📚 Year 8 AQA Statistics: Case Study Practice | Year 8 AQA 统计：案例分析实战演练
In this article, we will work through a complete statistics case study, applying the skills you have learned in Year 8 AQA Statistics. The case study focuses on analysing test scores to draw meaningful conclusions. We will move step by step from collecting and organising data to interpreting results. Both English and Chinese explanations are provided for every key point.
本文将通过一个完整的统计案例，演练你在 Year 8 AQA 统计学中掌握的技能。案例围绕分析测试成绩展开，最终得出有意义的结论。我们将按照步骤，从收集整理数据到解读结果。每个要点都配有中英文说明。
1. Introducing the Case Study | 案例介绍
A Year 8 mathematics teacher wants to understand how well her class performed in a recent end-of-topic test. She decides to carry out a small statistical investigation. The test was marked out of 50, and 20 students took the test. The teacher’s goal is to identify overall performance, spot any patterns, and suggest areas for improvement.
一位八年级数学老师想了解班级在最近一次单元测验中的表现。她决定开展一项小型统计调查。测验满分为50分，共有20名学生参加。老师的目的是了解整体成绩、发现规律并提出改进方向。
2. Understanding the Raw Data | 理解原始数据
Here are the raw scores collected: 34, 28, 41, 37, 29, 45, 38, 33, 42, 36, 31, 40, 39, 44, 35, 32, 30, 43, 27, 46. The data is unorganised. Before we can analyse it, we need to organise it properly. A list like this is called raw data because it has not been sorted or grouped.
以下是收集到的原始分数：34, 28, 41, 37, 29, 45, 38, 33, 42, 36, 31, 40, 39, 44, 35, 32, 30, 43, 27, 46。这些数据未经整理。在进行分析之前，我们需要将其妥善整理。像这样的列表被称为原始数据，因为它尚未排序或分组。
3. Organising Data: Frequency Tables | 整理数据：频数表
The first step is to sort the scores in ascending order and create a frequency table. Sorting helps us see the lowest and highest values quickly. A frequency table groups scores and shows how many students achieved each score.
第一步是将分数按升序排列，并制作频数表。排序能帮助我们快速看到最低分和最高分。频数表将分数分组，并显示每个分数对应有多少名学生。
| Score (分数) | Tally (计数) | Frequency (频数) |
|---|---|---|
| 27-29 | III | 3 |
| 30-32 | III | 3 |
| 33-35 | IIII | 4 |
| 36-38 | III | 3 |
| 39-41 | IIII | 4 |
| 42-44 | II | 2 |
| 45-47 | I | 1 |
Using equal class intervals makes the table easier to read. In this table, we used intervals of width 3. The frequency column shows that most students scored between 33-35 and 39-41.
使用相等的组距可以让表格更易于阅读。在这个表格中，我们使用了宽度为3的区间。频数列显示，大多数学生的分数分布在33-35和39-41之间。
4. Visual Representation: Bar Chart and Pie Chart | 可视化表示：柱状图和饼图
A bar chart is useful for showing frequencies of grouped data. Each bar’s height represents the frequency for that interval. The bars are separated because the data is discrete in intervals. The teacher draws a bar chart with the score groups on the horizontal axis and frequency on the vertical axis.
柱状图适合展示分组数据的频数。每个柱子的高度代表该区间的频数。柱子之间存在间隙，因为数据按离散的区间分组。老师绘制了一张柱状图，横轴为分数段，纵轴为频数。
A pie chart can also show the proportion of students in each score band. To draw a pie chart, we calculate the angle for each sector using: angle = (frequency ÷ total frequency) × 360°. For example, the angle for the 33-35 group is (4 ÷ 20) × 360° = 72°.
饼图同样可以展示各分数段中学生所占的比例。绘制饼图时，需要计算每个扇形的角度：角度 = (频数 ÷ 总频数) × 360°。例如，33-35分组的扇形角度为 (4 ÷ 20) × 360° = 72°。
5. Calculating Averages: Mean, Median, Mode | 计算平均值：平均数、中位数、众数
The mean is found by adding all scores and dividing by the number of students. Sum of all scores = 34+28+41+…+46 = 740. Mean = 740 ÷ 20 = 37.0. So the average test score is 37 out of 50.
平均数是将所有分数相加后除以学生人数。分数总和 = 34+28+41+…+46 = 740。平均数 = 740 ÷ 20 = 37.0。因此，测验的平均分为37分（满分50）。
The median is the middle value when data is ordered. With 20 values, the median lies between the 10th and 11th scores. Sorted scores: 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46. The 10th score is 36 and 11th is 37, so median = (36+37)/2 = 36.5.
中位数是将数据排序后位于中间的值。有20个数据时，中位数位于第10和第11个分数之间。排序后的分数：27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46。第10个分数是36，第11个是37，所以中位数 = (36+37) ÷ 2 = 36.5。
The mode is the most frequent score. From the raw data, we can see that scores 33, 34, 35, 39, 40, 41 each appear twice, so there are multiple modes. This is a multimodal dataset. Using grouped data, the modal class is 33-35 and 39-41 with frequency 4.
众数是出现频率最高的分数。从原始数据可以看出，33, 34, 35, 39, 40, 41 各出现了两次，因此存在多个众数，这是一个多峰数据集。对于分组数据，众数区间是33-35和39-41，频数均为4。
6. Measuring Spread: Range and Quartiles | 衡量分布范围：极差和四分位数
The range is the difference between the highest and lowest values: 46 – 27 = 19. This tells us the spread of all scores. A large range suggests uneven performance across the class.
极差是最大值与最小值的差：46 – 27 = 19。这反映了所有分数的分布范围。极差较大说明班级成绩的不均衡性较高。
Quartiles divide the ordered data into four equal parts. Q₁ (lower quartile) is the median of the first half of data. For our 20 values, Q₁ is the median of the first 10 numbers: (30+31)/2 = 30.5. Q₃ (upper quartile) is the median of the last 10: (41+42)/2 = 41.5. The interquartile range (IQR) = Q₃ – Q₁ = 41.5 – 30.5 = 11.
四分位数将有序数据分为四个等份。Q₁（下四分位数）是前一半数据的中位数。对于这20个数据，Q₁是前10个的中位数：(30+31)/2 = 30.5。Q₃（上四分位数）是后10个的中位数：(41+42)/2 = 41.5。四分位距（IQR）= Q₃ – Q₁ = 41.5 – 30.5 = 11。
7. Drawing a Box Plot (Box-and-Whisker Plot) | 绘制箱线图
A box plot visually summarises the five-number summary: minimum (27), Q₁ (30.5), median (36.5), Q₃ (41.5), maximum (46). The box spans from Q₁ to Q₃, with a vertical line at the median. Whiskers extend to the minimum and maximum, as long as there are no outliers.
箱线图直观总结了五数概括法：最小值（27）、Q₁（30.5）、中位数（36.5）、Q₃（41.5）、最大值（46）。箱体从Q₁延伸到Q₃，并在中位数处画一条垂直线。箱须延伸至最小值和最大值（假设没有异常值）。
We can see from the box plot that the middle 50% of scores lie between 30.5 and 41.5. The median is closer to the upper quartile, suggesting a slight concentration of higher scores. The left whisker is longer, indicating a wider spread in the lower half.
从箱线图可以看出，中间50%的分数位于30.5到41.5之间。中位数更接近上四分位数，表明较高分相对集中。左侧箱须较长，说明低分段分布更分散。
8. Interpreting the Results | 解读结果
The mean and median are close (37.0 vs 36.5), which suggests the data is roughly symmetric. However, the mode hints at two groups: one around 34 and another around 40. The range of 19 and IQR of 11 show moderate spread. The teacher notices that about half the class scored above 36.5, but a few students scored well below 30, pulling the lower whisker down. This may indicate the need for extra support for those students.
平均数和中位数很接近（37.0 与 36.5），表明数据大致对称。然而，众数暗示存在两个群体：一组约在34分，另一组约在40分。极差19和四分位距11显示适中的离散程度。老师注意到，大约一半的学生成绩在36.5分以上，但有少数学生远低于30分，从而拉长了低端箱须。这可能表明需要对这些学生提供额外支持。
9. Practical Exercise: Another Scenario | 实战练习：另一个场景
Now try applying these steps to a new dataset. A PE teacher recorded the number of push-ups completed by 20 students: 15, 22, 18, 25, 30, 20, 17, 28, 24, 19, 22, 26, 21, 29, 18, 23, 27, 16, 31, 20. Organise the data, draw a stem-and-leaf diagram, calculate mean, median, mode, range, quartiles, IQR, and draw a box plot. Then write a short interpretation.
现在请尝试将这些步骤应用到一个新的数据集上。一位体育老师记录了20名学生的俯卧撑完成次数：15, 22, 18, 25, 30, 20, 17, 28, 24, 19, 22, 26, 21, 29, 18, 23, 27, 16, 31, 20。请整理数据，绘制茎叶图，计算平均数、中位数、众数、极差、四分位数、四分位距，并绘制箱线图。然后写一段简短的解读。
For the stem-and-leaf diagram, use the tens digit as stem and units as leaf. Sorted: 15,16,17,18,18,19,20,20,21,22,22,23,24,25,26,27,28,29,30,31. Mean = 22.5, median = 22, modes: 18,20,22. Range = 16, Q₁ = 18.5, Q₃ = 26.5, IQR = 8. Interpretation: push-up performance is fairly symmetric with moderate spread.
绘制茎叶图时，以十位数作为茎，个位数作为叶。排序后：15,16,17,18,18,19,20,20,21,22,22,23,24,25,26,27,28,29,30,31。平均数 = 22.5，中位数 = 22，众数：18, 20, 22。极差 = 16，Q₁ = 18.5，Q₃ = 26.5，IQR = 8。解读：俯卧撑表现大致对称，离散程度适中。
10. Common Pitfalls 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. Always sort values first. Another pitfall is using the class midpoint incorrectly when calculating the mean from grouped data – in this case study we had the raw data so we didn’t need to estimate. Students also sometimes confuse the range with the interquartile range. Remember: range = max – min, IQR = Q₃ – Q₁, which is resistant to extreme values.
一个常见错误是求中位数前忘记将数据排序。一定要先排序。另一个易错点是在根据分组数据计算平均数时，错误地使用组中值——在本案例中我们拥有原始数据，所以不需要估算。学生有时也会混淆极差和四分位距。记住：极差 = 最大值 – 最小值，IQR = Q₃ – Q₁，后者不受极端值影响。
When drawing a pie chart, angles must add up to 360°. Double-check your calculations. Also, ensure that your diagrams are clearly labelled with titles and axis labels. A good diagram communicates the key message at a glance.
绘制饼图时，各扇形的角度总和必须为360°。务必核对计算。此外，要确保图表标题和坐标轴标签清晰。好的图表能让人一目了然地获取关键信息。
11. Writing a Statistical Conclusion | 撰写统计结论
A good conclusion should answer the original question and refer back to the statistics calculated. For the test score case study, the teacher might write: ‘The average test score was 37 out of 50, with half the class scoring above 36.5. The scores were fairly spread out (range 19). There seem to be two groups of students, one performing around 34 and another around 40. The box plot showed a slight skew towards higher scores. To improve, I will provide targeted revision for the lower-scoring group.’
一个好的结论应该回答最初的问题，并回头引用计算出的统计量。在测试成绩案例中，老师可能会这样写：“班级平均分为37分（满分50），一半学生的分数高于36.5分。成绩分布较广（极差19）。似乎存在两个群体，一部分在34分左右，另一部分在40分左右。箱线图略偏向高分段。为了提升成绩，我会为低分段学生提供针对性复习。”
12. Final Tips for Case Study Success | 案例成功要点总结
Always work systematically: collect, organise, visualise, calculate averages and measures of spread, then interpret. Use a variety of statistical diagrams to support your findings. Check your calculations and ensure your interpretations are grounded in the numbers, not assumptions. With practice, case studies become an enjoyable way to see statistics in action.
始终系统地开展工作：收集、整理、可视化、计算平均数和分布度量，然后解读。使用多种统计图表来支撑你的发现。检查计算过程，确保解读基于数据而非臆测。通过练习，案例分析将成为观察统计应用的有趣方式。
Published by TutorHao | Statistics Revision Series | aleveler.com
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