📚 Year 8 CIE Statistics: Case Study Practice | Year 8 CIE 统计：案例分析实战演练
Welcome to this case study practice session for Year 8 CIE Statistics. In this article, we will work through a real dataset step by step, applying key statistical skills including organising data, drawing charts, and calculating averages and spread. You will also learn to interpret results and explore basic probability from data.
欢迎来到 Year 8 CIE 统计的案例分析实战演练。本文将带领你一步一步地处理一个真实的数据集，运用关键的统计技能，包括整理数据、绘制图表、计算平均数和离散程度。你还将学习如何解读结果，并从数据中探索基础概率。
1. Introducing the Case Study | 案例介绍
Imagine a class of 20 students was asked about their daily screen time in hours (outside of schoolwork). The raw data collected is shown below: 1, 2, 1, 3, 0, 2, 1, 4, 2, 1, 3, 2, 0, 1, 2, 3, 1, 2, 4, 1.
假设有一个班级的 20 名学生被问及他们每天（课外）使用屏幕的时间（以小时为单位）。收集到的原始数据如下所示：1, 2, 1, 3, 0, 2, 1, 4, 2, 1, 3, 2, 0, 1, 2, 3, 1, 2, 4, 1。
2. Organising the Raw Data | 整理原始数据
The first step in any statistical analysis is to bring order to the numbers. We can list the values in ascending order to see the distribution clearly: 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4. This makes it much easier to count how many times each value appears.
任何统计分析的第一步都是把数字排列整齐。我们可以将数值从小到大列出，以便清晰地观察分布：0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4。这样能很容易地数出每个数值出现的次数。
3. Creating a Frequency Table | 制作频数表
A frequency table helps summarise the data by showing each distinct value, its tally and how many times it occurs. We will construct the table using the ordered list. The tally marks are grouped in fives for quick counting.
频数表通过列出每个不同的数值、计数符号及其出现次数来总结数据。我们将使用排序后的列表来构建表格。计数符号以五个为一组，便于快速计数。
| Screen Time (hours) | Tally | Frequency |
|---|---|---|
| 0 | || | 2 |
| 1 | |||| || | 7 |
| 2 | |||| | | 6 |
| 3 | ||| | 3 |
| 4 | || | 2 |
We can see at a glance that the most common screen time is 1 hour, reported by 7 students. The total number of data points is 20, which we will use as a check for all later calculations.
我们一眼就能看出，最常见的屏幕使用时间是 1 小时，有 7 名学生报告了该数值。数据总数为 20，我们将以此作为后续所有计算的核对依据。
4. Visualising Data: Bar Chart | 数据可视化：条形图
To draw a bar chart, label the horizontal axis with the screen time categories (0, 1, 2, 3, 4 hours) and the vertical axis with the frequency. Draw rectangular bars for each category. The height of each bar corresponds to the frequency from the table: bar for 0 hours is 2 units high, for 1 hour is 7 units, for 2 hours is 6 units, for 3 hours is 3 units and for 4 hours is 2 units. The bars must be of equal width and separated by equal gaps.
要绘制条形图，需在横轴上标出屏幕使用时间的类别（0、1、2、3、4 小时），在纵轴上标出频数。为每个类别画出矩形条。每个条的高度对应表中的频数：0 小时的条高 2 个单位，1 小时的高 7 个单位，2 小时的高 6 个单位，3 小时的高 3 个单位，4 小时的高 2 个单位。条必须等宽，且间隔相等。
This chart makes it very easy to compare the groups. The tallest bar is for 1 hour, confirming it is the most frequent value, while the shortest bars are for 0 and 4 hours, showing few students spend no time or very long on screens.
该图表让组间比较一目了然。最高的条是 1 小时，证实它是出现最频繁的数值；最矮的条是 0 小时和 4 小时，表明很少有学生不使用屏幕或使用时间很长。
5. Visualising Data: Pie Chart | 数据可视化：饼图
A pie chart displays each category as a sector of a circle. The angle of each sector is calculated using the formula: Angle = (Frequency / Total) × 360°. Let’s work out the angles for our screen time data.
饼图将每个类别显示为圆的一个扇形。每个扇形的角度用公式计算：角度 = (频数 / 总数) × 360°。我们来计算屏幕使用时间数据各扇形的角度。
0 hours: (2 / 20) × 360° = 0.1 × 360° = 36°
1 hour: (7 / 20) × 360° = 0.35 × 360° = 126°
2 hours: (6 / 20) × 360° = 0.3 × 360° = 108°
3 hours: (3 / 20) × 360° = 0.15 × 360° = 54°
4 hours: (2 / 20) × 360° = 0.1 × 360° = 36°
Adding these angles gives 36° + 126° + 108° + 54° + 36° = 360°, confirming our calculations are correct. To draw the pie chart, use a protractor to measure each angle from a starting radius. It provides a visual sense of proportions: the slice for 1 hour takes up over one third of the circle.
将这些角度相加得到 36° + 126° + 108° + 54° + 36° = 360°，证实了我们的计算是正确的。绘制饼图时，用量角器从起始半径测量每个角度。它能直观展示各部分的比例：1 小时的扇形占据了整个圆的三分之一以上。
6. Finding the Mode | 找到众数
The mode is the value that appears most often in a data set. From our frequency table, the highest frequency is 7, which belongs to the screen time of 1 hour. Therefore, the mode is 1 hour. It is the most typical amount of daily screen time for these students. The mode is especially useful for categorical data or when we want to know the most popular choice.
众数是数据集中出现次数最多的数值。从我们的频数表看，最高频数为 7，对应屏幕使用时间 1 小时。因此，众数是 1 小时。这是这些学生每天屏幕使用时间中最典型的值。众数在分类数据或我们想了解最普遍的选择时特别有用。
7. Finding the Median | 找到中位数
The median is the middle value when all data are arranged in order. We have 20 values (an even number), so the median is the average of the 10th and 11th values in the ordered list. Our ordered list is: 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4. The 10th value is 2, and the 11th value is also 2. Their average is (2 + 2) ÷ 2 = 2. Hence, the median screen time is 2 hours. The median tells us that half of the students spend 2 hours or less on screens, and half spend 2 hours or more.
中位数是所有数据按顺序排列后中间的那个值。我们有 20 个值（偶数个），所以中位数是排序列表中第 10 和第 11 个值的平均数。已排序的列表是：0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4。第 10 个值是 2，第 11 个值也是 2。它们的平均数为 (2 + 2) ÷ 2 = 2。因此，中位数屏幕使用时间是 2 小时。中位数告诉我们，一半学生花 2 小时或更少时间在屏幕上，另一半花 2 小时或更多。
8. Calculating the Mean | 计算平均数
The mean (or average) is found by adding up all the values and then dividing by how many values there are. We can use the frequency table to speed up the calculation. Multiply each screen time by its frequency, sum these products, and divide by 20.
平均数（或均值）是通过将所有数值相加，再除以数值的个数得到的。我们可以利用频数表来加快计算。将每个屏幕使用时间乘以其频数，把这些乘积相加，再除以 20。
Sum = (0×2) + (1×7) + (2×6) + (3×3) + (4×2)
Sum = 0 + 7 + 12 + 9 + 8 = 36
Mean = 36 ÷ 20 = 1.8 hours
The mean screen time is 1.8 hours. Notice that the mean (1.8) is slightly less than the median (2) because the data is a bit skewed by the few students with 0 hours. The mean gives a ‘balance point’ of the data but can be influenced by extreme values.
平均屏幕使用时间为 1.8 小时。注意平均数（1.8）略小于中位数（2），因为数据受少数几个 0 小时学生的影响而略微偏斜。平均数给出了数据的“平衡点”，但可能受极端值的影响。
9. Understanding the Range | 理解极差
The range measures how spread out the data is. It is the difference between the largest and smallest values. Here, the maximum screen time is 4 hours, and the minimum is 0 hours. Range = 4 – 0 = 4 hours. A range of 4 hours indicates that there is quite a wide variation in screen habits among these 20 students – from none to 4 hours daily.
极差衡量数据的分散程度。它是最大值与最小值的差。这里，最大屏幕使用时间是 4 小时，最小是 0 小时。极差 = 4 – 0 = 4 小时。4 小时的极差表明这 20 名学生的屏幕使用习惯差异相当大——从每天 0 小时到 4 小时不等。
10. Interpreting the Results | 结果解读
We have now calculated several summary statistics: mode = 1 hour, median = 2 hours, mean = 1.8 hours, and range = 4 hours. The mode tells us that the most common daily screen time is 1 hour. However, because the median and mean are higher, we know that many students use screens for longer periods. The range of 4 hours suggests that while some students have very low screen time, others have relatively high usage. In a report, we could suggest that most students spend between 1 and 2 hours per day on recreational screens, but there are a few outliers on both ends.
我们已经计算出了几个汇总统计量：众数 = 1 小时，中位数 = 2 小时，平均数 = 1.8 小时，极差 = 4 小时。众数告诉我们每天最常见的屏幕时间是 1 小时。然而，由于中位数和平均数更高，我们知道许多学生使用屏幕的时间更长。4 小时的极差表明，尽管一些学生屏幕时间很少，但另一些学生使用时间相对较高。在一份报告中，我们可以指出大多数学生每天在娱乐屏幕上花费 1 到 2 小时，但两端都存在少数异常值。
11. Probability from Data | 从数据中看概率
We can use the data to estimate probabilities. Suppose we pick one student at random from the class. What is the probability that this student’s daily screen time is greater than 2 hours? From the frequency table, the students with screen time > 2 hours are those with 3 hours (3 students) and 4 hours (2 students). Total favourable outcomes = 3 + 2 = 5. Total outcomes = 20. So the probability is 5/20 = 1/4 = 0.25 or 25%.
我们可以利用数据来估计概率。假设我们从班级中随机选一名学生。这名学生每天屏幕使用时间大于 2 小时的概率是多少？根据频数表，屏幕时间 > 2 小时的学生是那些 3 小时（3 人）和 4 小时（2 人）的学生。有利结果总数 = 3 + 2 = 5。总结果数 = 20。所以概率为 5/20 = 1/4 = 0.25 或 25%。
Probability = (Number of favourable outcomes) / (Total number of outcomes) is a fundamental concept linked directly to frequency. This shows how data can be used to make predictions about an individual.
概率 = (有利结果数) / (总结果数) 是直接与频数相关的基本概念。这表明了数据可如何用于对个体进行预测。
12. Drawing Conclusions | 得出结论
In this case study, we turned a raw list of numbers into meaningful information. We learned that in this class, the typical daily recreational screen time is around 1 to 2 hours, with an average of 1.8 hours. The data is quite varied, but only 25% of students exceed 2 hours per day. If we were giving advice, we might recommend looking closer at why some students have 0 or 4 hours, and perhaps encourage balanced screen habits. Always remember to check your calculations, present charts neatly, and explain what the numbers actually mean in real life.
在这个案例研究中，我们将一组原始的数字列表转化为了有意义的信息。我们了解到，在这个班级中，典型的每天娱乐屏幕时间在 1 到 2 小时左右，平均为 1.8 小时。数据差异较大，但只有 25% 的学生每天超过 2 小时。如果我们要给出建议，我们可能会建议仔细探究为什么有些学生是 0 小时或 4 小时，或许鼓励平衡的屏幕使用习惯。请始终记得检查计算，整洁地呈现图表，并解释这些数字在现实生活中真正意味着什么。
Published by TutorHao | Statistics Revision Series | aleveler.com
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