📚 SQA Statistics: Case Study Practical Exercise | SQA 统计：案例分析实战演练
Statistical reasoning is not just about crunching numbers; it is about understanding the story behind the data. In this article, we will work through a complete case study that mirrors the kind of investigation you might carry out in a Year 7 classroom. By following the steps of the statistical enquiry cycle – pose a question, collect data, analyse and interpret – you will see how all the separate topics fit together in a real-world context. This hands-on approach will strengthen your confidence for SQA assessments.
统计推理不仅仅是摆弄数字，更是理解数据背后的故事。在本文中，我们将通过一个完整的案例研究，模拟七年级课堂上可能进行的调查。遵循统计探究循环的步骤——提出问题、收集数据、分析并解读——你会看到各个独立的知识点如何在实际情境中相互关联。这种动手实践的方法将增强你应对 SQA 评测的信心。
1. Case Background and Data Collection | 案例背景与数据收集
Our case study is based on a survey carried out in a Year 7 class of 24 students. The teacher wanted to find out about the pupils’ physical activity habits outside school. Two main questions were asked: ‘How many hours do you spend exercising or playing sport in a typical week?’ and ‘What is your favourite type of sport or physical activity?’ The responses were recorded on a simple tally sheet, ensuring that every student gave honest answers. Data collection is the crucial first step – if the data is not gathered carefully, any conclusions drawn later could be misleading.
我们的案例研究基于一个七年级班级（24 名学生）开展的调查。老师想了解学生们在校外的体育锻炼习惯。主要问了两个问题：“你通常每周花多少小时锻炼身体或进行体育运动？”以及“你最喜欢的运动或身体活动类型是什么？”回答记录在一张简单的计数表上，确保每名学生都给出了诚实的答案。数据收集是至关重要的第一步——如果数据收集不仔细，后续得出的任何结论都可能产生误导。
2. Organising the Data: Frequency Distribution Table | 整理数据：频数分布表
Once all the responses were in, the hours of exercise needed to be organised. Raw data is messy, so we group it. Because the hours varied from 0 to 10, we chose class intervals of width 2: 0-1, 2-3, 4-5, 6-7, 8-9, 10-11. A tally was used to count the number of students falling into each interval. The completed frequency table is shown below. Notice that the intervals do not overlap, and every possible value can be allocated to exactly one group.
所有回答收集完后，需要整理锻炼小时数。原始数据杂乱无序，因此我们进行分组。由于小时数从 0 到 10 不等，我们选择了宽度为 2 的组距：0–1、2–3、4–5、6–7、8–9、10–11。使用划线计数来统计落入每个区间的学生人数。完成的频数表如下所示。请注意，区间不重叠，且每个可能的值都能恰好归入一个组。
| Hours of exercise per week | Tally | Frequency |
|---|---|---|
| 0–1 | ||| | 3 |
| 2–3 | |||| || | 7 |
| 4–5 | |||| ||| | 8 |
| 6–7 | ||| | 3 |
| 8–9 | || | 2 |
| 10–11 | | | 1 |
The frequency distribution table immediately reveals patterns: the most common amount of exercise is between 4 and 5 hours per week, while very few students exercise for more than 8 hours. Organising data in this way makes it much easier to see the shape of the distribution.
频数分布表立刻揭示了规律：最常见的锻炼时长是每周 4 至 5 小时，而很少有学生锻炼超过 8 小时。以这种方式整理数据，能让我们更容易看清分布的形状。
3. Data Visualisation: Bar Chart | 数据可视化：条形图
To communicate the findings clearly, we draw a bar chart. The horizontal axis shows the class intervals of exercise hours, and the vertical axis shows the frequency. Bars are drawn with equal width and separated by gaps, because the data is categorical (grouped into intervals). The height of each bar tells us how many students fall into that range. A quick glance at the bar chart confirms that the 4–5 hour bar is the tallest, representing the modal class.
为了清晰地传达调查结果，我们绘制了条形图。横轴表示锻炼小时数的组距，纵轴表示频数。条形宽度相等，彼此之间有间隙，因为数据是分类的（按区间分组）。每个条形的高度告诉我们该区间内有多少学生。快速浏览条形图便可确认，4–5 小时的条形最高，代表众数所在组。
When constructing a bar chart, remember to label both axes, give the chart a title, and use a sensible scale that makes the differences easy to compare. For example, if you made the vertical scale go up in steps of 0.5, the chart would be too crowded; using steps of 1 works well here.
构建条形图时，记得标注两条坐标轴、为图表加上标题，并选用合适的刻度，使差异易于比较。例如，若纵轴以 0.5 为步长，图表会显得过于拥挤；这里以 1 为步长就很合适。
4. Data Visualisation: Pie Chart | 数据可视化：饼图
For the second survey question about favourite sport, a bar chart could also be used, but a pie chart is excellent for showing proportions. The 24 students chose from five categories: Football (8), Swimming (5), Dance (4), Cycling (4) and Other (3). To draw a pie chart, we calculate the angle for each sector: (Frequency ÷ Total) × 360°. For Football, that is (8 ÷ 24) × 360° = 120°. The full set of angles is: Football 120°, Swimming 75°, Dance 60°, Cycling 60°, Other 45°. Each slice then represents the relative popularity of the activity.
对于关于最喜欢运动的第二个调查问题，也可以使用条形图，但饼图在展示比例方面非常出色。24 名学生从五个类别中做出选择：足球（8 人）、游泳（5 人）、舞蹈（4 人）、骑自行车（4 人）和其他（3 人）。要绘制饼图，我们需要计算每个扇形的角度：（频数 ÷ 总数）× 360°。以足球为例，(8 ÷ 24) × 360° = 120°。完整的扇角为：足球 120°、游泳 75°、舞蹈 60°、骑自行车 60°、其他 45°。然后每个扇形就代表了该活动的相对受欢迎程度。
A limitation of pie charts is that they do not easily show exact frequencies unless the numbers are written on the slices. However, they are brilliant for answering questions like ‘More than half the class chose a single sport?’ because the 120° football slice is exactly one third, not a half. Always interpret pie charts by comparing angles to 180° (half) or 90° (quarter).
饼图的一个局限性是，除非在扇区上标注数字，否则不易显示精确的频数。但它们在回答 “是否有超过一半的学生选择了同一种运动？” 这类问题时非常出色，因为 120° 的足球扇区正好是三分之一，而不是一半。解读饼图时，始终将角度与 180°（一半）或 90°（四分之一）进行比较。
5. Measures of Central Tendency: The Mean | 集中趋势的量数：平均数
From the original (ungrouped) list of exercise hours, we can calculate the mean to find the average amount of weekly exercise. The raw data for the 24 students is: 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 9, 10. To find the mean, we add all the values together and then divide by 24.
从原始（未分组）的锻炼小时数列表中，我们可以计算平均数，以了解每周锻炼的平均时长。24 名学生的原始数据为：0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 9, 10。要计算平均数，先将所有数值相加，然后除以 24。
Mean = Sum of all data values ÷ Number of data values
平均数 = 所有数据值的总和 ÷ 数据个数
The sum is 0+1+1+2+2+2+3+3+3+3+4+4+4+4+5+5+5+5+6+6+7+8+9+10 = 102. So the mean is 102 ÷ 24 = 4.25 hours. This tells us that, on average, students in this class exercise for about 4 hours and 15 minutes per week. However, the mean can be affected by extreme values: the student who exercises 10 hours pulls the mean upwards slightly. In a case with a very large outlier, the mean might not represent the typical student well.
总和为：0+1+1+2+2+2+3+3+3+3+4+4+4+4+5+5+5+5+6+6+7+8+9+10 = 102。因此平均数为 102 ÷ 24 = 4.25 小时。这意味着该班级学生平均每周锻炼约 4 小时 15 分钟。然而，平均数会受到极端值的影响：那位锻炼 10 小时的学生轻微地拉高了平均数。当存在非常大的异常值时，平均数可能无法很好地代表典型学生。
6. Measures of Central Tendency: Median and Mode | 集中趋势的量数：中位数与众数
The median is the middle value when the data is arranged in order. With 24 values (an even number), the median is the average of the 12th and 13th values. Ordering the data: the 12th value is 4, and the 13th value is also 4. Therefore, the median is (4 + 4) ÷ 2 = 4 hours. The median is not influenced by the 10-hour student, making it a more robust measure of central tendency when outliers are present.
中位数是数据按顺序排列后的中间值。由于有 24 个值（偶数个），中位数是第 12 和第 13 个值的平均数。将数据排序：第 12 个值是 4，第 13 个值也是 4。因此，中位数为 (4 + 4) ÷ 2 = 4 小时。中位数不受那位锻炼 10 小时的学生的影响，因此在存在异常值时，它是更稳健的集中趋势度量。
The mode is simply the value that occurs most frequently. Scanning the list, we see that 3, 4 and 5 each appear four times. So this data set has three modes, making it multimodal. In grouped data, the modal class is the interval 4–5 hours, which contains 8 students. Reporting the mode or modal class gives a different insight: the most common amount of exercise is between 4 and 5 hours.
众数是最频繁出现的值。扫描列表可以发现，3、4 和 5 各出现了四次。因此这个数据集有三个众数，属于多峰数据。在分组数据中，众数所在组是 4–5 小时区间，包含 8 名学生。报告众数或众数组能提供不同的洞察：最常见的锻炼时长在 4 至 5 小时之间。
7. Measure of Spread: The Range | 离差的量数：范围
To understand how spread out the exercise hours are, we calculate the range. The range is the difference between the largest and smallest data values. Here, the maximum is 10 hours and the minimum is 0 hours, so the range = 10 − 0 = 10 hours. A large range indicates high variability: while some students do no exercise outside school, one student dedicates 10 hours a week. This might prompt the teacher to investigate reasons for such extreme differences.
为了解锻炼小时数的分散程度，我们计算范围。范围是最大值与最小值之差。这里，最大值为 10 小时，最小值为 0 小时，因此范围 = 10 − 0 = 10 小时。较大的范围表明变异程度高：一些学生根本不锻炼，而一位学生每周投入 10 小时。这可能促使老师调查造成这种极端差异的原因。
Although the range is quick to compute, it only looks at the two extremes and ignores how the rest of the data behaves. For example, if the 10-hour student moved away, the range would suddenly drop to 9 (from 0 to 9). In later years, you will learn about interquartile range, which gives a better picture of spread by focusing on the middle 50% of data.
虽然范围计算快捷，但它仅关注两个极端值，忽略了其余数据的表现。例如，如果那位锻炼 10 小时的学生转学了，范围会突然降至 9（从 0 到 9）。在更高的年级，你将学习四分位距，它通过关注中间 50% 的数据，能更好地展示离散程度。
8. Analysing Favourite Sport and Drawing Conclusions | 分析最喜欢的运动并得出结论
Returning to the favourite sport data, we can use both the frequency table and the pie chart to draw conclusions. Football is the clear favourite with 8 votes, accounting for one third of the class. Swimming and Dance/Cycling tie at 5 and 4 votes respectively. The teacher might note that team sports and individual activities are well balanced. However, we must be cautious: these results only apply to this specific class of 24 students. We cannot automatically assume that all Year 7 pupils have the same preferences. Sampling variability means a different class might give a different outcome.
回到最喜欢的运动数据，我们可以利用频数表和饼图得出结论。足球以 8 票明显最受欢迎，占全班的三分之一。游泳以 5 票随后，舞蹈和骑自行车并列，各 4 票。老师可能会注意到团队运动和个人活动之间相当平衡。但我们必须谨慎：这些结果仅适用于这特定的 24 名学生。我们不能自动假设所有七年级学生都有相同的偏好。抽样变异性意味着不同的班级可能会得出不同的结果。
When writing conclusions for SQA tasks, always refer back to the original question. For instance: ‘The survey suggests that within this class, football is the most popular sport, and the typical student exercises approximately 4 to 5 hours per week.’ Such a statement is supported by our mean, median and mode analyses.
在为 SQA 任务撰写结论时，务必回扣最初的问题。例如：“调查表明，在这个班级中，足球是最受欢迎的运动，典型学生每周锻炼约 4 至 5 小时。” 这样的陈述得到了我们平均数、中位数和众数分析的支持。
9. Exploring Possible Relationships | 探索可能的关系
Good statistical investigations often probe deeper: is there a link between favourite sport and the number of hours exercised? It might be that students who favour football tend to exercise more hours because of team training sessions. Without collecting that linked data, we cannot say for sure in this case study. However, you could sketch a comparative dot plot or a back-to-back stem-and-leaf diagram to explore such a question. For now, simply asking the question shows a high level of statistical thinking.
优秀的统计调查往往会深入探究：最喜欢的运动与锻炼小时数之间是否存在关联？可能喜欢足球的学生因为团队训练而锻炼时间更长。不收集相关的关联数据，我们在本案例研究中无法确定。但你可以绘制比较点图或背靠背茎叶图来探索这个问题。目前，仅提出这个问题本身就展示了较高层次的统计思维。
If we had collected data on both variables for each student (e.g. Sarah: Football, 4 hours), we could group the hours by sport and see whether the distributions overlap. This type of thinking sets the groundwork for scatter graphs and correlation in later years.
如果我们为每名学生收集了两个变量的数据（例如莎拉：足球，4 小时），就可以按运动分组小时数，观察分布是否重叠。这类思维为今后学习散点图和相关性奠定了基础。
10. Practical Tips for Your Own Case Study | 开展个人案例研究的实用建议
When you tackle a case study in class or for homework, follow this checklist: First, phrase your question clearly so it cannot be misinterpreted. Second, design a data collection sheet with space for tallies and totals. Third, always organise raw data into a frequency table before drawing charts. Fourth, calculate at least two measures of average (mean and median or mode) and the range. Finally, write a few sentences that tell the story behind the numbers. Practise explaining what the statistics mean in everyday language – this skill is highly valued in SQA assessments.
当你在课堂上或家庭作业中处理案例研究时，请遵循以下清单：第一，清晰地表述问题，确保不会被误解。第二，设计一份数据收集表，留出划记和总计的空间。第三，绘制图表前，始终将原始数据整理成频数表。第四，至少计算两种平均数度量（平均数、中位数或众数）以及范围。最后，写几句话讲述数字背后的故事。练习用日常语言解释统计量的含义——这项技能在 SQA 评测中备受重视。
Remember that no single statistic tells the whole truth. The mean gives the arithmetic centre, the median the positional centre, the mode the most typical category, and the range the spread. Together, they build a comprehensive picture.
请记住，没有任何单一统计量能反映全部真相。平均数给出算术中心，中位数给出位置中心，众数给出最典型类别，范围给出离散度。它们共同构建起全面的图景。
11. Reflecting on the Statistical Enquiry Cycle | 统计探究循环的反思
Let’s step back and see how this case study followed the full cycle. We started with a problem (What are the exercise habits of this class?), planned and collected data, processed and presented it using tables and graphs, then analysed and interpreted the results. Finally, we considered limitations and possible next questions. This cycle is at the heart of statistical literacy. Whether you are investigating traffic outside school or the most common birthday month, the same logical steps apply.
让我们退后一步，看看这个案例研究如何遵循了完整的循环。我们从一个问题开始（这个班级的锻炼习惯是怎样的？），计划并收集了数据，使用表格和图表处理和呈现了数据，然后分析并解读了结果。最后，我们考虑了局限性和可能的后续问题。这个循环是统计素养的核心。无论你是调查校外交通流量还是最常见的生日月份，同样的逻辑步骤都适用。
Being able to critique your own work is also important. Could the class intervals have been chosen differently? Yes, using width 1 would give more detail but make the table longer. Is there any bias? Students might have overestimated their exercise hours. Recognising such issues shows mature understanding.
能够批判自己的作品也很重要。组距是否可以选得不同？可以，使用宽度 1 会给出更多细节，但会使表格变长。是否有偏差？学生可能高估了自己的锻炼时长。认识到这类问题体现出成熟的理解。
12. Summary and Key Takeaways | 总结与关键要点
Through this case study, we have seen how raw survey data can be transformed into meaningful statistics. Key skills covered include constructing frequency tables, drawing bar and pie charts, calculating mean, median, mode and range, and writing evidence-based conclusions. The ability to select the right statistical tool for the question is just as important as being able to perform the calculation. As you practise, try designing your own mini-surveys – the more you engage with real data, the more confident you will become.
通过这个案例研究，我们看到了原始调查数据如何转化为有意义的统计量。涵盖的关键技能包括：构建频数表、绘制条形图和饼图、计算平均数、中位数、众数和范围，以及撰写基于证据的结论。为问题选择合适的统计工具的能力，与进行计算的能力同等重要。在你练习时，尝试设计自己的小型调查——你接触真实数据越多，就会越自信。
Keep a ‘statistics diary’ where you note the different graphs and averages you encounter in everyday life, from weather reports to sports scores. This will sharpen your intuition and help you spot misleading graphs. Statistics is a powerful tool for making sense of the world, and your SQA journey is just the beginning.
坚持写一本 “统计日记”，记下你在日常生活中遇到的不同图表和平均数，从天气预报到体育比分。这将增强你的直觉，并帮助你识别误导性的图表。统计是理解世界的有力工具，你的 SQA 之旅才刚刚开始。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导