Case Study Practical Exercises in KS3 OCR Statistics | KS3 OCR 统计：案例分析实战演练

📚 Case Study Practical Exercises in KS3 OCR Statistics | KS3 OCR 统计：案例分析实战演练

Statistics is not just about numbers and formulas — it is a powerful tool for understanding the world around us. At Key Stage 3, the OCR curriculum encourages learners to apply statistical methods to real‑life situations through case studies. This article will guide you through a series of hands‑on practical exercises, from collecting data to interpreting probability, all built around realistic scenarios. By working through these case studies, you will develop confidence in calculating averages, constructing charts, comparing data sets, and evaluating the likelihood of events.

统计学不仅仅是数字和公式，它还是我们理解周围世界的有力工具。在关键阶段三（KS3），OCR 课程鼓励学习者通过案例学习将统计方法应用于现实生活情境。本文将通过一系列动手实践的练习，从收集数据到解释概率，全部围绕真实场景展开。通过完成这些案例分析，你将建立计算平均数、绘制图表、比较数据集以及评估事件可能性的信心。

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1. Introduction to Case Studies in Statistics | 统计案例学习简介

In OCR KS3 Statistics, case studies help you see how data handling skills are used in everyday decision‑making. A case study is simply a real‑world scenario where we ask a question, gather data, analyse it, and draw conclusions. Common topics include school surveys, sports results, weather patterns, and consumer choices. The key is to practise the full statistical cycle: posing a question, collecting data, organising it, representing it graphically, calculating summary statistics, and interpreting the findings.

在 OCR 的 KS3 统计学中，案例学习帮助你了解数据处理技能如何应用于日常决策。案例学习就是一个现实场景，我们提出问题、收集数据、分析数据并得出结论。常见主题包括学校调查、运动成绩、天气模式和消费者选择。关键在于实践完整的统计循环：提出问题、收集数据、整理数据、用图形表示、计算汇总统计量以及解释结果。

Throughout these practical exercises, always think about the context. Why are we collecting this data? Who will use the results? Being able to explain what your calculations mean in words is just as important as getting the right numbers.

在这些实践练习中，始终要考虑背景。我们为什么要收集这些数据？谁会使用这些结果？能够用语言解释你的计算结果意味着什么，和得到正确的数字同样重要。

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2. Collecting and Organising Data: The Oakwood Maths Test | 收集与整理数据：奥克伍德数学测验

Imagine you are a Year 8 student at Oakwood School. Your teacher gives a maths test out of 50 marks to 20 students. The raw scores are as follows: 32, 45, 28, 36, 42, 38, 45, 30, 44, 35, 40, 42, 28, 33, 45, 37, 41, 29, 34, 45. The first task in your case study is to organise this data so that patterns become visible.

想象你是奥克伍德学校的一名八年级学生。老师对 20 名学生进行了一次满分 50 分的数学测验。原始分数如下：32, 45, 28, 36, 42, 38, 45, 30, 44, 35, 40, 42, 28, 33, 45, 37, 41, 29, 34, 45。你这个案例学习中的第一个任务就是整理这些数据，以便看出规律。

A tidy list sorted in ascending order is the first step. Sort the scores from smallest to largest: 28, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 42, 44, 45, 45, 45, 45.

第一步是将数据按升序排列整齐。把分数从小到大排序：28, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 42, 44, 45, 45, 45, 45。

With ordered data, you can quickly spot the lowest and highest values, and whether any scores repeat. Here the lowest mark is 28 and the highest is 45. Notice that 45 appears four times — a detail that will be useful later when finding the mode.

有了排序后的数据，你就可以快速找出最低值和最高值，以及是否有分数重复。这里最低分是 28，最高分是 45。注意 45 出现了四次——这个细节在后面找众数时会很有用。

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3. Creating Frequency Tables | 制作频率表

A frequency table helps to summarise data clearly. For the Oakwood test scores, we can group individual marks into a tally and frequency table. Using tallies makes counting easier and reduces mistakes.

频率表有助于清晰汇总数据。对于奥克伍德测验的分数，我们可以将单个分数整理成一张计分和频率表。使用划记法可以更容易计数，并减少错误。

Score	Tally	Frequency
28	||	2
29	|	1
30	|	1
32	|	1
33	|	1
34	|	1
35	|	1
36	|	1
37	|	1
38	|	1
40	|	1
41	|	1
42	||	2
44	|	1
45	||||	4

Always check the sum of the frequencies: 2+1+1+1+1+1+1+1+1+1+1+1+2+1+4 = 20, which matches the number of students. This confirms the table is correct. Grouped frequency tables can also be used when there are many different values. For instance, you might group scores into intervals like 20–29, 30–39, 40–50.

一定要检查频率总和：2+1+1+1+1+1+1+1+1+1+1+1+2+1+4 = 20，与学生的数量相符。这确认了表格的正确性。当数值很多时，也可以使用分组的频率表。例如，你可能会把分数分成 20–29、30–39、40–50 这样的区间。

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4. Drawing Bar Charts and Pictograms | 绘制条形图和象形图

Visual representations make data easier to understand. For the Oakwood scores, a bar chart is ideal because the data is discrete. On the horizontal axis we place the test scores, and on the vertical axis the frequency. Make sure the bars are evenly spaced and clearly labelled.

图形表示使数据更易于理解。对于奥克伍德的分数，条形图是理想的选择，因为数据是离散的。我们在横轴上放置测验分数，在纵轴上放置频率。确保条形间距均匀，并清晰标注。

When drawing by hand, use a ruler and a sharp pencil. Include a title, e.g., ‘Year 8 Maths Test Scores’, and label both axes. The vertical scale must be suitable — here the maximum frequency is 4, so a scale of 1 cm per student works well. If you were using intervals, the bar chart would look different because the width of each bar would represent an interval range.

手绘图表时，请使用直尺和尖铅笔。加上标题，例如“八年级数学测验分数”，并给两个坐标轴都加上标签。纵轴刻度必须合适——这里的最大频率是 4，因此每厘米代表 1 个学生的刻度就很合适。如果使用区间，条形图看起来会不同，因为每个条形的宽度代表一个区间范围。

Pictograms can also be used to represent the same data. You might use one book symbol to represent one student. For a score of 45 with frequency 4, you would draw four identical book symbols in that row. Always include a key showing what one symbol represents.

象形图也可用于表示相同的数据。你可以用一个书本符号代表一个学生。对于频率为 4 的 45 分，你就在那一行画四个相同的书本符号。始终要提供一个图例，说明一个符号代表什么。

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5. Calculating Mean, Median, Mode | 计算平均数、中位数和众数

The three measures of central tendency are essential for summarising a data set. Let’s calculate each one for the 20 maths scores.

集中趋势的三种度量对于汇总数据集至关重要。我们来为这 20 个数学成绩分别计算它们。

Mode: The mode is the value that appears most often. From the frequency table, the score 45 occurs 4 times — more than any other mark. Therefore, the modal score is 45. A set can have more than one mode, but here there is only one.

众数：众数是出现次数最多的值。根据频率表，45 分出现 4 次——比任何其他分数都多。因此，众数是 45。一组数据可以有多个众数，但这里只有一个。

Median: The median is the middle value when the data is ordered. With 20 values (an even number), the median lies between the 10th and 11th values. The ordered list: 1st = 28, 2nd = 28, 3rd = 29, …, 10th = 37, 11th = 38. So the median = (37 + 38) ÷ 2 = 37.5.

中位数：中位数是排序后位于中间的值。因为有 20 个值（偶数），中位数位于第 10 和第 11 个值之间。排序列表：第 1 个=28，第 2 个=28，第 3 个=29，……第 10 个=37，第 11 个=38。因此中位数 = (37 + 38) ÷ 2 = 37.5。

Mean: To find the mean, add all the scores together and divide by the total number of scores. The sum of all scores is 32+45+28+…+45 = 750. (Try adding them yourself to verify.) Then divide by 20. So the mean x̄ = 750 ÷ 20 = 37.5.

平均数：要计算平均数，先把所有分数相加，然后除以分数的总个数。所有分数的总和是 32+45+28+…+45 = 750。（自己加总验证一下。）然后除以 20。因此平均数 x̄ = 750 ÷ 20 = 37.5。

x̄ = Σx ÷ n

In this case, the mean and median happen to be the same, but this is not always true. When the mean and median are close, the data is fairly symmetrical. If the mean is much higher than the median, the data may be positively skewed by a few high values.

在本例中，平均数和中位数碰巧相同，但这并非总是如此。当平均数和中位数相近时，数据是对称的。如果平均数远高于中位数，数据可能由于少数高值而呈正偏态。

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6. Understanding the Range | 理解极差

The range is a simple measure of spread or dispersion. It is calculated by subtracting the smallest value from the largest value. For our data set, the maximum score is 45 and the minimum is 28, so the range = 45 − 28 = 17.

极差是衡量离散或分散程度的一个简单指标。它等于最大值减去最小值。对于我们的数据集，最高分是 45，最低分是 28，因此极差 = 45 − 28 = 17。

A larger range indicates that the scores are more spread out, while a smaller range means they are clustered closely together. However, the range only takes into account the two extreme values, so it can be affected by outliers. An extremely low or high score can make the range very large, even if the rest of the data is consistent.

极差越大，说明分数分布越分散；极差越小，则表示分数聚集得更紧密。但是，极差只考虑了两个极端值，因此容易受异常值的影响。一个极低或极高的分数会使极差变得非常大，即使其余数据很一致。

In the Oakwood test, a range of 17 out of 50 shows moderate spread. Suppose another class had scores ranging from 40 to 48 — range 8. That class’s scores are more consistent and clustered at the top end, while Oakwood’s marks are more varied.

在奥克伍德的测验中，满分 50 分的情况下极差为 17，显示出中等程度的离散。假设另一班级的分数在 40 到 48 之间——极差为 8。那个班级的成绩更一致，聚集在高端，而奥克伍德的分数则差异更大。

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7. Comparing Two Data Sets | 比较两组数据

Case studies often involve comparing two or more groups. Let’s add a second set of maths test scores from another Year 8 class, Class B: 42, 41, 39, 44, 43, 38, 45, 40, 42, 41, 44, 39, 43, 40, 42, 45, 43, 41, 44, 42. We can compare the two classes using averages and range.

案例学习通常涉及比较两个或更多组。我们新增另一个八年级班级——B 班的数学测验分数：42, 41, 39, 44, 43, 38, 45, 40, 42, 41, 44, 39, 43, 40, 42, 45, 43, 41, 44, 42。我们可以用平均数和极差来比较这两个班级。

For Class B, the ordered data is: 38, 39, 39, 40, 40, 41, 41, 41, 42, 42, 42, 42, 43, 43, 43, 44, 44, 44, 45, 45. The mode is 42 (frequency 4), the median is the average of the 10th and 11th values: (42 + 42) ÷ 2 = 42. The sum of scores is 832, so mean x̄ = 832 ÷ 20 = 41.6. The range = 45 − 38 = 7.

对于 B 班，排序后的数据为：38, 39, 39, 40, 40, 41, 41, 41, 42, 42, 42, 42, 43, 43, 43, 44, 44, 44, 45, 45。众数是 42（出现 4 次），中位数是第 10 个和第 11 个值的平均数：(42 + 42) ÷ 2 = 42。总和是 832，平均数 x̄ = 832 ÷ 20 = 41.6。极差 = 45 − 38 = 7。

Now we can write a comparison: Oakwood’s class has a higher mean (37.5 vs 41.6? Wait, recalc: Oakwood mean 37.5, Class B mean 41.6 — actually Class B mean is higher. So Oakwood’s class scored lower on average but had a much larger spread (range 17 vs 7). This suggests Class B performed more consistently and generally better. Using these statistics, you can form evidence‑based conclusions.

现在我们可以进行比较：奥克伍德班的平均数（37.5）低于 B 班（41.6），但奥克伍德班的分数分布更广（极差 17 对 7）。这表明 B 班的表现更加稳定，整体更好。利用这些统计量，你可以形成基于证据的结论。

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8. Introduction to Probability with Dice | 用骰子介绍概率

Probability is the branch of mathematics that deals with chance. In OCR KS3, you learn to express probability on a scale from 0 (impossible) to 1 (certain). A fair six‑sided die is a classic tool for exploring theoretical probability. The probability of rolling any specific number, say 3, is 1 out of 6, written as 1/6.

概率是数学中处理随机性的分支。在 OCR 的 KS3 阶段，你要学习用 0（不可能）到 1（必然）的尺度来表示概率。一颗均匀的六面骰子是探索理论概率的经典工具。掷出任何一个特定数字，比如 3 的概率是 1/6。

You can conduct an experiment as a case study: roll a die 60 times and record the outcomes. Suppose your results show: 1 appeared 8 times, 2 appeared 11 times, 3 appeared 9 times, 4 appeared 13 times, 5 appeared 10 times, 6 appeared 9 times. The experimental probability of rolling a 4 is 13/60, while the theoretical probability is 1/6 ≈ 0.167. The experimental value is close to 0.217 — a slight difference due to random variation.

你可以用一个实验作为案例研究：掷骰子 60 次并记录结果。假设你的结果如下：1 出现 8 次，2 出现 11 次，3 出现 9 次，4 出现 13 次，5 出现 10 次，6 出现 9 次。掷出 4 的实验概率是 13/60，而理论概率是 1/6 ≈ 0.167。实验值接近 0.217——差异不大，由随机波动引起。

As a case study exercise, plot the experimental results on a bar chart and compare visually with the theoretical expectation (each outcome 10 times). Discuss why the two might differ and what would happen if you increased the number of trials to 600 or 6000 (the law of large numbers).

作为案例研究练习，将实验结果绘制在条形图上，并与理论预期（每个结果 10 次）进行视觉比较。讨论两者为何会不同，以及如果你将试验次数增加到 600 或 6000 次会发生什么（大数定律）。

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9. Case Study: Sports Day Events | 案例：运动会项目分析

Imagine your school’s Sports Day features a 100 m sprint for Year 8 boys. The recorded times (in seconds) for 25 runners are: 14.2, 13.8, 15.1, 14.5, 13.9, 14.0, 15.2, 14.3, 14.7, 13.7, 14.6, 14.1, 14.9, 15.0, 14.4, 13.6, 14.8, 15.3, 14.2, 13.5, 14.5, 14.3, 15.1, 13.9, 14.6. This is a new data set to practise your skills.

假设你们学校的运动会上有八年级男生的 100 米短跑比赛。记录的 25 名选手的时间（秒）如下：14.2, 13.8, 15.1, 14.5, 13.9, 14.0, 15.2, 14.3, 14.7, 13.7, 14.6, 14.1, 14.9, 15.0, 14.4, 13.6, 14.8, 15.3, 14.2, 13.5, 14.5, 14.3, 15.1, 13.9, 14.6。这是一个新数据集，用来练习你的技能。

First, sort the times in ascending order: 13.5, 13.6, 13.7, 13.8, 13.9, 13.9, 14.0, 14.1, 14.2, 14.2, 14.3, 14.3, 14.4, 14.5, 14.5, 14.6, 14.6, 14.7, 14.8, 14.9, 15.0, 15.1, 15.1, 15.2, 15.3. Find the fastest time (13.5 s) and the slowest (15.3 s). The range is 15.3 − 13.5 = 1.8 seconds.

首先将时间按升序排列：13.5, 13.6, 13.7, 13.8, 13.9, 13.9, 14.0, 14.1, 14.2, 14.2, 14.3, 14.3, 14.4, 14.5, 14.5, 14.6, 14.6, 14.7, 14.8, 14.9, 15.0, 15.1, 15.1, 15.2, 15.3。找出最快时间（13.5 秒）和最慢时间（15.3 秒）。极差为 15.3 − 13.5 = 1.8 秒。

With 25 values, the median is the 13th value in the ordered list: 14.4 seconds. The mean: sum all times (you can practise adding decimals) — the total is 362.0, so mean = 362.0 ÷ 25 = 14.48 seconds. The mode is not straightforward here because several times appear twice, but 13.9, 14.2, 14.3, 14.5, 14.6 and 15.1 all appear twice, so the data set is multimodal. You could group the times into intervals (e.g., 13.5−13.9, 14.0−14.4, 14.5−14.9, 15.0−15.4) to see the distribution.

有 25 个值，中位数是排序列表中的第 13 个值：14.4 秒。平均数：将所有时间相加（你可以练习小数加法）——总和为 362.0，因此平均数 = 362.0 ÷ 25 = 14.48 秒。这里的众数不唯一，因为有好几个时间出现了两次（13.9、14.2、14.3、14.5、14.6、15.1），所以这组数据是复众数的。你可以把时间分组（例如 13.5−13.9、14.0−14.4、14.5−14.9、15.0−15.4）来看分布情况。

A box plot could be used to display the five‑number summary: minimum (13.5), lower quartile (14.0), median (14.4), upper quartile (14.9), and maximum (15.3). This is a valuable skill for OCR KS3 and helps compare performance across year groups.

可以使用箱形图来显示五数概括：最小值 (13.5)、下四分位数 (14.0)、中位数 (14.4)、上四分位数 (14.9) 和最大值 (15.3)。这是 OCR KS3 阶段的一项宝贵技能，有助于比较不同年级的表现。

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10. Case Study: School Travel Survey | 案例：学校出行方式调查

A school travel survey asks 200 students how they travel to school. The results are: walk 45%, cycle 25%, bus 20%, car 10%. This categorical data is perfect for a pie chart. To draw it, you need the angle for each sector: walk = 0.45 × 360° = 162°, cycle = 0.25 × 360° = 90°, bus = 0.20 × 360° = 72°, car = 0.10 × 360° = 36°. Check total: 162°+90°+72°+36° = 360°.

Published by TutorHao | KS3 统计 Revision Series | aleveler.com

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