📚 Statistics Case Study: From Data to Decisions | 统计案例分析：从数据到决策
In Year 9 Statistics, real understanding comes from applying concepts to actual data. This case study follows a complete investigation, from posing a question to drawing conclusions, using test scores from two classes. You will practise organising data, calculating averages and spread, drawing charts, interpreting stem-and-leaf diagrams and linking data to basic probability. Every step mirrors the kind of thinking required in the Cambridge checkpoint and beyond.
在九年级统计中，真正的理解来自将概念应用于实际数据。本案例研究从一个问题出发，使用两个班级的测验成绩，完整演示一次统计调查。你将练习整理数据、计算平均数和离散程度、绘制图表、解读茎叶图，并将数据与基本概率联系起来。每一步都反映了剑桥 checkpoint 及更高阶段所需要的思维方式。
1. Setting the Scene: The Investigation | 场景设定：调查研究
Imagine you are a data analyst for your school. The head of mathematics wants to know whether Class 9A or Class 9B performed better in the recent algebra test. Each class has 20 students, and the marks are out of 100. Your task is to collect the scores, organise them, present them visually, compare central tendency and spread, and finally advise which class showed stronger overall performance.
假设你是学校的数学数据分析师。数学组长想知道在最近的代数测验中，9A 班和 9B 班哪个班表现更好。每个班有 20 名学生，满分 100 分。你的任务是收集成绩、整理数据、用图形展示、比较集中趋势和离散程度，最后就哪个班级整体表现更强提出建议。
The data below will be used throughout this case study. Before reading on, copy the two sets of scores into your notebook so you can work alongside the analysis.
以下数据将贯穿整个案例研究。在继续阅读之前，请将两组分数抄到你的笔记本中，以便边学边做。
Class 9A scores: 65, 70, 75, 80, 85, 90, 55, 60, 95, 100, 40, 72, 88, 78, 82, 68, 92, 53, 77, 83
Class 9B scores: 58, 63, 72, 81, 88, 45, 66, 74, 80, 85, 50, 62, 79, 84, 91, 48, 71, 76, 82, 90
2. Collecting the Data | 收集数据
heir accuracy. Always note the source, the number of data items (sample size n = 20 for each class) and whether any scores are missing. Here there are no gaps, and the type of data is discrete numerical (whole number marks).
良好的统计实践始于可靠的数据收集。在这项研究中，分数直接从测验记录中提取，因此我们可以相信其准确性。务必注明数据来源、数据个数（每个班样本量 n = 20）以及是否有缺失值。此处没有缺失，数据类型为离散数值（整数分数）。
You might ask: is the sample size large enough? For a class comparison, 20 students each gives a reasonable snapshot, but we must be careful not to over-generalise. We are only comparing these two groups, not all Year 9 students.
你可能要问：样本量是否足够大？在进行班级比较时，每个班 20 名学生提供了一个合理的快照，但我们必须谨慎，不要过度推广。我们只是比较这两个组，而不是所有九年级学生。
3. Organising Data with Frequency Tables | 用频数表整理数据
Raw scores are difficult to interpret. The first step is to group the data into equal class intervals. For marks out of 100, intervals of 10 are convenient: 40–49, 50–59, 60–69, 70–79, 80–89 and 90–100. Tally the scores and record frequencies.
原始分数很难解读。第一步是将数据分组到等宽的组距中。对于满分 100 的分数，采用组距 10 比较方便：40–49、50–59、60–69、70–79、80–89 和 90–100。用划记法统计并记录频数。
| Interval | Class 9A Frequency | Class 9B Frequency |
|---|---|---|
| 40–49 | 1 | 3 |
| 50–59 | 2 | 3 |
| 60–69 | 3 | 3 |
| 70–79 | 5 | 5 |
| 80–89 | 5 | 4 |
| 90–100 | 4 | 2 |
From the table, you can already see that Class 9A has more scores in the highest interval (4 vs 2), while Class 9B has slightly more at the lower end. But a table alone is not enough — we need visuals.
从表格中你已经可以看出，9A 班在最高区间有更多学生（4 人对 2 人），而 9B 班低分段人数稍多。但仅靠表格还不够，我们需要图形。
4. Visualising Data: Bar Charts and Pie Charts | 数据可视化：条形图和饼图
Bar charts are ideal for grouped discrete data. Draw a bar chart for each class with the intervals on the horizontal axis and frequency on the vertical. Use equal bar widths and label axes clearly. Your chart will show 9A’s bars rising towards the higher end, while 9B’s distribution is more even.
条形图非常适用于分组离散数据。为每个班绘制一个条形图，横轴为组距，纵轴为频数。条形宽度相等，并清晰标注坐标轴。你的图表会显示 9A 班的条形在高分段更高，而 9B 班的分布更加平均。
de band. Calculate the angle for each sector: (frequency / 20) × 360°. For example, the 80– p to see relative sizes but lose the order of intervals, so use them alongside bar charts.
饼图可以显示每个分数段的学生比例。计算每个扇形的角度：（频数 / 20）× 360°。例如，9A 班 80–89 区间的角度为 (5/20) × 360° = 90°。画出扇形并添加图例。饼图有助于观察相对大小，但会丢失区间的顺序，因此应与条形图配合使用。
A quick comparison: 9A’s pie chart will have a larger slice for 90–100, while 9B has bigger slices for 40–49 and 50–59. This already suggests differences, but we need numerical summaries.
快速对比：9A 班的饼图中 90–100 扇形更大，而 9B 班在 40–49 和 50–59 扇形更大。这已表明存在差异，但我们还需要数值概括。
5. Exploring Central Tendency: Mean, Median and Mode | 探索集中趋势：平均数、中位数与众数
9B, the sum is 1488, mean = 1488 ÷ 20 = 74.4. On average, Class 9A scored 6 marks higher.
平均数（均值）通过将所有分数相加再除以 n 得到。对于 9A 班，总和为 1608，因此均值 = 1608 ÷ 20 = 80.4。9B 班总和为 1488，均值 = 1488 ÷ 20 = 74.4。平均而言，9A 班分数高出 6 分。
20 values, the median is the average of the 10th and 11th: (77 + 78) ÷ 2 = 77.5. For 9B: 45, 48, 50, 58, 62, 63, 66, 71, 72, 74, 76, 79, 80, 81, 82, 84, 85, 88, 90, 91. Median = (74 + 76) ÷ 2 = 75. So the medians are much closer: 77.5 vs 75.
为求中位数，将分数按顺序排列。对于 9A，排序后为：40, 53, 55, 60, 65, 68, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 90, 92, 95, 100。有 20 个值，中位数是第 10 和第 11 个值的平均数：(77 + 78) ÷ 2 = 77.5。9B 班为：45, 48, 50, 58, 62, 63, 66, 71, 72, 74, 76, 79, 80, 81, 82, 84, 85, 88, 90, 91。中位数 = (74 + 76) ÷ 2 = 75。可见中位数很接近：77.5 对 75。
The mode is the most frequent score. In 9A, no score repeats clearly, so there is no mode. In 9B, likewise no value appears more than once. When no mode exists, we report that the data set has no mode and rely on mean and median instead. Sometimes grouped data gives a modal class — the interval with the highest frequency. For 9A, the modal classes are 70–79 and 80–89 (both frequency 5). For 9B, the modal class is 70–79 (frequency 5).
众数是最常出现的分数。在 9A 班，没有分数明显重复，因此没有众数。9B 班同样没有出现次数超过一次的值。当没有众数时，我们就报告该数据集无众数，转而依赖平均数和中位数。有时分组数据可以给出众数所在组——频数最高的组距。9A 班的众数所在组是 70–79 和 80–89（频数均为 5）。9B 班的众数所在组是 70–79（频数为 5）。
Choosing the best average depends on the distribution. The mean is affected by the very low score of 40 in 9A, pulling it down slightly; 9B's mean is also lowered by three scores in the 40s. The medians are less affected by extremes, so they may better represent a 'typical' student.
选择最佳平均数取决于分布情况。9A 班中非常低的 40 分影响了均值，使其略微偏低；9B 班的均值也因三个 40 多分的分数而降低。中位数受极端值影响较小，因此可能更能代表“典型”学生的水平。
6. Measuring Spread: Range | 衡量离散程度：极差
Range = highest score – lowest score. For 9A, range = 100 – 40 = 60. For 9B, range = 91 – more variability; the top students did very well, but one student scored only 40. Class 9B's scores are more bunched together.
极差 = 最高分 − 最低分。9A 班的极差 = 100 − 40 = 60。9B 班的极差 = 91 − 45 = 46。9A 班的极差更大，显示出更大的变异性；顶尖学生表现非常好，但有一名学生只得了 40 分。9B 班的分数更集中。
y outliers. If we ignore the 40 in 9A, the next lowest is 53, making the trimmed range 100 – 53 = 47, which is similar to 9B' lier is important to note. We will examine spread more deeply with stem-and-leaf diagrams.
然而，极差只用到两个值，并且可能被异常值扭曲。如果我们忽略 9A 班的 40 分，下一个最低分是 53，修剪后的极差为 100 − 53 = 47，与 9B 班类似。因此这个异常值很值得注意。我们将通过茎叶图更深入地考察离散情况。
7. Stem-and-Leaf Diagrams: Keeping Data Orderly | 茎叶图：保持数据有序
A stem-and-leaf diagram preserves every original value while showing the shape of the distribution. Use the tens digit as the stem and the units as the leaf. For 9A:
茎叶图在展示分布形状的同时保留了每个原始数值。用十位数作为茎，个位数作为叶。9A 班如下：
4 | 0 5 | 3 5 6 | 0 5 8 7 | 0 2 5 7 8 8 | 0 2 3 5 8 9 | 0 2 5 10 | 0
Include a key: e.g., 4|0 means 40. For 9B:
4 | 5 8 5 | 0 8 6 | 2 3 6 7 | 1 2 4 6 9 8 | 0 1 2 4 5 8 9 | 0 1
The diagram immediately reveals that 9A has two peaks in the 70s and 80s and a tail at the low end. 9B's scores peak in the 80s but the leaves are more evenly spread across the 40s to 90s, with no extreme 100. You can also see the median clearly: for 9A, count to the 10th and 11th leaves (77 and 78).
该图立即显示，9A 班在 70 分段和 80 分段有两个高峰，低分段有一条尾巴。9B 班的分数高峰在 80 分段，但叶片更均匀地分布在 40 到 90 分段之间，没有极端的 100 分。你还能清晰地看出中位数：9A 班数到第 10 和第 11 个叶片（77 和 78）。
Stem-and-leaf diagrams are excellent for finding quartiles later on, but at this stage focus on their ability to compare spreads without losing data detail.
茎叶图以后用于求四分位数也很方便，但在现阶段，重点关注其在不丢失数据细节的情况下比较分布的能力。
8. Comparing Two Data Sets | 比较两个数据集
When comparing data sets, always comment on both a measure of central tendency and a measure of spread. For our classes:
在比较数据集时，务必同时评论集中趋势的度量和离散程度的度量。对于我们的班级：
- Mean: 9A (80.4) > 9B (74.4).
- Median: 9A (77.5) > 9B (75).
- Range: 9A (60) > 9B (46).
中文：
- 平均数：9A (80.4) > 9B (74.4)。
- 中位数：9A (77.5) > 9B (75)。
- 极差：9A (60) > 9B (46)。
On average and at the median, 9A performed better. However, the larger range for 9A indicates that while some students excelled, others struggled significantly. In 9B, the lower range suggests a more consistent performance, with fewer very high and very low scores. Therefore, if the school wants high achievers, 9A appears stronger; if consistency matters more, 9B may be preferable.
从平均数和位数看，9A 班表现更好。然而，9A 班较大的极差表明，虽然部分学生表现优异，但另一些学生相当吃力。在 9B 班，较小的极差表明表现更加稳定，高分和低分都较少。因此，如果学校追求高分学生，9A 显得更强；如果更看重一致性，9B 可能更合适。
You could also compare the proportion of students scoring above 80. In 9A, 9 out of 20 (45%) achieved 80 or above; in 9B, 6 out of 20 (30%) did. This supports the interpretation that 9A has a stronger top end.
你还可以比较得分在 80 分以上的学生比例。9A 班有 9 人（45%）达到 80 分或以上；9B 班有 6 人（30%）。这进一步支持了 9A 班高分段更强的解读。
9. Introducing Probability from Data | 从数据引入概率
9A student at random from this sample, the estimated probability that their score is above 80 is 9/20 = 0.45 or 45%.
统计与概率紧密相连。利用数据中的相对频数，我们可以估计概率。例如，如果从这个样本中随机挑选一名 9A 班的学生，其分数高于 80 的估计概率为 9/20 = 0.45，即 45%。
8 give P(low) = 2/20 = 0.1. These probabilities are derived from past data and assume the sample represents the class fairly.
对于 9B 班，P(分数 > 80) = 6/20 = 0.3。得分低于 50 的概率：9A 班只有 40 分符合，因此 P(低分) = 1/20 = 0.05。9B 班有 45 和 48 两个分数，P(低分) = 2/20 = 0.1。这些概率由历史数据得出，并假设样本能较好地代表班级。
Be careful: probability based on a small sample like 20 students can be unreliable for predictions, but it gives a useful starting point for discussion. We say 'estimated probability' rather than true probability.
要注意：基于 20 名学生这样的小样本得出的概率用于预测可能不可靠，但它为讨论提供了一个有用的起点。我们说“估计概率”而非真实概率。
10. Considering Bias and Reliability | 考虑偏差与可靠性
Could there be bias in our data? The scores are objective, but we must consider whether the test itself was fair, whether all students took it under the same conditions, and if any students were absent. Also, the sample only includes these two classes; results might differ with another cohort.
我们的数据可能存在偏差吗？分数是客观的，但我们必须考虑测验本身是否公平，所有学生是否在相同条件下参加测验，以及是否有学生缺席。此外，样本只包含这两个班级；换成另一组学生结果可能会不同。
The sample size of 20 per group is typical for a class but too small to detect subtle differences with high confidence. Always mention that conclusions apply to these classes only, and repeating the investigation with more data would improve reliability.
每组 20 人的样本量对于班级来说是典型的，但太小，不足以高置信度地发现细微差异。一定要说明结论仅适用于这些班级，用更多数据重复调查会提高可靠性。
11. Drawing Conclusions and Making Decisions | 得出结论并做出决策
Based on our analysis, Class 9A has a higher mean and median, a larger proportion of high scores, but also a wider spread with one very low outlier. Class 9B shows more consistency and a slightly lower central tendency. The stem-and-leaf diagrams confirm these patterns visually.
根据我们的分析，9A 班平均数和位数更高，高分比例更大，但分布也更广，且有一个非常低的异常值。9B 班表现出更高的一致性，集中趋势略低。茎叶图直观地证实了这些模式。
A report to the head of mathematics might read: "Class 9A outperformed Class 9B in mean score by 6 marks, and 45% of its students achieved marks above 80, compared with 30% in 9B. However, 9A's range was 14 marks wider. I recommend celebrating 9A's high achievers while providing extra support to lower-attaining students in both classes."
提交给数学组长的报告可以这样写：“9A班的平均分比9B班高6分，45%的学生取得80分以上，而9B班为30%。但9A班的极差大14分。我建议表彰9A班的高分学生，同时为两个班中成绩较低的学生提供额外支持。”
12. Presenting Your Findings | 展示你的发现
Effective communication is key in statistics. Your final output should include: a clear statement of the question, a table of frequencies, at least one bar chart or pie chart, the mean, median, mode (or modal class), range, a stem-and-leaf diagram, probability estimates, and a written conclusion that compares the two sets and suggests action.
在统计中，有效的沟通至关重要。你的最终作品应包括： 明确的问题陈述、频数表、至少一幅条形图或饼图、平均数、中位数、众数（或众数所在组）、极差、茎叶图、概率估计，以及比较两组数据并提出行动建议的书面结论。
You could present this as a poster, a slide deck or a report. Use headings, label graphs fully, and write in full sentences. Always relate your numbers back to the original question: which class performed better, and why?
你可以用海报、幻灯片或报告的形式来展示。使用标题，为图形添加完整标签，并用完整的句子写作。始终将数字与最初的问题联系起来：哪个班表现更好，为什么？
Published by TutorHao | Statistics Revision Series | aleveler.com
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