📚 Case Study Practice in Statistics | 统计学案例分析实战演练
Statistics is not just about learning formulas—it is about applying your skills to real-world problems. In CAIE Year 8 Mathematics, you will often be asked to work through a case study: a practical scenario where you collect, organise, display and interpret data. This article gives you two complete case studies to practice, along with key techniques and common mistakes to avoid. By working through these examples, you will build confidence in handling statistical questions that combine graphs, averages and data comparison.
统计学不仅仅是学习公式，更重要的是将你的技能应用到实际问题中。在 CAIE 八年级数学中，你经常会遇到案例分析题——也就是要求你收集、整理、展示并解读数据的实际场景。本文为你提供两个完整的案例进行练习，并介绍重要的方法和常见错误。通过一步步完成这些例子，你将更有信心应对那些结合了图表、平均数和数据比较的统计问题。
1. What Is a Statistical Case Study? | 什么是统计案例研究？
A statistical case study is a real-life situation where you use data to answer a question. You might need to decide which data to collect, how to record it, which graph best shows the pattern, and what the numbers tell you. In Year 8, case studies often involve comparing groups, spotting trends over time, or summarising a large set of measurements.
统计案例研究指的是利用数据来回答问题的真实情境。你可能需要确定收集哪些数据、如何记录、哪类图表最能展示规律，以及数字背后说明了什么。在八年级，案例研究常常涉及比较不同组别、发现随时间变化的趋势，或者总结一组较大的测量数据。
2. Case Study 1: School Sports Day Results | 案例一：学校运动会成绩
Imagine you help the PE teacher analyse the long jump results of 15 students. Their jumps, measured to the nearest centimetre, are shown below:
假设你帮助体育老师分析 15 名学生的跳远成绩。他们的成绩（精确到厘米）如下：
| 120, 135, 140, 125, 130, 145, 150, 128, 132, 138, 122, 136, 141, 127, 131 |
You need to organise the data, draw a graph, and calculate average and spread. This is a typical case study task.
你需要整理数据、绘制图表，并计算平均数和离散程度。这是一项典型的案例分析任务。
3. Collecting and Organising Data | 数据的收集与整理
First, put the raw data in order from smallest to largest. This makes it easier to find the median and range. The ordered list is: 120, 122, 125, 127, 128, 130, 131, 132, 135, 136, 138, 140, 141, 145, 150.
首先，将原始数据从小到大排序。这样更容易找到中位数和极差。排序后的数据为：120, 122, 125, 127, 128, 130, 131, 132, 135, 136, 138, 140, 141, 145, 150。
Next, group the data into class intervals to create a frequency table. Suitable intervals could be 120-124, 125-129, and so on. Good grouping helps you see the shape of the distribution without losing too much detail.
接下来，将数据分组并制作频数表。合适的分组区间可以是 120-124，125-129 等。合理的分组能帮助你看出分布的形状，同时不会丢失太多细节。
- 120-124: 3 (120, 122, 125? Wait 125 is 125, so 120-124 includes 120,122. Actually need consistent boundaries. Let’s set intervals 120-124, 125-129, 130-134, 135-139, 140-144, 145-149, 150-154. Then count: 120,122 (2); 125,127,128 (3); 130,131,132 (3); 135,136,138 (3); 140,141 (2); 145 (1); 150 (1). This is messy; better to keep simpler: use 120-129, 130-139, 140-149, 150-159. Count: 120,122,125,127,128 (5); 130,131,132,135,136,138 (6); 140,141,145 (3); 150 (1). For Year 8, ungrouped is fine, but we’ll demonstrate frequency table. I’ll adjust intervals in article to match typical Year 8: 120-124, 125-129 etc with correct boundaries: 120-124 includes 120,121,122,123,124. Our data: 120,122 -> 2; 125,127,128 -> 3; 130,131,132 -> 3; 135,136,138 -> 3; 140,141 -> 2; 145 -> 1; 150 -> 1. So the frequency table will be: 120-124: 2; 125-129: 3; 130-134: 3; 135-139: 3; 140-144: 2; 145-149: 1; 150-154: 1. This is correct and allows bar chart. In the article I’ll present that. So I’ll write the table as described.
A frequency table for the long jump data might look like this:
跳远数据的频数表可能如下：
| Class interval (cm) | Tally | Frequency |
|---|---|---|
| 120-124 | || | 2 |
| 125-129 | ||| | 3 |
| 130-134 | ||| | 3 |
| 135-139 | ||| | 3 |
| 140-144 | || | 2 |
| 145-149 | | | 1 |
| 150-154 | | | 1 |
Recording data in a table is the first step in any case study. It helps you see how the values are spread.
在表格中记录数据是所有案例分析的第一步。它能帮你看出数据是如何分布的。
4. Frequency Tables and Bar Charts | 频率表和条形图
From the frequency table you can draw a bar chart. Use the class intervals on the horizontal axis and frequency on the vertical axis. Make sure the bars are of equal width and clearly labelled. A bar chart makes it easy to compare how many students jumped within each distance range.
根据频数表你可以画出条形图。将分组区间放在横轴，频数放在纵轴。确保柱子的宽度相等并清楚标注。条形图能让你更容易比较每个距离范围内有多少学生。
When you describe the bar chart, mention which interval has the highest frequency (the modal class) and whether the data is symmetrical or skewed. For our data, the modal class is 125-129, 130-134 and 135-139 all with 3, so the distribution is fairly flat in the middle.
当你描述条形图时，要指出哪个区间的频数最高（众数所在的组），以及数据是对称的还是偏斜的。对于我们这组数据，众数所在组是 125-129、130-134 和 135-139，频数均为 3，因此分布的中部较为平坦。
5. Calculating Averages: Mean, Median, Mode | 计算平均数：均值、中位数、众数
Now calculate the three measures of central tendency. The mean is found by adding all the values and dividing by the number of values:
现在计算三种集中趋势的量度。均值通过将所有数值相加再除以数值个数得出：
Mean = (120 + 135 + 140 + 125 + 130 + 145 + 150 + 128 + 132 + 138 + 122 + 136 + 141 + 127 + 131) ÷ 15
Sum = 2000? Let’s actually sum: 120+135=255, +140=395, +125=520, +130=650, +145=795, +150=945, +128=1073, +132=1205, +138=1343, +122=1465, +136=1601, +141=1742, +127=1869, +131=2000. Yes sum is 2000. So mean = 2000 ÷ 15 ≈ 133.3 cm.
总和为 2000，除以 15 得到均值约为 133.3 厘米。
The median is the middle value when the data is ordered. With 15 values, the 8th value is the median: 132 cm.
中位数是排序后位于中间的数值。15 个数据中，第 8 个是中位数：132 厘米。
The mode is the most frequent value. Here, no single jump appears more than once, so there is no mode. However, the modal class from the grouped table is the interval 125-129, 130-134, 135-139 all with 3. When a data set has no repeated number, we usually say it has no mode, or we use the modal class.
众数是出现次数最多的数值。这里每个成绩都只出现一次，所以没有众数。不过，根据分组表，众数所在的组是 125-129、130-134 和 135-139，频数均为 3。当数据集没有重复数字时，我们通常说没有众数，或者使用众数组。
6. Understanding Range | 了解极差
The range tells you how spread out the data is. It is the difference between the largest and smallest values:
极差告诉你数据的分散程度。它是最大值与最小值之差：
Range = 150 − 120 = 30 cm
A small range means the jumps are close together; a large range indicates more variation. In a case study, always comment on what the range tells you about consistency.
极差小说明跳远成绩很接近；极差大则表示变化较大。在案例分析中，一定要针对极差所反映的一致性情况加以评论。
7. Case Study 2: Monthly Temperature Data | 案例二：月度气温数据
A second case study involves time-series data. The table below shows the average monthly temperature in a city over one year.
第二个案例涉及时间序列数据。下表显示了某城市一年中各月的平均气温。
| Month | Temp (°C) |
|---|---|
| Jan | 5 |
| Feb | 7 |
| Mar | 10 |
| Apr | 15 |
| May | 20 |
| Jun | 25 |
| Jul | 28 |
| Aug | 27 |
| Sep | 23 |
| Oct | 17 |
| Nov | 11 |
| Dec | 6 |
Your task is to display the data in a line graph, find the mean temperature, and describe the trend.
你的任务是绘制折线图展示数据，计算平均气温，并描述变化趋势。
8. Drawing and Interpreting Line Graphs | 绘制与解读折线图
A line graph is ideal for time-series data. Plot the months on the horizontal axis and temperature on the vertical axis. Join the points with straight lines. Make sure both axes are labelled and the scale is even.
折线图非常适合时间序列数据。将月份放在横轴，气温放在纵轴，用直线连接各点。确保两个坐标轴都有标注，且刻度间隔均匀。
From the line graph, you can see that the temperature rises steadily from January to July, peaks in July at 28°C, and then falls gradually to December. This clear seasonal pattern is a key feature to mention in your case study report.
从折线图中可以看出，气温从一月到七月稳步上升，在七月达到峰值 28°C，然后逐渐下降到十二月。这种清晰的季节规律是你在案例分析报告里需要提到的重要特征。
9. Comparing Data Sets | 比较数据集
Often, a case study asks you to compare two sets of data. For example, you might compare the temperatures of two different cities, or compare the long jump results of boys and girls. When comparing, always calculate the same statistics for both sets and then comment on differences in mean, median, range and shape of the graph.
案例分析经常要求你比较两组数据。例如，你可能会比较两个不同城市的气温，或者比较男生和女生的跳远成绩。比较时，要为两组数据计算相同的统计量，然后评论它们在均值、中位数、极差以及图形形状上的差异。
Use comparative phrases like ‘The mean jump for boys was higher than that for girls, but the girls’ range was smaller, showing more consistent performance’. This shows you can interpret numbers, not just calculate them.
要使用比较性的语言，比如“男生的平均跳远成绩高于女生，但女生的极差更小，表明表现更为稳定”。这证明你不仅能计算数字，还能进行解读。
10. Making Conclusions and Predictions | 得出结论与预测
Every case study ends with a conclusion. Summarise what the data shows and whether it answers the original question. In the temperature case, you might conclude that the warmest months are June to August and suggest that the city is in the northern hemisphere.
每个案例研究都以结论收尾。总结数据说明了什么，以及它是否回答了最初的问题。在气温案例中，你可能会得出最温暖的月份是六月到八月这一结论，并推断该城市位于北半球。
You can also make simple predictions. For instance, based on the trend, you could predict that the following January would again be around 5°C, as long as the pattern stays the same. This is called extrapolation, but be careful not to predict too far beyond the data.
你还可以做出简单的预测。例如，根据趋势可以预测，如果规律保持不变，明年一月份的气温仍将在 5°C 左右。这称作外推，但要注意不要对过于遥远的数据做预测。
11. Common Mistakes in Case Studies | 案例分析中的常见错误
One common mistake is forgetting to order the data before finding the median. Another is using the grouped frequency table to calculate the mean without considering midpoints, although in Year 8 you usually work with raw data. Also, do not confuse the mode with the modal class; always state clearly which one you are referring to.
一个常见的错误是求中位数前忘记将数据排序。另一个错误是，在使用分组频数表计算均值时没有考虑组中值，不过在八年级你通常处理原始数据。另外，不要混淆众数和众数组，要清楚说明你指的是哪一个。
When drawing graphs, avoid squashing the scale or forgetting to label axes. In case study reports, always give units and write a brief comment interpreting each statistic. Marks are often lost for simply stating numbers without explanation.
画图时，要避免压缩刻度或者忘记标注坐标轴。在案例分析报告中，始终写明单位，并对每个统计量给出简要的解读。如果只列出数字而没有解释，通常会被扣分。
12. Practice Questions and Tips | 练习题与技巧
To master case studies, practise with real data. Try collecting your own data, such as the shoe sizes of your classmates, the number of goals scored in a football season, or daily screen time. Go through the full cycle: organise, graph, calculate averages and range, and then write a short paragraph explaining your findings.
要掌握案例分析，就用真实数据来练习。试着收集自己的数据，比如同学的鞋码、足球赛季的进球数，或者每天的屏幕使用时间。完整地走一遍流程：整理数据、画图、计算平均数和极差，然后写一小段话解释你的发现。
When working under time pressure, read the question carefully to identify what data is given and what statistics are needed. Use a highlighter to mark key words like ‘compare’, ‘explain’ or ‘predict’. Show all your working clearly, because even if your final answer is wrong, you can still earn marks for correct method.
在时间紧迫时做题，要仔细读题，找出给定了哪些数据、需要哪些统计量。用荧光笔标出“比较”、“解释”或“预测”等关键词。清晰展示所有的计算过程，因为即使最后答案错了，正确的方法也能得分。
Published by TutorHao | Statistics Revision Series | aleveler.com
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