offering more diverse activities to engage students who currently have low participation. The report should also note limitations: small sample size and self‑reported data can introduce bias.
一个好的统计结论要回归到最初的问题。学校也许应考虑提供更多样化的活动，以吸引目前参与度较低的学生。报告还应指出局限性：样本量较小，且自我报告的数据可能引入偏差。
This case study has walked you through the practical application of Year 8 statistics — from questionnaire design to probability. Each step builds skills assessed by CIE, and practicing with real data sharpens your statistical reasoning.
这个案例研究带你走过了 Year 8 统计学的实际应用——从问卷设计到概率。每一步都练就了 CIE 评估的技能，用真实数据进行练习能增强你的统计推理能力。
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