Case Study Practical Exercise in Statistics | 统计案例分析实战演练

📚 Case Study Practical Exercise in Statistics | 统计案例分析实战演练

Welcome to your first real-world statistical investigation! In this case study, you will follow the complete statistical cycle – from posing a question to collecting data, organising it, presenting graphs, calculating averages, and drawing conclusions. The scenario focuses on a Year 7 class investigating how many hours students spend reading for pleasure each week. By working through each stage, you will strengthen the data-handling skills that form the core of the Cambridge Year 7 statistics curriculum.

欢迎来到你的第一次真实统计调查！在这个案例研究中，你将经历完整的统计循环——从提出问题、收集数据、整理数据、绘制图表、计算平均数，一直到得出结论。场景设定为七年级的一个班级调查学生每周课外阅读多少小时。通过逐步推进每一个阶段，你将夯实剑桥七年级统计课程核心的数据处理技能。

1. Introduction to the Case Study | 案例介绍

A group of Year 7 students wants to find out: ‘How many hours per week do students in our class spend reading for enjoyment?’ They suspect that reading habits vary a lot, and they want to use statistics to describe the class’s behaviour clearly. This question is perfect for a statistical investigation because the data is numerical (discrete) and easy to gather through a simple survey.

一群七年级学生想知道：“我们班同学每周花多少小时进行课外阅读？”他们猜测阅读习惯差异很大，希望用统计方法清楚地描述全班的情况。这个问题非常适合开展统计调查，因为数据是数值型（离散）的，而且只需通过简单的问卷就能收集。

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2. Data Collection: Designing a Survey | 数据收集：设计调查问卷

The students designed a short questionnaire asking each classmate: ‘How many whole hours did you spend reading for fun last week?’ They made sure the question was clear and that everyone understood it meant leisure reading, not schoolwork. They collected responses from all 20 students in the class, ensuring nobody was left out, which gives a complete population data set for their class.

学生们设计了一份简短的问卷，向每位同学提问：“你上周花了多少完整小时进行消遣性阅读？”他们确保问题清晰，每个人都明白是指休闲阅读，不是作业阅读。他们收集了全班20名学生的回答，不漏掉任何人，因此获得了该班级的总体数据集。

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3. Organising Raw Data | 整理原始数据

The raw responses were recorded as follows (in hours): 2, 3, 1, 5, 4, 2, 3, 6, 0, 1, 3, 4, 2, 5, 3, 1, 2, 4, 3, 2. At first glance, it is hard to spot any pattern. The first step is to sort the data in ascending order, which helps to see the spread and to count frequencies later.

收集到的原始回答（以小时计）如下：2, 3, 1, 5, 4, 2, 3, 6, 0, 1, 3, 4, 2, 5, 3, 1, 2, 4, 3, 2。乍看之下很难发现规律。第一步是将数据按升序排列，这有助于观察数据的分布，也方便后续计算频数。

Sorted data: 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6. Now we can quickly see the smallest and largest values and prepare to group identical values.

排序后的数据：0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6。现在我们能快速看到最小值和最大值，并准备将相同的值归组。

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4. Frequency Tables: Tally and Count | 频数表：画记与计数

A frequency table shows how often each value appears. We use tally marks to count systematically, then record the frequency. Below is the completed frequency table for our reading hours data.

频数表显示每个数值出现的次数。我们使用画记法系统地计数，然后记录频数。下面是阅读小时数数据的频数表。

Hours (h)	Tally	Frequency
0	|	1
1	|||	3
2	||||	5
3	||||	5
4	|||	3
5	||	2
6	|	1

From the table, the total frequency adds up to 20, which matches the class size. The mode is the value with the highest frequency – here both 2 hours and 3 hours share the highest frequency of 5.

从表中可以看出，总频数加起来为20，与班级人数相符。众数是出现次数最多的数值——这里2小时和3小时的频数最高，均为5。

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5. Visualising Data: Bar Charts | 数据可视化：条形图

A bar chart is perfect for displaying discrete numerical data like our reading hours. The horizontal axis shows the number of hours, and the vertical axis shows the frequency. Each bar is drawn with equal width, and the height corresponds to the frequency. Bars do not touch, because the data is categorical-like integers.

条形图非常适合展示像阅读小时数这样的离散数值数据。横轴表示小时数，纵轴表示频数。每个条形的宽度相等，高度与频数对应。条形之间不接触，因为该数据类似分类整数值。

If we sketched the bar chart: bar for 0 hours at height 1; 1 hour at height 3; 2 hours at 5; 3 hours at 5; 4 hours at 3; 5 hours at 2; 6 hours at 1. The graph would show a shape that rises and then falls, with a peak around 2–3 hours. This gives an immediate visual impression of the most common reading times.

如果我们画出条形图：0小时对应的条形高度为1；1小时高度为3；2小时高度为5；3小时高度为5；4小时高度为3；5小时高度为2；6小时高度为1。图形会呈现先升后降的形状，在2–3小时附近达到峰值。这能让人瞬间直观感受到最常见的阅读时长。

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6. Visualising Data: Pie Charts | 数据可视化：饼图

A pie chart shows proportions of the whole. To construct it, we calculate the angle for each sector using the formula: Angle = (Frequency / Total frequency) × 360°. For example, for 2 hours: (5 ÷ 20) × 360° = 90°. We can present the calculations in a table.

饼图能显示各部分的占比。要绘制饼图，我们需用公式计算每个扇形的角度：角度 = (频数 ÷ 总频数) × 360°。例如，2小时对应的角度：(5 ÷ 20) × 360° = 90°。我们可以用表格展示计算过程。

Hours	Frequency	Calculation	Angle (°)
0	1	(1/20)×360	18°
1	3	(3/20)×360	54°
2	5	(5/20)×360	90°
3	5	(5/20)×360	90°
4	3	(3/20)×360	54°
5	2	(2/20)×360	36°
6	1	(1/20)×360	18°

Notice that the two equal largest frequencies (2 and 3 hours) both take up 90°, each covering a quarter of the pie. This visual makes it easy to compare categories – the ‘2–3 hour’ region clearly dominates.

请注意，两个相同的最高频数（2小时和3小时）各占90°，均覆盖四分之一饼图。这种可视化方式便于比较各类别——”2–3小时”区域明显占主导地位。

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7. Measures of Central Tendency: Mean | 集中趋势度量：平均数

The mean (often called the average) is found by adding all the data values together and then dividing by the number of values. For our data set we sum all 20 hours carefully.

平均数（通常所说的平均值）是将所有数据值相加，再除以数据个数得到的。对于我们的数据集，需仔细加总20个小时数。

Sum = 0 + 1×3 + 2×5 + 3×5 + 4×3 + 5×2 + 6×1 = 0 + 3 + 10 + 15 + 12 + 10 + 6 = 56

Mean = 56 ÷ 20 = 2.8 hours

So the average reading time is 2.8 hours per week. The mean is useful because it takes every student’s time into account, but it can be influenced by very high or low values – here the 6 and 0 stretch it slightly.

因此平均阅读时间为每周2.8小时。平均数很有用，因为它考虑到了每位学生的时间，但它容易受极端高值或低值的影响——这里的6和0稍微拉低或拉高了平均值。

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8. Median and Mode | 中位数与众数

The median is the middle value when the data is ordered. With 20 values (an even number), the median is the mean of the 10th and 11th values. Looking at our sorted list: positions 1–20, the 10th value is 3 (since 1–3: 0,1,1,1; 4–8: five 2s; 9–13: five 3s). The 10th is 3, the 11th is also 3, so median = (3 + 3) ÷ 2 = 3 hours.

中位数是数据排序后位于中间的值。因为有20个值（偶数个），中位数是第10个和第11个值的平均数。查看排序后的列表：第1–3位为0,1,1,1；第4–8位为五个2；第9–13位为五个3。第10个值是3，第11个值也是3，因此中位数 = (3 + 3) ÷ 2 = 3小时。

The mode is the value that appears most often. From the frequency table, both 2 and 3 hours occur 5 times each. So the data set is bimodal: modes are 2 hours and 3 hours. Mode is especially useful for spotting the most typical reading duration.

众数是出现次数最多的值。从频数表来看，2小时和3小时各出现5次。因此这个数据集是双峰的：众数为2小时和3小时。众数特别适合识别最典型的阅读时长。

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9. Understanding Range | 理解极差

The range measures how spread out the data is. It is calculated as: Range = Maximum value – Minimum value. For the reading hours, the maximum is 6 and the minimum is 0, so Range = 6 – 0 = 6 hours. A range of 6 indicates a fairly wide spread within a 0–6 hour scale, showing that reading habits differ considerably among students.

极差衡量数据的离散程度。计算公式为：极差 = 最大值 – 最小值。对于阅读小时数，最大值为6，最小值为0，因此极差 = 6 – 0 = 6小时。6小时的极差表明在0–6小时的尺度内分布相当广泛，说明学生之间的阅读习惯差异很大。

Although the range is quick to compute, it only uses the two extreme values and does not tell us about the clustering in the middle. For that, we look at the interquartile range in more advanced work, but for Year 7 the range is a good starting point.

尽管极差计算起来很快，但它只用了两个极端值，无法告诉我们中间值的聚集情况。在更高年级的学习中我们会看四分位距，但对七年级而言，极差是一个很好的起点。

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10. Interpreting the Results | 解读结果

Now we bring all the statistics together. The typical reading time is around 2–3 hours (mode = 2 and 3, median = 3). The mean of 2.8 hours is very close to these, which tells us the distribution is fairly symmetric without extreme skew. The range of 6 reminds us there are outliers – one student reads 0 hours, another reads 6 hours.

现在我们汇总所有统计量。典型的阅读时间在2–3小时左右（众数为2和3，中位数为3）。平均数2.8小时与它们非常接近，这表明分布相当对称，没有极端偏斜。6小时的极差提醒我们存在离群值——一名学生阅读0小时，另一名阅读6小时。

From the bar chart and pie chart, it is clear that very few students read 0 or 6 hours. The bulk of the class (16 out of 20) reads between 1 and 5 hours, with the largest concentration at 2 and 3 hours. The class might conclude that most students do spend some time reading for pleasure each week.

从条形图和饼图可以清楚看出，几乎没有学生阅读0小时或6小时。班上大多数学生（20人中有16人）的阅读时间在1至5小时之间，最集中的是2小时和3小时。班级可以得出结论：大多数学生每周确实会花一些时间进行消遣性阅读。

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11. Drawing Conclusions and Reflection | 得出结论与反思

Based on the investigation, the Year 7 students can answer their initial question: the weekly leisure reading hours typically fall between 2 and 3 hours, with a class average of 2.8 hours. However, they also note the wide range and recommend a follow-up survey to explore why some students read very little – perhaps access to books or time spent on other hobbies.

根据调查结果，七年级学生们可以回答最初的问题：每周课外阅读小时数通常在2到3小时之间，班级平均为2.8小时。然而，他们也注意到极差较大，并建议进行后续调查，以探究为什么有些学生阅读很少——可能是获取书籍的渠道或花在其他爱好上的时间所致。

Reflecting on the process, students realise how important it is to collect data carefully, organise it, and use more than one average to describe the centre. The case study also shows that drawing a graph helps to spot patterns instantly that tables of numbers might hide.

反思整个过程，学生们意识到仔细收集数据、整理数据，以及使用多个平均数来描述中心趋势是何等重要。本案例也表明，绘制图表有助于迅速发现数据表可能隐藏的规律。

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12. Summary of Key Skills | 关键技能总结

This case study has given you hands-on practice with the essential Year 7 statistics toolkit: designing a simple survey, creating frequency tables with tallies, drawing bar charts and pie charts, calculating the mean, median, mode, and range, and interpreting results in context. These skills will be built upon in future years, but mastering them now will give you confidence in handling data in any subject.

本案例让你亲身实践了七年级统计学的核心工具包：设计简单问卷、用画记法制作频数表、绘制条形图和饼图、计算平均数、中位数、众数和极差，以及结合情境解读结果。这些技能将在今后的学习中不断深化，但现在就掌握它们将使你有信心处理任何学科中的数据。

Remember to always check your calculations, label graphs clearly, and relate your findings back to the original question. Statistics is not just about numbers – it’s about telling the story behind the data.

请记住，一定要检查计算、为图表添加清晰的标签，并将发现与最初的问题联系起来。统计学不仅仅是关于数字——它讲述的是数据背后的故事。

Published by TutorHao | 统计 Revision Series | aleveler.com

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