KS3 Cambridge Statistics: A Case Study in Action | KS3 剑桥统计：案例分析实战演练

📚 KS3 Cambridge Statistics: A Case Study in Action | KS3 剑桥统计：案例分析实战演练

In this article, we will work through a complete KS3 statistics case study. You will act as a school data analyst investigating the relationship between how much Year 7 students read outside school and their latest English test scores. This hands-on example allows you to revise all the key statistical skills: designing surveys, collecting data, making tables and charts, calculating averages, interpreting scatter graphs, and drawing conclusions.

在这篇文章里，我们将完成一个完整的 KS3 统计案例研究。你将扮演学校的“数据分析师”，调查七年级学生课外阅读时长与最近一次英语测验成绩之间的关系。这个动手实例会帮助你复习所有重要的统计技能：设计调查、收集数据、制作表格和图表、计算平均值、解读散点图并得出结论。

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1. Introducing the Case Study | 案例介绍

Imagine your class wants to find out whether students who read more often tend to get higher marks in English. To answer this question, you decide to survey 30 Year 7 pupils. You will record two pieces of information for each student: the number of hours they read per week (including fiction and non-fiction) and their most recent English test score out of 50. This investigation brings together data collection, presentation, summary statistics and the idea of correlation – all core parts of the Cambridge KS3 statistics syllabus.

设想你们班想弄清楚一个问题：阅读时间更长的学生，英语成绩是否往往更高。为了回答这个问题，你们决定调查 30 名七年级学生。你们将记录每位学生的两项信息：每周课外阅读时长（包括虚构和非虚构类书籍）以及他们最近一次英语测验的分数（满分 50 分）。这项调查融合了数据收集、图表展示、概括统计和相关性概念，这些都是剑桥 KS3 统计大纲的核心内容。

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2. Designing the Survey | 设计调查

Before collecting any data, you need a clear plan. First, decide on your target population: all the Year 7 students in your school. Since it is not practical to survey everyone, you select a sample of 30 pupils using a random method, such as picking names from a hat. You choose two variables: ‘Reading hours per week’, which is a continuous variable because time can be measured to fractions of an hour, and ‘English test score’, which is a discrete variable because marks are whole numbers out of 50. Having a clear design makes the data reliable and the analysis meaningful.

在收集任何数据之前，你需要一份清晰的计划。首先，明确目标总体：全校七年级学生。因为调查所有人并不现实，你采用随机方法（例如从帽子里抽签）选出 30 名学生作为样本。你选择了两个变量：“每周阅读小时数”属于连续变量，因为时间可以被测量到几分之一小时；“英语测验成绩”是离散变量，因为分数是满分为 50 的整数。清晰的设计能让数据更加可靠，分析也更有意义。

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3. Collecting the Data | 收集数据

The table below shows the data collected from the 30 students. Each row gives a student’s ID, their weekly reading hours and their test score. For example, Student 1 reads 2 hours per week and scored 40 out of 50.

下表展示了从 30 位学生那里收集到的数据。每一行给出学生编号、每周阅读时长和测验分数。例如，学生 1 每周阅读 2 小时，得分 40（满分 50）。

ID	Reading (h)	Score (/50)
1	2.0	40
2	1.0	28
3	3.0	45
4	0.5	22
5	4.0	48
6	2.5	38
7	1.5	30
8	5.0	47
9	2.0	35
10	3.5	42
11	0.0	15
12	6.0	50
13	1.0	26
14	2.5	40
15	4.5	46

The full data set continues with 15 more students whose reading times range between 0 and 7 hours, with scores varying similarly. The table above is enough for practising the skills that follow.

完整的数据集还包括另外 15 名学生，他们的阅读时间在 0 到 7 小时之间，分数也相应变化。上表足以进行接下来的技能练习。

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4. Organising Data in Tables | 数据表格整理

A frequency table helps to group the scores and see how many students fall into each interval. Below, the English test scores are grouped in class intervals of width 10.

频率表有助于对成绩进行分组，并看出每个区间里有多少名学生。下面的表格将英语测验分数按组距 10 进行了分组。

Score interval	Frequency
0 – 9	0
10 – 19	1
20 – 29	5
30 – 39	9
40 – 50	15

Notice that the intervals do not overlap, and the total frequency is 30. Using a grouped frequency table makes it much easier to spot that most students scored in the highest band, while very few scored below 20. This is the kind of summary that raw data cannot show at a glance.

注意这些区间互不重叠，且总频数为 30。使用分组频率表后，更容易发现大多数学生处于最高分数段，而得分低于 20 分的人很少。这种概括是原始数据无法一眼看出的。

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5. Displaying Data: Bar Chart of Scores | 数据展示：成绩条形图

A bar chart is ideal for showing the frequency of scores in each interval. The height of each bar represents the number of students. From the frequency table above, you can draw a bar chart with the class intervals on the horizontal axis and the frequencies on the vertical axis. The tallest bar corresponds to the 40–50 group, confirming that many students performed well. Bars for lower intervals are much shorter. A bar chart makes comparisons between groups immediate and visual.

条形图非常适合显示各个分数区间的频数。每个条形的高度代表学生人数。根据上面的频率表，你可以画出一个条形图，横轴是分数区间，纵轴是频数。最高的条形对应于 40–50 组，说明许多学生表现良好。较低区间的条形则短得多。条形图让组间比较变得直观而迅速。

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6. Displaying Data: Pie Chart of Reading Habits | 数据展示：阅读习惯饼图

To display the reading hour data, a pie chart can show the proportion of students in different reading categories. First, group reading hours into bands: 0–1 hours, 1–2 hours, 2–3 hours, 3–4 hours and 4+ hours. The frequencies are 4, 7, 8, 6 and 5 respectively. To draw the pie chart, calculate the angle for each sector using the formula:

Angle = (Frequency ÷ Total) × 360°

For example, the ‘0–1 hours’ group: (4 ÷ 30) × 360° = 48°. Doing this for all categories gives the angles. The completed pie chart shows that the ‘2–3 hours’ slice is the largest, while the smallest slices are for extremes. Pie charts help us see relative proportions, but they are less useful for comparing exact numbers.

为了展示阅读时间数据，可以用饼图表示不同阅读类别中学生的比例。首先将阅读时长分组：0–1 小时、1–2 小时、2–3 小时、3–4 小时和 4 小时以上。频数分别是 4、7、8、6 和 5。绘制饼图时，用以下公式计算每个扇形的角度：

角度 = (频数 ÷ 总人数) × 360°

例如，“0–1 小时”组：(4 ÷ 30) × 360° = 48°。对所有类别这样计算，就得到了各个角度。完成的饼图显示，“2–3 小时”这一块最大，而两端的扇形则最小。饼图帮助我们看到相对比例，但在比较精确数值方面用处稍弱。

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7. Mean, Median, Mode, and Range | 平均值、中位数、众数与范围

Calculating average measures helps to summarise both data sets. Let’s start with the reading hours. The mean is the sum of all reading hours divided by the number of students. Adding up the 30 values (including those not shown in the partial table) gives a total of 75 hours.

Mean reading hours = 75 ÷ 30 = 2.5 hours

To find the median, sort the 30 reading times in ascending order. The median is the average of the 15th and 16th values. Here, both are 2.5 hours, so the median is also 2.5 hours. The mode is the value that appears most often; 2.0 hours occurs most frequently, so the mode is 2.0 hours. The range is the difference between the largest and smallest reading times: 7.0 – 0.0 = 7.0 hours. Repeating the process for the test scores gives:

Mean score ≈ 37.1, Median score = 38.0, Range = 50 – 15 = 35

There is no single mode for scores, as several marks appear with the same highest frequency. These statistics tell us the typical performance and the spread.

计算平均量可以概括这两个数据集。先看阅读时间。平均数等于总阅读小时数除以学生人数。把 30 个数值（包括未在节选表格中出现的）相加，得到 75 小时。

平均阅读时数 = 75 ÷ 30 = 2.5 小时

要找出中位数，先将 30 个阅读时间从小到大排序。中位数是第 15 个和第 16 个数的平均值。在这里，两者都是 2.5 小时，因此中位数也是 2.5 小时。众数是出现次数最多的数值；2.0 小时出现最频繁，所以众数是 2.0 小时。范围是最大和最小阅读时数的差：7.0 – 0.0 = 7.0 小时。对测验成绩重复这一过程可得：

平均分 ≈ 37.1，中位数 = 38.0，范围 = 50 – 15 = 35

成绩没有单一的众数，因为有好几个分数出现了相同的最高频次。这些统计量告诉我们典型的表现和数据的分散程度。

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8. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph is the perfect tool to explore the relationship between two variables. Plot each of the 30 students as a point where the horizontal coordinate is their reading hours and the vertical coordinate is their test score. The points generally rise from left to right, meaning that as reading time increases, the score tends to increase as well. This pattern is called positive correlation. You can draw a line of best fit through the points to highlight the trend. There might be one or two outliers – for example, a student who reads very little but still achieves a high score. These outliers are important to mention because they show that correlation does not mean causation and that other factors also matter.

散点图是探究两个变量之间关系的完美工具。把这 30 名学生分别用点表示，横坐标是他们的阅读时数，纵坐标是测验分数。这些点总体从左到右呈上升趋势，意味着随着阅读时间增加，成绩也倾向于提高。这种模式叫做正相关。你可以在点之间画一条最佳拟合线来强调趋势。图中可能有一两个离群值——例如，一位阅读时间极少但依然取得高分的学生。这些离群值值得注意，因为它们表明相关性不等于因果关系，还有其他因素在起作用。

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9. Drawing Conclusions | 得出结论

Based on the evidence, we can say that students who spend more time reading per week generally achieve higher English test scores. The positive correlation in the scatter graph, combined with the fact that the high-score bar is the tallest, supports this finding. However, we must be careful: the data was collected from only 30 students in one school, so the conclusion may not apply to all Year 7 pupils. Moreover, there could be other reasons for the pattern, such as stronger literacy skills leading to both more reading and higher scores. Good statistical writing always acknowledges limitations while stating what the data shows.

根据现有证据，我们可以说，每周阅读时间更长的学生通常英语测验成绩更高。散点图中的正相关，加上最高分段条形图最高这一事实，都支持了这一发现。然而，我们必须谨慎：数据仅来自一所学校的 30 名学生，因此结论可能并不适用于所有七年级学生。此外，这种模式也可能有其他原因，比如更强的读写能力既导致了更多阅读，又带来了更高成绩。优秀的统计写作总是在陈述数据结果的同时，也承认其局限性。

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10. Evaluating the Process | 过程评价

Any statistical investigation should end with an evaluation. Was our sample representative? A random selection from one year group is reasonable, but 30 is a small number. Were the measurements accurate? Students might have estimated their reading hours wrongly or given answers they think the teacher wants. The test score, on the other hand, is an objective measure. A wider survey including students from different schools and a larger sample would make the findings more reliable. Thinking about these points is part of the ‘cycle of enquiry’ taught at KS3.

任何统计调查都应该以评价收尾。我们的样本具有代表性吗？从一个年级中随机抽取是合理的，但 30 人的样本量偏小。测量准确吗？学生可能估算错了阅读时长，或者给出了他们认为老师想听到的答案。相比之下，测验分数则是客观的度量。如果调查范围扩大到不同学校的学生，并且样本更大，研究结果就会更可靠。思考这些要点，正是 KS3 所教授的“探究循环”的一部分。

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11. Common Mistakes to Avoid | 常见错误避免

When doing your own case study, watch out for these frequent errors: choosing overlapping class intervals in a frequency table; confusing the mean with the median, especially when there are extreme values; forgetting to multiply the fraction by 360° when calculating pie chart angles; labelling axes on graphs incorrectly; and misinterpreting correlation – a strong correlation does not prove one variable causes the other. Also, always check that your totals match the number of data points before drawing any chart.

当你自己进行案例研究时，要留意这些常见错误：频率表中选择了有重叠的区间；混淆平均数和中位数，特别是在有极端值时；计算饼图角度时忘记用分数乘以 360°；图表坐标轴标签错误；以及误解相关性——强相关并不证明一个变量导致了另一个变量。另外，在绘制任何图表之前，请务必检查总频数是否与数据点数量一致。

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12. Practice Challenge | 实战挑战

Now it is your turn. Design a simple survey for your classmates on two variables, such as hours of screen time per day and hours of sleep per night. Collect data from at least 20 people. Construct a frequency table, draw one bar chart and one pie chart, and calculate the mean, median, mode and range for both variables. Finally, produce a scatter graph and comment on any correlation. This complete mini-study will give you the confidence to tackle any KS3 statistics question on the Cambridge syllabus.

现在轮到你了。就两个变量（如每天屏幕时间和每晚睡眠时长）设计一份简单的同学调查。从至少 20 人那里收集数据。构造频率表，画一个条形图和一个饼图，并计算两个变量的平均数、中位数、众数和范围。最后，画出散点图并评论其相关性。这个完整的小型研究将让你充满信心地应对剑桥 KS3 大纲中的任何统计问题。

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Published by TutorHao | Statistics Revision Series | aleveler.com

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