Year 10 CIE Statistics: Case Study Practical Exercises | Year 10 CIE 统计：案例分析实战演练

📚 Year 10 CIE Statistics: Case Study Practical Exercises | Year 10 CIE 统计：案例分析实战演练

This article guides you through two real-world case studies designed to strengthen your application of CIE IGCSE Statistics concepts. You will work with actual student performance data, practising data presentation, summary statistics, probability and bivariate analysis – exactly the skills tested in your exams.

本文通过两个真实案例研究，指导你应用 CIE IGCSE 统计概念。你将使用实际的学生成绩数据，练习数据呈现、汇总统计量、概率和双变量分析——正是考试中考查的技能。

1. Introduction to the Case Study | 案例介绍

Case Study A focuses on the mathematics test scores of 30 Year 10 students, each marked out of 50. The aim is to describe the distribution using tables, charts and summary measures. Case Study B then pairs the same students’ maths scores with their English scores (also out of 50) to explore the relationship between performance in the two subjects. The first ten rows of the dataset are shown below.

案例 A 聚焦 30 名 Year 10 学生的数学测试成绩（满分 50），目标是使用表格、图表和汇总量数描述其分布。案例 B 则将这批学生的数学成绩与英语成绩（满分 50）配对，探究两科成绩之间的关系。以下列出数据集前 10 行。

Student	Maths Score	English Score
1	23	25
2	25	28
3	28	30
4	30	32
5	31	33
6	32	34
7	34	36
8	35	38
9	35	38
10	36	39

You will learn to calculate averages, draw charts, find probabilities and interpret correlation – all by working through the steps as if you were the statistician.

你将学习计算平均数、绘制图表、求概率和解释相关性——全部通过像统计学家一样逐步操作来完成。

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2. Data Collection and Types | 数据收集与类型

The data were collected from a classroom test administered to 30 students. Each student’s maths and English raw marks were recorded as whole numbers. In statistics, we must first classify the type of data: maths scores are discrete numerical data because they can only take certain values (here integers from 0 to 50).

数据来自对 30 名学生进行的课堂测试，记录每名学生的数学和英语原始分数，均为整数。在统计中，我们首先要对数据类型分类：数学成绩是离散数值数据，因为它们只能取特定值（此处是 0 到 50 的整数）。

If we had recorded the time taken to complete the test, that would be continuous data. Recognising data type guides our choice of graph – bar charts for discrete data and histograms for continuous grouped data.

如果记录的是完成测试所用时间，那将是连续数据。识别数据类型可以帮助我们选择图表——离散数据用条形图，连续分组数据用直方图。

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3. Organising Data: Frequency Tables | 数据整理：频数表

To summarise the 30 maths scores, we group them into equal-width intervals. Using a class width of 5, we construct a frequency table:

为了汇总 30 个数学成绩，我们按等宽组距分组。采用组距 5，构建如下频数表：

Score Interval	Tally	Frequency
20 – 24	|	1
25 – 29	||	2
30 – 34	||||	4
35 – 39	|||| |	6
40 – 44	|||| |||	8
45 – 50	|||| ||||	9

A tally column helps to count accurately. The grouped data show that scores are clustered in the higher intervals, with the modal class being 45–50.

计数列有助于准确清点。分组数据显示成绩集中在较高区间，众数组为 45–50。

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4. Visual Representation: Bar Charts and Histograms | 可视化表示：条形图与直方图

Because maths score is discrete and we have ungrouped raw data, we could draw a bar chart of individual scores. However, with grouped data, a histogram is more appropriate. A histogram plots frequency on the vertical axis and score intervals on the horizontal axis, with bars touching to reflect continuous class boundaries (e.g. 19.5–24.5, 24.5–29.5, …).

由于数学成绩是离散的且我们有未分组原始数据，我们可以绘制单个分数的条形图。但对于已分组数据，直方图更合适。直方图以纵轴表示频数，横轴表示分数区间，条形紧靠以反映连续的分组边界（例如 19.5–24.5, 24.5–29.5 ……）。

In our histogram, the tallest bar is for 45–50, confirming the modal class. Always label axes clearly and give the chart a title.

在此直方图中，最高的条形是 45–50，确认了众数组。务必清晰标记坐标轴并为图表命名。

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5. Measures of Central Tendency | 集中趋势度量

We calculate three averages for the 30 maths scores. The mean is the sum of all scores divided by 30.

我们对 30 个数学成绩计算三种平均数。平均值是所有成绩总和除以 30。

x̄ = Σx / n = 1194 / 30 = 39.8

To find the median, list the scores in order. The 15th and 16th values are 40 and 41, so the median is (40 + 41) / 2 = 40.5. The mode is the most frequent score, which is 50 (appearing 4 times). The mean and median are close, indicating a fairly symmetric distribution with a slight negative skew.

求中位数时，将成绩升序排列。第 15 和第 16 个数值是 40 和 41，因此中位数为 (40+41)/2 = 40.5。众数是出现次数最多的分数，即 50（出现 4 次）。平均值和中位数接近，表明分布大致对称，略带负偏态。

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6. Measures of Spread: Range and IQR | 离散程度：极差与四分位距

Spread tells us how varied the data are. The range is simply the maximum minus the minimum: 50 – 23 = 27. A more robust measure is the interquartile range (IQR).

离散程度说明数据有多分散。极差即最大值减最小值：50 – 23 = 27。更稳健的度量是四分位距 (IQR)。

To find IQR, we first locate the lower quartile Q₁ (25th percentile) and upper quartile Q₃ (75th percentile). For n = 30, Q₁ is the value between the 7th and 8th ordered scores: (34 + 35)/2 = 34.5. Q₃ is between the 23rd and 24th scores: (46 + 47)/2 = 46.5. Hence IQR = Q₃ – Q₁ = 12. The IQR ignores extreme values and shows that the middle 50% of students scored within a 12-mark range.

为求 IQR，先确定下四分位数 Q₁（第 25 百分位数）和上四分位数 Q₃（第 75 百分位数）。n=30 时，Q₁ 位于第 7 和第 8 个排序成绩之间：(34+35)/2 = 34.5。Q₃ 位于第 23 和第 24 个之间：(46+47)/2 = 46.5。因此 IQR = Q₃ – Q₁ = 12。IQR 忽略极端值，显示中间 50% 学生的成绩落在 12 分的区间内。

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7. Probability from Data | 从数据计算概率

We can use the frequency table to estimate probabilities. If a student is chosen at random from the 30, what is the probability their maths score is at least 40?

我们可以利用频数表估计概率。如果从 30 人中随机选一名学生，其数学成绩不低于 40 的概率是多少？

From the table, the intervals 40–44 and 45–50 together contain 8 + 9 = 17 students. So P(score ≥ 40) = 17/30 ≈ 0.567. What about scoring below 30? Only the first two intervals (20–24 and 25–29) count, giving 1 + 2 = 3 students, so probability = 3/30 = 0.1. These are experimental probabilities derived directly from the sample.

由表可知，40–44 和 45–50 两组合计 8+9=17 人。因此 P(成绩 ≥ 40) = 17/30 ≈ 0.567。若问成绩低于 30 的概率，只有前两组（20–24 和 25–29）符合，共 1+2=3 人，概率 = 3/30 = 0.1。这些是从样本直接得到的试验概率。

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

Case Study B investigates whether high maths scores tend to go with high English scores. We plot each student as a point with maths on the x-axis and English on the y-axis. The complete dataset of 30 points shows an upward pattern from bottom-left to top-right, suggesting positive correlation.

案例 B 探究数学高分是否通常伴随英语高分。我们将每名学生以点标出，x 轴为数学成绩，y 轴为英语成绩。全部 30 个点显示从左下到右上的上升模式，表明存在正相关。

The points are fairly closely packed around an imaginary straight line, so the correlation is strong positive. We describe it as ‘strong, positive linear correlation’. Always remember: correlation does not imply causation.

这些点相当紧密地分布在一条虚拟直线周围，因此是强正相关。我们描述为“强正线性相关”。务必牢记：相关性不意味着因果关系。

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9. Line of Best Fit and Predictions | 最佳拟合线与预测

A line of best fit can be drawn by eye, balancing points above and below the line. It should pass through the mean point (x̄, ȳ). For our data, x̄ = 39.8 and the mean English score ȳ is approximately 41.2.

我们可以凭目测画出最佳拟合线，使线上下两侧的点数大致均衡。该线应通过均值点 (x̄, ȳ)。这里 x̄=39.8，英语平均分 ȳ 约为 41.2。

Suppose we want to predict the English

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