Case Study in Statistics: A Practical Workout | 统计案例分析实战演练

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

A school is planning to improve its sports facilities and wants to understand what activities students enjoy most. This case study will take you through a complete statistical investigation, from designing a survey to drawing conclusions. You will learn how to work with real data, create charts, calculate averages and even make simple predictions using probability.

一所学校计划改善体育设施，希望了解学生最喜欢哪些运动。本案例将带你经历一个完整的统计调查过程，从设计问卷到得出结论。你将学会如何处理真实数据、制作图表、计算平均值，甚至用概率进行简单预测。

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1. Introduction to the Case Study | 案例介绍

Imagine you are helping the school council collect information about sports preferences. Your task is to find out which sport is the most popular among Year 8 students and how much time they spend on physical activity each week. The results will be used to decide whether to build a new basketball court or improve the swimming pool. Good data analysis means the school can make a better decision.

想象你正在帮助学校学生会收集运动偏好的信息。你的任务是找出八年级学生中最受欢迎的运动，以及他们每周运动多长时间。结果将用于决定是新建一个篮球场还是改善游泳池。好的数据分析能帮助学校做出更好的决定。

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2. Designing the Survey | 设计调查

We designed a short questionnaire with two questions: ‘What is your favourite sport?’ and ‘How many hours do you exercise per week?’ The first question is categorical (nominal data), while the second is numerical (discrete data). We decided to survey 30 students chosen randomly from the Year 8 register. The sample size is small but manageable for a classroom investigation.

我们设计了一份简短问卷，包含两个问题：“你最喜欢的运动是什么？”和“你每周运动几小时？”第一个问题是分类数据（名义数据），第二个是数值数据（离散数据）。我们决定从八年级名册中随机抽取30名学生进行调查。样本虽小，但在课堂调查中便于操作。

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3. Collecting Data | 收集数据

Here are the responses from the 30 students. The favourite sports recorded were: Football, Basketball, Swimming, Tennis, and Running. The weekly exercise hours for each student, in the same order, were: 3, 4, 5, 2, 6, 1, 4, 5, 3, 2, 7, 4, 6, 3, 5, 2, 4, 1, 6, 5, 4, 3, 7, 2, 5, 4, 3, 6, 4, 5. We now have a raw data set ready to be organised.

以下是30名学生的回答。记录的最喜爱运动有：足球、篮球、游泳、网球和跑步。每名学生对应的每周运动小时数（相同顺序）为：3, 4, 5, 2, 6, 1, 4, 5, 3, 2, 7, 4, 6, 3, 5, 2, 4, 1, 6, 5, 4, 3, 7, 2, 5, 4, 3, 6, 4, 5。现在我们有了原始数据集，可以开始整理了。

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4. Organising Data: Frequency Tables | 整理数据：频数表

For categorical data, a frequency table shows how many students chose each sport. Our tally gives: Football 12, Basketball 7, Swimming 5, Tennis 4, Running 2. This quickly reveals Football as the most popular choice. For the numerical hours, we can group the data or simply list the frequencies of each value. A frequency table for hours helps us see the most common exercise amounts.

对于分类数据，频数表显示了每种运动的选择人数。计数结果为：足球12人，篮球7人，游泳5人，网球4人，跑步2人。这迅速揭示了足球是最受欢迎的选择。对于数值型的小时数，我们可以分组，或直接列出每个数值的频数。小时数的频数表可以帮助我们看到最常见的运动时长。

Hours frequency: 1 hour: 2 students, 2 hours: 4 students, 3 hours: 5 students, 4 hours: 7 students, 5 hours: 6 students, 6 hours: 4 students, 7 hours: 2 students.

小时数频数：1小时2人，2小时4人，3小时5人，4小时7人，5小时6人，6小时4人，7小时2人。

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5. Bar Charts and Pictograms | 条形图和象形图

A bar chart is perfect for displaying the favourite sport data. The horizontal axis shows the sport categories, and the vertical axis shows the frequency. The height of each bar represents the count. Using the same data, a pictogram could use a football icon to represent, say, 2 students, making the comparison visual and fun. Bar charts are easy to read and are a key part of KS3 statistics.

条形图非常适合展示最喜欢运动的数据。横轴表示运动类别，纵轴表示频数。每个条的高度代表数量。对于同样的数据，象形图可以用一个足球图标代表例如2名学生，使对比更直观有趣。条形图易于阅读，是KS3统计学的关键内容。

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6. Pie Charts | 饼图

To create a pie chart, we calculate the angle for each sector using the formula: sector angle = (frequency / total) × 360°. For Football: (12/30) × 360° = 144°. Basketball: (7/30) × 360° = 84°. Swimming: (5/30) × 360° = 60°. Tennis: (4/30) × 360° = 48°. Running: (2/30) × 360° = 24°. The pie chart clearly shows Football occupies the largest slice, making it an effective visual for proportions.

要制作饼图，我们使用公式计算每个扇形的角度：扇形角度 = (频数 / 总数) × 360°。足球：(12/30) × 360° = 144°；篮球：(7/30) × 360° = 84°；游泳：(5/30) × 360° = 60°；网球：(4/30) × 360° = 48°；跑步：(2/30) × 360° = 24°。饼图清晰地显示足球占据了最大的一块，使其成为展示比例的直观方式。

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7. Measures of Central Tendency: Mean, Median, Mode | 集中趋势度量：平均数、中位数、众数

We use the exercise hours data to find three averages. The mean is the sum divided by the count. Sum of hours = 3+4+5+2+6+1+4+5+3+2+7+4+6+3+5+2+4+1+6+5+4+3+7+2+5+4+3+6+4+5 = 121. Mean = 121 / 30 ≈ 4.03 hours.

我们用运动小时数据求三种平均数。平均数是总和除以个数。小时数总和=121。平均数=121/30≈4.03小时。

To find the median, we order the data: 1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7. The median is the middle value; with 30 values, it is the average of the 15th and 16th numbers. Both are 4, so median = 4 hours. The mode is the value that appears most often, which is 4 hours (7 students). So the typical student exercises about 4 hours per week.

求中位数时，我们把数据排序：1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7。中位数是中间值；有30个数，取第15和第16个数的平均。两者都是4，所以中位数=4小时。众数是出现最多的值，即4小时（7人）。因此，典型学生每周大约运动4小时。

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8. Range and Spread | 极差与分布

The range tells us how spread out the exercise hours are. Range = maximum – minimum = 7 – 1 = 6 hours. A large range means there is a big difference between the least and most active students. In this case, some students exercise only 1 hour, while others do 7 hours. Range is a simple measure of spread, but it doesn’t tell us about the pattern in between.

极差告诉我们运动小时的分布有多广。极差 = 最大值 – 最小值 = 7 – 1 = 6 小时。较大的极差意味着最不活跃和最活跃的学生之间差异很大。在这里，有些学生只运动1小时，而有些达到7小时。极差是一个简单的离散程度度量，但它不能反映中间的模式。

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9. Comparing Groups with Dual Bar Charts | 用双条形图比较组别

Suppose we also recorded the gender of each student. The boys’ preferences: Football 8, Basketball 3, Swimming 2, Tennis 2, Running 1. The girls’ preferences: Football 4, Basketball 4, Swimming 3, Tennis 2, Running 1. A dual bar chart places bars side by side for each sport, making it easy to compare. We can see Football is more popular among boys, while Basketball is equally favoured. This helps the school understand different group interests.

假设我们还记录了每名学生的性别。男生偏好：足球8，篮球3，游泳2，网球2，跑步1。女生偏好：足球4，篮球4，游泳3，网球2，跑步1。双条形图将每个运动的条形并排放置，便于比较。我们可以看到足球在男生中更受欢迎，而篮球受欢迎程度相当。这有助于学校了解不同群体的兴趣。

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10. Introduction to Probability from Data | 从数据中引入概率

We can use relative frequency to estimate probabilities. If we pick one student at random, the probability their favourite sport is Football is 12/30 = 0.4 or 40%. The probability of choosing a student who exercises more than 5 hours a week: count of students with 6 or 7 hours = 4+2=6, so probability = 6/30 = 0.2 = 20%. These are experimental probabilities based on our sample.

我们可以用相对频率估计概率。如果随机抽取一名学生，其最喜欢足球的概率是12/30=0.4，即40%。选择一名每周运动超过5小时的学生的概率：6小时和7小时的学生数为4+2=6，概率为6/30=0.2=20%。这些都是基于样本的实验概率。

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11. Drawing Conclusions and Making Predictions | 得出结论与做出预测

Our analysis shows Football is the clear favourite, with a mean exercise time of about 4 hours per week. The range of 6 hours indicates varied activity levels. The school could now predict that if a new Year 8 cohort is similar, about 40% would prefer Football. They might invest in football facilities while also catering to swimmers and basketball players. Predictions are always uncertain because they rely on the sample being representative.

我们的分析显示足球是明显的最爱，平均每周运动时间约4小时。6小时的极差表明活动水平差异大。学校现在可以预测，如果新一届八年级学生情况相似，约40%会偏爱足球。学校或许会投资足球设施，同时兼顾游泳和篮球爱好者。预测总是不确定的，因为它们依赖于样本的代表性。

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12. Reflection on the Statistical Process | 统计过程反思

This case study demonstrates the complete statistical cycle: posing a question, collecting and processing data, presenting it visually, analysing with averages and spread, and interpreting results. We learned that no single measure tells the whole story; using mean, median, mode and range together gives a fuller picture. Limitations include a small sample size and possible biased responses. Nevertheless, the investigation provides a solid foundation for data-based decisions.

本案例展示了完整的统计循环：提出问题，收集和处理数据，用图表呈现，用平均数和离散程度进行分析，并解释结果。我们认识到没有单一的度量能说明所有问题；同时使用平均数、中位数、众数和极差能提供更全面的图景。局限性包括样本量小和可能存在回答偏差。尽管如此，这次调查为基于数据的决策提供了坚实基础。

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Published by TutorHao | Statistics Revision Series | aleveler.com

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