📚 Case Study Practical Exercises | 案例分析实战演练
In statistical investigations, case studies provide a powerful way to apply concepts to real data. This article guides you through three practical exercises typical of the Year 10 CIE Statistics course. You will learn how to formulate a problem, collect and organise data, present it visually, calculate summary measures, and draw conclusions. By working through heights, favourite sports, and test scores, you will deepen your understanding of descriptive statistics and data analysis.
在统计调查中，案例研究是将概念应用于真实数据的有力方式。本文将带领你完成三个典型的10年级CIE统计课程实战练习。你将学习如何提出问题、收集与整理数据、可视化展示、计算概括性指标并得出结论。通过分析身高、喜爱的运动和考试分数这些实例，你会加深对描述性统计和数据分析的理解。
1. Introduction to Case Study Analysis | 案例分析简介
A statistical case study begins with a clear question, such as ‘What is the typical height of Year 10 students?’ or ‘Is there a relationship between maths and physics scores?’ We then collect data, analyse it, and interpret the results. The process mirrors real-world research and helps you master the statistical enquiry cycle: problem, plan, data, analysis, conclusion.
统计案例分析始于一个明确的问题，例如“10年级学生的典型身高是多少？”或“数学和物理成绩之间是否存在关系？”接着我们收集数据，进行分析并解读结果。这个过程模拟了真实世界的研究，帮助你掌握统计探究循环：问题、计划、数据、分析、结论。
In this article, we will work through three mini case studies: a survey of student heights (continuous data), a poll of favourite school sports (categorical data), and an investigation linking mathematics and physics marks (bivariate data). Each exercise highlights essential skills tested in the CIE examination.
在本文中，我们将完成三个小型案例研究：学生身高调查（连续数据）、校园最喜爱运动投票（分类数据），以及数学与物理成绩关联性研究（双变量数据）。每个练习都突出了CIE考试所考查的关键技能。
2. Defining the Problem and Data Collection | 明确问题与数据收集
For our first case study, imagine a school that wants to understand the heights of its Year 10 cohort. The statistical problem is: ‘What is the average height and how much do individual heights vary?’ To answer this, we need a representative sample. We randomly select 30 students and measure their heights to the nearest centimetre. The raw data recorded is:
针对第一个案例，假设一所学校希望了解其10年级学生的身高情况。统计问题是：“平均身高是多少？个体身高差异有多大？”为回答该问题，我们需要一个有代表性的样本。我们随机抽取30名学生，测量他们的身高并精确到厘米。记录到的原始数据如下：
150, 155, 160, 158, 162, 165, 153, 167, 170, 172, 175, 168, 164, 160, 159, 163, 166, 169, 171, 154, 157, 161, 156, 173, 176, 152, 174, 168, 162, 160
Always note the units and the context. Here, data is continuous and we might later group it for a clearer overview.
务必注明单位和背景。此处数据是连续的，我们后续可能会将其分组以便更清晰地展示全貌。
3. Organising Data: Frequency Tables | 数据整理：频数表
First, we sort the heights in ascending order: 150, 152, 153, 154, 155, 156, 157, 158, 159, 160, 160, 160, 161, 162, 162, 163, 164, 165, 166, 167, 168, 168, 169, 170, 171, 172, 173, 174, 175, 176. Because height is continuous, a grouped frequency table is more useful. We choose class intervals of width 5 cm, starting at 150–154, 155–159, and so on. Tallying the data gives the following table.
首先，我们将身高按升序排列：150, 152, 153, 154, 155, 156, 157, 158, 159, 160, 160, 160, 161, 162, 162, 163, 164, 165, 166, 167, 168, 168, 169, 170, 171, 172, 173, 174, 175, 176。由于身高是连续数据，分组频数表更为实用。我们选择组距为5厘米，从150–154开始，接着155–159，以此类推。将数据划记后得到下表。
| Height (cm) | Tally | Frequency |
|---|---|---|
| 150 – 154 | |||| | 4 |
| 155 – 159 | |||| || | 7 |
| 160 – 164 | |||| |||| | 9 |
| 165 – 169 | |||| || | 7 |
| 170 – 174 | ||| | 3 |
| 175 – 179 | | | 1 |
From the table, the modal class is 160–164 cm. A grouped table condenses the data and reveals the shape of the distribution. We can also add cumulative frequency for later use.
由表可知，众数所在组为160–164厘米。分组表浓缩了数据并揭示出分布形态。我们还可以添加累积频数以备后用。
4. Visual Representation: Histogram and Frequency Polygon | 可视化：直方图与频数多边形
With grouped continuous data, a histogram is appropriate. The horizontal axis shows the class boundaries (e.g., 149.5–154.5, 154.5–159.5, etc.) and the vertical axis shows frequency density if the intervals are unequal; here all intervals are equal, so frequency can be plotted directly. Adjacent bars touch to indicate continuity. The histogram below is constructed from our grouped table.
对于分组连续数据，直方图是合适的图形。横轴显示组界（如149.5–154.5, 154.5–159.5等），若组距不等则纵轴为频率密度；这里所有组距相等，可直接绘制频率。相邻的条形紧贴在一起以表示连续性。下面的直方图是根据我们的分组表绘制的。
We can also overlay a frequency polygon by plotting the midpoints of each class (152, 157, 162, 167, 172, 177) against the frequencies and joining the points with straight lines. The polygon makes it easier to compare distributions and spot the peak around 162 cm.
我们还可以通过描出各组中点（152, 157, 162, 167, 172, 177）对应的频数，并用直线连接这些点，来叠加频数多边形。多边形更易于比较分布并识别出在162厘米附近的峰值。
A histogram should always have labelled axes, a title, and sensible scales. In the exam, you may be asked to complete a histogram from a given table or to interpret one.
直方图必须有轴标签、标题以及合理的刻度。考试中，你可能需要根据给出的频数表补全直方图，或解读一幅直方图。
5. Measures of Central Tendency: Mean, Median, Mode | 集中趋势量数：平均数、中位数、众数
For the ungrouped height data, we calculate the three common averages. The mean is the sum of all values divided by the number of observations. Summing the 30 heights gives 4910 cm (verify: 150 + 152 + 153 + … + 176 = 4910). Therefore, mean x̄ = 4910 ÷ 30 ≈ 163.67 cm. The mean is sensitive to extreme values.
对于未分组的身高数据，我们计算三种常见的平均数。平均数是所有数值之和除以数据个数。将30个身高求和得到4910厘米（可验算：150+152+153+…+176=4910）。因此，平均数 x̄ = 4910 ÷ 30 ≈ 163.67厘米。平均数容易受极端值影响。
The median is the middle value when data is ordered. With 30 values, the median lies between the 15th and 16th ordered values. The 15th value is 162 and the 16th is 163, so the median = (162 + 163) ÷ 2 = 162.5 cm. The median is not affected by outliers.
中位数是排序后位于中间的数据。对于30个数据，中位数位于第15和第16个排序值之间。第15个值为162，第16个为163，因此中位数 = (162 + 163) ÷ 2 = 162.5厘米。中位数不受异常值影响。
The mode is the most frequent value. Looking at the sorted list, 160 cm appears three times, more than any other value, so the mode is 160 cm. A data set can have more than one mode.
众数是出现次数最多的值。观察排序后的列表，160厘米出现了三次，比任何其他值都多，因此众数为160厘米。一组数据可以有多个众数。
6. Measures of Spread: Range and Interquartile Range | 离散程度量数：范围与四分位距
Spread tells us how consistent or varied the data are. The simplest measure is the range: maximum – minimum = 176 – 150 = 26 cm. The range is easy to compute but depends only on the two extreme values.
离散程度告诉我们数据的一致性或差异性如何。最简单的衡量指标是范围：最大值 – 最小值 = 176 – 150 = 26厘米。范围易于计算，但仅依赖于两个极值。
A more robust measure is the interquartile range (IQR), which covers the middle 50% of the data. To find the quartiles, we locate the positions. The lower quartile Q₁ is at the 25th percentile. Using the formula (n+1)/4 = (30+1)/4 = 7.75, Q₁ lies between the 7th and 8th ordered values. The 7th value is 157, the 8th is 158, so Q₁ = 157 + 0.75 × (158 – 157) = 157.75 cm.
一个更稳健的指标是四分位距（IQR），它涵盖中间50%的数据。为找到四分位数，我们确定其位置。下四分位数 Q₁ 位于第25百分位。用公式 (n+1)/4 = (30+1)/4 = 7.75，Q₁ 介于第7和第8个排序值之间。第7个值为157，第8个为158，因此 Q₁ = 157 + 0.75 × (158 – 157) = 157.75厘米。
The upper quartile Q₃ is at 3(n+1)/4 = 23.25. The 23rd value is 169 and the 24th is 170, so Q₃ = 169 + 0.25 × (170 – 169) = 169.25 cm. The IQR = Q₃ – Q₁ = 169.25 – 157.75 = 11.5 cm. The IQR ignores extreme values and gives a better picture of the core spread.
上四分位数 Q₃ 位于 3(n+1)/4 = 23.25。第23个值为169，第24个为170，因此 Q₃ = 169 + 0.25 × (170 – 169) = 169.25厘米。四分位距 IQR = Q₃ – Q₁ = 169.25 – 157.75 = 11.5厘米。IQR忽略极值，能更好地反映核心数据的离散情况。
7. Box-and-Whisker Plots | 箱线图
A box plot (or box-and-whisker diagram) uses the five-number summary: minimum (150), Q₁ (157.75), median (162.5), Q₃ (169.25), maximum (176). The box is drawn from Q₁ to Q₃ with a line at the median. Whiskers extend to the minimum and maximum, provided there are no declared outliers.
箱线图（或箱须图）利用五数概括：最小值（150）、Q₁（157.75）、中位数（162.5）、Q₃（169.25）、最大值（176）。从 Q₁ 到 Q₃ 画一个矩形，并在中位数处画一条线。须线延伸至最小值和最大值，前提是没有被声明的异常值。
Box plots are excellent for comparing distributions. They show centre, spread, and skewness at a glance. For our heights, the slightly longer upper whisker and the positioning of the median inside the box suggest a mild positive skew, meaning there are a few taller students pulling the distribution to the right.
箱线图非常适合比较分布。它们能一眼展示中心、离散度和偏态。就我们的身高数据而言，上须略长且中位数在矩形内的位置表明有轻微正偏，意味着有一些较高的学生将分布向右拉长。
When drawing a box plot, use a scaled axis and label it clearly. The CIE exam often expects you to construct or interpret a box plot from a set of summary statistics.
绘制箱线图时，应使用有刻度的坐标轴并清晰标注。CIE考试通常期望你根据一组概括性数据构建或解读箱线图。
8. Case Study 2: Categorical Data – Favourite Sports | 案例研究2：分类数据 – 最喜爱的运动
In a second mini case study, 70 students were asked to name their favourite after-school sport. The responses are categorical and mutually exclusive. The frequency table is shown below.
在第二个小型案例研究中，70名学生被问及他们最喜爱的课外运动是什么。回答是分类数据且互斥。频数表如下所示。
| Sport | Frequency |
|---|---|
| Football | 25 |
| Basketball | 15 |
| Cricket | 10 |
| Tennis | 8 |
| Swimming | 12 |
For categorical data, a bar chart is suitable. The bars are separated because categories are distinct. The vertical axis shows frequency, and each bar is labelled with the sport name. A bar chart can be drawn horizontally as well. The mode is Football, with the highest frequency of 25.
对于分类数据，条形图是合适的。条形之间分开，因为各类别是独立的。纵轴显示频数，每个条形标注运动名称。条形图也可以绘制成水平形式。众数是足球，其频数最高，为25。
A pie chart is another visual tool. Each sector’s angle is calculated by (frequency ÷ total) × 360°. For Football, the angle is (25/70) × 360° ≈ 128.6°. The pie chart shows the relative popularity of each sport at a glance.
饼图是另一种可视化工具。每个扇形的角度由 (频数 ÷ 总数) × 360° 计算。对于足球，角度为 (25/70) × 360° ≈ 128.6°。饼图能一眼看出各项运动的相对受欢迎程度。
Remember that pie charts should only be used when the categories form a meaningful whole, and it is good practice to include percentages or frequencies on each sector.
请记住，饼图只应在各分类构成一个有意义的整体时使用，并且最好在每个扇区标出百分比或频数。
9. Case Study 3: Bivariate Data – Maths and Physics Scores | 案例研究3：双变量数据 – 数学与物理成绩
Our third case explores the relationship between mathematics and physics marks for 10 students. Bivariate data pairs two numerical variables. The scores are listed below:
我们第三个案例探索10名学生的数学与物理成绩之间的关系。双变量数据将两个数值变量配对。成绩列表如下：
| Maths (%) | Physics (%) |
|---|---|
| 45 | 48 |
| 56 | 55 |
| 67 | 70 |
| 78 | 74 |
| 89 | 92 |
| 90 | 88 |
| 55 | 60 |
| 65 | 63 |
| 75 | 80 |
| 85 | 82 |
A scatter diagram plots maths on the x-axis and physics on the y-axis. Each point represents one student. The pattern shows that higher maths scores are generally associated with higher physics scores, indicating a positive correlation.
散点图将数学作为x轴，物理作为y轴绘制。每个点代表一名学生。图形显示，较高的数学分数通常与较高的物理分数相对应，表明存在正相关。
We can describe the correlation as strong, moderate, or weak by observing how closely the points follow a straight line. Here the points lie near a line with few deviations, so the correlation is strong positive. CIE exams may ask you to draw the ‘line of best fit’ by eye and use it for interpolation or extrapolation.
我们可以通过观察点围绕一条直线的紧密程度，将相关性描述为强、中等或弱。这里各点接近一条直线且偏离较少，因此是强正相关。CIE考试可能会要求你凭目测画出“最佳拟合线”，并用它进行内插或外推。
In statistics, we often calculate Spearman’s rank correlation coefficient for such data, but for Year 10, visual assessment and qualitative description are stressed.
在统计学中，我们通常为此类数据计算斯皮尔曼等级相关系数，但对于10年级，主要强调可视评估和定性描述。
10. Identifying Outliers | 识别
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