📚 Cambridge Year 8 Statistics: A Case Study in Practice | 剑桥八年级统计：案例分析实战演练
In this article, we will work through a complete statistical investigation designed for Cambridge Year 8 learners. Using a case study based on screen time and reading habits, we will demonstrate how to collect, present, analyse, and interpret data, linking each step to the curriculum requirements for statistics and probability.
在这篇文章中，我们将进行一次完整的统计调查，专为剑桥八年级学生设计。通过一个关于屏幕时间和阅读习惯的案例研究，我们将展示如何收集、展示、分析和解读数据，每一步都紧扣统计与概率的课程要求。
1. Defining the Problem and Collecting Data | 定义问题与数据收集
Our case study asks: 'Is there a relationship between daily recreational screen time and weekly reading hours among Year 8 students?' To answer this, we designed a questionnaire to gather both quantitative and categorical data. The quantitative variables are screen time (hours per day) and reading time (hours per week). The categorical variable is favourite device type: smartphone, tablet, laptop, or games console.
我们的案例研究问题是：“八年级学生每天的娱乐屏幕时间与每周阅读时长之间是否存在关系？”为了回答这个问题，我们设计了一份问卷来收集定量数据和分类数据。定量变量为屏幕时间（小时/天）和阅读时间（小时/周）。分类变量是最喜欢的设备类型：智能手机、平板电脑、笔记本电脑或游戏主机。
We selected a random sample of 30 Year 8 students by choosing every 4th name on the school register. This gave a representative mix. Each student recorded their average daily screen time for recreation, their total weekly reading hours, and their single favourite device. A small section of the raw data is shown below.
我们通过从学生名册中每隔3个名字选取一人，抽取了30名八年级学生的随机样本，确保了代表性。每位学生记录了自己每日娱乐屏幕时间的平均值、每周总阅读时长以及最喜欢的一种设备。下方展示了部分原始数据。
| Student | Screen time (h/day) | Reading time (h/week) | Favourite device |
|---|---|---|---|
| Alice | 2.0 | 7.5 | Tablet |
| Ben | 4.5 | 3.0 | Smartphone |
| Chloe | 3.2 | 5.0 | Laptop |
| David | 5.8 | 1.5 | Games console |
| Eva | 1.5 | 9.0 | Tablet |
2. Organising Data and Frequency Tables | 整理数据与频率表
We first organised the screen time data into a grouped frequency table. The values ranged from 1.2 h to 6.5 h, so we chose class intervals of 0-2, 2-4, 4-6, and 6-8 hours. This makes patterns easier to see than a long list of raw numbers.
我们首先将屏幕时间数据整理成分组频率表。数值范围从1.2小时到6.5小时，因此我们选择0–2、2–4、4–6、6–8小时的组距。这样做比罗列一串原始数字更容易看出规律。
| Screen time (h) | Tally | Frequency |
|---|---|---|
| 0–2 | |||| | 4 |
| 2–4 | |||| |||| || | 12 |
| 4–6 | |||| |||| | 10 |
| 6–8 | |||| | 4 |
For the categorical data 'favourite device', we used a tally chart to record frequencies without grouping. Smartphone was most popular with 12 votes, followed by Tablet (8), Laptop (6) and Games console (4). These frequencies will be used for bar and pie charts.
对于分类数据“最喜欢的设备”，我们使用划记表记录频数，无需分组。智能手机最受欢迎，获12票，其次为平板电脑（8）、笔记本电脑（6）和游戏主机（4）。这些频数将用于条形图和饼图。
3. Drawing Bar Charts and Comparing Data | 绘制条形图与比较数据
A bar chart is perfect for displaying the device preference data. We draw vertical bars for each device, all of equal width, with heights proportional to frequency. The horizontal axis labels the device, and the vertical axis shows frequency from 0 to 14. The chart makes it instantly clear that Smartphone is the favourite, almost twice as popular as Games console.
条形图非常适合展示设备偏好数据。我们为每种设备绘制垂直的等宽条形，高度与频数成正比。横轴标注设备类型，纵轴显示从0到14的频数。图表一目了然地表明智能手机最受欢迎，几乎比游戏主机流行一倍。
Key features of a good bar chart include labelled axes, a title, and consistent gaps between bars. All bars must start from zero so that the comparison is fair. We would write a title like 'Favourite devices of Year 8 students'.
一张好的条形图的关键要素包括标注的坐标轴、标题以及条形之间一致的间隔。所有条形必须从零开始，以便公平比较。我们会写上标题，如“八年级学生最喜欢的设备”。
From the chart we can quickly read off the frequency for Laptop as 6. We can also compare two categories: there are 6 more students choosing Smartphone than Games console.
从图表中我们可以快速读出笔记本电脑的频数为6。我们也可以比较两个类别：选择智能手机的学生比游戏主机多出6人。
4. Constructing Pie Charts to Show Proportions | 构建饼图展示比例
l number of students is 30, so one student represents 360° ÷ 30 = 12°. We use the formula:
为了展示相对比例，我们将频数转换为饼图的角度。学生总数为30，因此一名学生代表360° ÷ 30 = 12°。我们使用公式：
Angle (°) = (Frequency ÷ 30) × 360
12° = 48°. The sum of angles is 144° + 96° + 72° + 48° = 360°, which confirms correctness.
将此应用到每种设备：智能手机 → 12 × 12° = 144°，平板电脑 → 8 × 12° = 96°，笔记本电脑 → 6 × 12° = 72°，游戏主机 → 4 × 12° = 48°。角度总和为144° + 96° + 72° + 48° = 360°，验证正确。
A well-drawn pie chart uses a protractor and compass, with each sector labelled clearly. The chart shows that the Smartphone sector is by far the largest, occupying 144° out of 360°. Pie charts help us see that 40% of students chose Smartphone, while only about 13% chose Games console.
画饼图需使用量角器和圆规，每个扇区应清晰标注。图表显示智能手机的扇区最大，占据360°中的144°。饼图帮助我们看出40%的学生选择了智能手机，而只有约13%选择游戏主机。
5. Line Graphs for Time Trends | 折线图揭示时间趋势
To explore variation over time, one student, John, recorded his screen time every day for a week. The data are:
为了探究随时间的变化，学生约翰记录了自己一周内每天的屏幕时间。数据如下：
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Screen time (h) | 3.0 | 2.5 | 4.0 | 3.5 | 5.0 | 6.0 | 4.5 |
We plot these as points on a graph with 'Day' on the x-axis and 'Screen time' on the y-axis and connect them with straight lines. A line graph is ideal when the horizontal axis shows time or a continuous sequence.
我们将这些点画在图上，x轴为“星期几”，y轴为“屏幕时间”，并用直线连接。当横轴表示时间或连续序列时，折线图是理想选择。
From the graph, we see screen time increased towards the weekend, peaking on Saturday at 6 hours, then falling slightly on Sunday. The trend helps John reflect on his weekly routine and consider a more balanced approach.
从图中我们看到屏幕时间在周末增加，周六达到峰值6小时，周日略有回落。这一趋势帮助约翰反思他的每周作息，并考虑更平衡的安排。
6. Scatter Graphs and Correlation | 散点图与相关性
The main question is about the relationship between screen time and reading time. We plot each of the 30 students as a point on a scatter graph, with screen time on the x-axis and reading time on the y-axis. A sample of coordinate pairs is (2.0, 7.5), (4.5, 3.0), (3.2, 5.0), (5.8, 1.5), (1.5, 9.0).
主要问题是关于屏幕时间和阅读时间之间的关系。我们将30名学生各自作为一个点描在散点图上，x轴为屏幕时间，y轴为阅读时间。部分坐标对样本为(2.0, 7.5)、(4.5, 3.0)、(3.2, 5.0)、(5.8, 1.5)、(1.5, 9.0)。
The points show a general downward trend from top-left to bottom-right. This indicates a negative correlation: as screen time increases, weekly reading hours tend to decrease. The relationship appears moderate because the points are not very tightly clustered around a straight line.
这些点显示出从左上到右下的大致下降趋势。这表明负相关：随着屏幕时间增加，每周阅读时长趋于减少。这种关系似乎是中等强度，因为点并没有紧密聚集在一条直线附近。
We can draw a line of best fit through the points, which passes through (3.0, 5.5) approximately. This line helps us predict that a student with 4 hours of screen time might read about 4 hours per week. However, correlation does not imply causation; other factors like homework or hobbies also play a role.
我们可以通过各点画一条最佳拟合线，它大致穿过(3.0, 5.5)。这条线有助于预测屏幕时间为4小时的学生可能每周阅读约4小时。但相关不等于因果；家庭作业或爱好等其他因素也起作用。
7. Calculating Central Tendency: Mean, Median, Mode | 计算集中趋势：均值、中位数、众数
Let us summarise the screen time data. The 30 values sum to 105 hours. The mean is calculated as:
现在总结屏幕时间数据。30个数值总和为105小时。均值计算如下：
Mean = Σx ÷ n = 105 ÷ 30 = 3.5 hours
ese are 3.4 h and 3.6 h, so the median is (3.4 + 3.6) ÷ 2 = 3.5 hours. Here the mean and median are equal, suggesting the distribution is fairly symmetric.
为求中位数，我们将数据从小到大排序。有30个数值，中位数位于第15和第16个值之间。在我们的排序列表中，这两个值分别为3.4小时和3.6小时，因此中位数为(3.4 + 3.6) ÷ 2 = 3.5小时。这里均值与中位数相等，暗示分布相当对称。
The mode is the most frequent value. In the grouped data, the modal class interval is 2–4 hours with a frequency of 12. For individual values, 3.2 hours appears 3 times, making it the mode among exact readings. We report both to describe typical screen use.
众数是最常出现的值。在分组数据中，众数组区间为2–4小时，频数为12。在具体数值中，3.2小时出现3次，成为精确读数中的众数。我们两者都报告，以描述典型的屏幕使用情况。
For reading time, the mean is 4.2 hours per week, the median is 4.0 hours, and the modal class is 4–6 hours. Comparing the averages, students tend to read slightly more than they spend on recreational screens daily, though individual variation is large.
对于阅读时间，均值为每周4.2小时，中位数为4.0小时，众数组为4–6小时。比较这些平均数，学生每周阅读时间略多于每日娱乐屏幕时间，但个体差异很大。
8. Measuring Spread: Range | 测量数据变异性：极差
The range tells us how spread out the data are. For screen time, the minimum is 1.2 hours and the maximum is 6.5 hours:
极差告诉我们数据的离散程度。对于屏幕时间，最小值为1.2小时，最大值为6.5小时：
Range = Maximum – Minimum = 6.5 – 1.2 = 5.3 hours
ave very low screen time, while others are heavy users. For reading time, the range is 8.0 – 1.0 = 7.0 hours, showing even greater spread in reading habits.
极差为5.3小时，表明差异相当大。一些学生屏幕时间很短，而另一些则是重度用户。关于阅读时间，极差为8.0 – 1.0 = 7.0小时，显示阅读习惯的差异更大。
Together with the averages, the ranges paint a fuller picture: the typical student has around 3.5 h screen time and 4.2 h reading, but there are always extremes. Analysing spread helps us understand that averages alone can be misleading without knowing how varied the data are.
结合平均值，极差描绘出更完整的图景：典型学生有约3.5小时屏幕时间和4.2小时阅读，但总存在极端值。分析离散程度有助于我们理解：如果不知道数据有多大差异，仅靠平均值可能会产生误导。
9. Interpreting Data and Making Inferences | 数据解读与推断
Returning to our original question, the scatter graph shows a negative correlation between screen time and reading time. The mean screen time (3.5 h) is less than the mean reading time (4.2 h), but the correlation suggests that students with higher screen time tend to read less. The bar chart and pie chart revealed that smartphones dominate, which may explain why screen time is so prevalent.
回到我们最初的问题，散点图显示屏幕时间与阅读时间呈负相关。屏幕时间的均值（3.5小时）小于阅读时间的均值（4.2小时），但相关性表明屏幕时间较高的学生往往读书更少。条形图和饼图揭示了智能手机占主导地位，这或许能解释屏幕时间为何如此普遍。
We can infer that encouraging a reduction in recreational screen time might be associated with increased reading, but we cannot prove a cause. Other factors, like availability of books at home or parental rules, likely influence both variables. A well-designed investigation must acknowledge its limitations.
我们可以推断，鼓励减少娱乐屏幕时间可能与阅读增加有关，但我们无法证明因果关系。其他因素，如家中书籍的可获得性或家长的规定，很可能影响这两个变量。一个设计良好的调查必须承认其局限性。
For decision-making, the school might use this data to launch a reading challenge or a 'screen-free' hour. The statistics provide evidence, not certainty.
在做决策时，学校或许可以利用这些数据发起阅读挑战或“无屏幕”一小时活动。统计提供的是证据，而非确定性。
10. Introduction to Probability: From Data to Chance | 概率入门：从数据到机会
We can use the collected data to think about chance. If we pick one student from the sample at random, what is the probability their favourite device is a
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