📚 Case Study Practical Drills in Year 7 Statistics | 七年级统计案例分析实战演练
Working with real data is the best way to build confidence in statistics. In this article, we will walk through several practical case studies that cover the key topics in the Year 7 CAIE Statistics syllabus. Each case study shows you how to collect data, organise it into tables, draw graphs, and calculate averages and the range. You will also learn how to interpret results, spot mistakes, and apply these skills to exam-style questions. The aim is to turn dry numbers into a story you can understand and use.
用真实数据来练习是建立统计信心的最佳方法。本文将带你逐步完成几个实用的案例分析，这些案例覆盖了七年级 CAIE 统计大纲的核心知识点。每个案例都展示了如何收集数据、将数据整理成表格、绘制图表，以及如何计算平均数和极差。你还将学习如何解读结果、发现错误，并将这些技能运用到考试题型中。目的是把枯燥的数字变成你能理解并运用的故事。
1. Case Study 1: Favourite Colours Survey | 案例一：最喜欢的颜色调查
A class of 30 Year 7 students was asked, "What is your favourite colour?" The choices given were Red, Blue, Green, Yellow, and Other. The raw responses were noted on the board as follows: Red, Blue, Blue, Green, Red, Yellow, Red, Blue, Green, Red, Blue, Red, Green, Blue, Yellow, Red, Blue, Green, Red, Blue, Red, Other, Blue, Green, Red, Blue, Red, Green, Yellow, Blue.
某七年级班级的 30 名学生被问道："你最喜欢什么颜色？" 选项有红色、蓝色、绿色、黄色和其他。原始回答写在白板上如下：Red, Blue, Blue, Green, Red, Yellow, Red, Blue, Green, Red, Blue, Red, Green, Blue, Yellow, Red, Blue, Green, Red, Blue, Red, Other, Blue, Green, Red, Blue, Red, Green, Yellow, Blue。
The first step is to sort this mess into a frequency table. Tally marks help you count accurately. Go through the list and for each colour, draw a tally stroke. Group them in fives for easy counting. When you finish, write the total frequency.
第一步是将这堆杂乱的数据整理成频数表。划记符号可以帮助你准确计数。逐一浏览列表，为每种颜色画一道记数线。每五个划为一组方便统计。完成后，写出总频数。
| Colour | Tally | Frequency |
|---|---|---|
| Red | |||| |||| | | 11 |
| Blue | |||| |||| | 10 |
| Green | |||| | 5 |
| Yellow | ||| | 3 |
| Other | | | 1 |
| Total | 30 |
rrect. From the table we can immediately see that Red is the most popular colour and Other is the least common.
一定要检查频数之和等于数据总数。这里 11+10+5+3+1=30，因此表格正确。从表中我们可以立刻看出红色是最受欢迎的颜色，"其他"选项最少见。
2. Visualising Data: Bar Chart and Pictogram | 数据可视化：条形图和象形图
A frequency table is useful, but a graph makes the pattern easier to spot. A bar chart is perfect for categorical data like favourite colours. Draw axes: the horizontal axis shows the colour categories, the vertical axis shows the frequency. The bars must have equal width and gaps between them. Label the axes and give the chart a title. For this data, the bar for Red reaches 11, Blue reaches 10, Green 5, Yellow 3, and Other 1.
频数表很有用，但图形使得规律更容易被发现。条形图非常适合像喜爱颜色这样的分类数据。绘制坐标轴：横轴表示颜色类别，纵轴表示频数。条形必须有相同的宽度且条形之间要有间隙。给坐标轴加上标签，并为图表标上标题。针对这组数据，红色条到达 11，蓝色条到达 10，绿色条 5，黄色条 3，其他条 1。
udents. Then Red would have 5 and a half smileys, Blue 5 smileys, Green 2 and a half, Yellow 1 and a half, and Other a half smiley. A key must always be included, e.g. 😊 = 2 students.
另一种有用的图是象形图。选择一个符号代表一定数量的学生。例如，一个笑脸代表 2 名学生。那么红色将有 5 个半笑脸，蓝色 5 个，绿色 2 个半，黄色 1 个半，其他半个笑脸。必须包含图例，例如 😊 = 2 名学生。
Practice: Draw the bar chart and pictogram in your notebook. Which one makes it easier to see that Red and Blue are almost tied as favourites? Compare the heights instantly in the bar chart; in the pictogram you have to count symbols carefully.
练习：在你的笔记本上画出条形图和象形图。哪种图更容易看出红色和蓝色几乎并列第一？在条形图中可以立即比较高度；在象形图中则需要仔细数符号。
3. Case Study 2: Class Heights Measurement | 案例二：班级身高测量
A group of 15 students measured their heights in centimetres. The results were: 142, 138, 145, 140, 148, 139, 142, 144, 138, 141, 146, 140, 142, 137, 143. This is numerical continuous data. The first task is to order the data from smallest to largest: 137, 138, 138, 139, 140, 140, 141, 142, 142, 142, 143, 144, 145, 146, 148. Ordering helps you find the median and range quickly.
一组 15 名学生测量了他们的身高，单位为厘米。结果如下：142, 138, 145, 140, 148, 139, 142, 144, 138, 141, 146, 140, 142, 137, 143。这是数值型连续数据。第一个任务是将数据从小到大排序：137, 138, 138, 139, 140, 140, 141, 142, 142, 142, 143, 144, 145, 146, 148。排序能帮助你快速找到中位数和极差。
When you have an ordered list, it is much harder to miss a value. Double-check that you have the same number of data items (15) after ordering. Now let's compute the three averages: mode, median, and mean.
当你有了排序后的列表，就不太容易漏掉数值了。仔细检查排序后你仍有相同数量的数据（15个）。现在我们来计算三种平均数：众数、中位数和平均数。
4. Calculating Mode, Median, Mean, and Range | 计算众数、中位数、平均数和极差
The mode is the value that appears most often. In our height list, 142 cm appears three times. All other values appear once or twice. Therefore the mode is 142 cm.
众数是出现次数最多的数值。在我们的身高列表中，142 cm 出现了三次。其他数值出现一次或两次。因此众数是 142 cm。
The median is the middle value when data is ordered. Since we have 15 values, the median is the 8th value. Count along: the 1st is 137, 2nd 138, 3rd 138, 4th 139, 5th 140, 6th 140, 7th 141, 8th is 142. So the median height is 142 cm.
中位数是数据排序后中间的那个值。因为有 15 个数值，中位数就是第 8 个值。数过去：第 1 个是 137，第 2 个 138，第 3 个 138，第 4 个 139，第 5 个 140，第 6 个 140，第 7 个 141，第 8 个是 142。因此中位数身高是 142 cm。
= 142.5 cm. Always remember to add the two middle numbers and divide by two in that case.
如果有偶数个数据值，比如 16 个，中位数将是第 8 和第 9 个值的平均数。例如，(142+143) ÷ 2 = 142.5 cm。在这种情况下一定要记住把中间两个数相加然后除以 2。
The mean 48. Use a calculator or add step by step. The total is 2125 cm. Now divide by 15: 2125 ÷ 15 = 141.666... We round this to 141.7 cm (1 decimal place). So the mean height is approximately 141.7 cm.
平均数是所有身高之和除以学生人数。把 15 个身高全部相加：137+138+138+139+140+140+141+142+142+142+143+144+145+146+148。用计算器或逐步相加。总和是 2125 cm。现在除以 15：2125 ÷ 15 = 141.666... 我们四舍五入到 141.7 cm（一位小数）。因此平均身高大约是 141.7 cm。
The range 137 cm. Range = 148 − 137 = 11 cm. The range gives an idea of how spread out the data is.
极差是最大值与最小值之间的差。最大值 = 148 cm，最小值 = 137 cm。极差 = 148 − 137 = 11 cm。极差能让你了解数据的分散程度。
Mean = 141.7 cm, Median = 142 cm, Mode = 142 cm, Range = 11 cm
Notice that in this set, the median and mode are the same, while the mean is slightly lower. This tells us the data is fairly symmetric but slightly pulled down by the smaller values.
请注意在这组数据中，中位数和众数相同，而平均数略低一些。这告诉我们数据大致对称，但被偏小的值略微拉低了。
5. Case Study 3: Weekly Sales of a Tuck Shop | 案例三：小吃店一周销售数据
A school tuck shop recorded the number of healthy snack packs sold each day from Monday to Friday. The data is: Monday 24 packs, Tuesday 30 packs, Wednesday 35 packs, Thursday 28 packs, Friday 42 packs. This is time-based data, so a line graph is the best choice to show the trend across the week.
学校小吃店记录了周一到周五每天售出的健康零食包数量。数据是：周一 24 包，周二 30 包，周三 35 包，周四 28 包，周五 42 包。这是基于时间的数据，因此线形图是展示一周趋势的最佳选择。
First, present the data in a table for clarity:
首先，用表格清晰地呈现数据：
| Day | Packs Sold |
|---|---|
| Monday | 24 |
| Tuesday | 30 |
| Wednesday | 35 |
| Thursday | 28 |
| Friday | 42 |
To draw a line graph, plot each day on the horizontal axis (evenly spaced) and the number of packs on the vertical axis. Plot the points (Mon, 24), (Tue, 30), (Wed, 35), (Thu, 28), (Fri, 42). Join the points with straight lines. Don't forget to label the axes and give a title such as "Healthy Snack Sales During the Week".
要绘制线形图，在横轴上标出每天（等距），在纵轴上标出销售包数。描出各点：(周一, 24), (周二, 30), (周三, 35), (周四, 28), (周五, 42)。用直线连接各点。别忘了给坐标轴加标签，并加上标题，比如"一周健康零食销售情况"。
The line graph reveals an overall upward trend, although Thursday shows a dip. Perhaps Thursday's sales were affected by a school trip or a special event. On Friday, sales peaked, maybe because students wanted a treat before the weekend. Analysing the 'why' behind trends is a key statistical thinking skill.
线形图揭示了一个总体上升的趋势，尽管周四有所下降。也许周四的销售受到了学校旅行或某个特别活动的影响。周五销售达到顶峰，可能是因为学生们想在周末前犒劳一下自己。分析趋势背后的 "为什么" 是一项关键的统计思维能力。
6. Calculating Total and Average Weekly Sales | 计算周销售总量和平均销量
= 31.8 packs. This average helps the tuck shop plan how many snacks to prepare each day. If they expect about 32 packs daily, they can avoid waste or running out of stock.
根据表格，我们可以计算出一周售出的总包数：24 + 30 + 35 + 28 + 42 = 159 包。日均销量是总量除以 5 天：159 ÷ 5 = 31.8 包。这个平均值有助于小吃店规划每天准备多少零食。如果预计每天大约需要 32 包，就可以避免浪费或缺货。
that demand can vary quite a lot, so flexibility is needed, especially on busy days like Friday.
日销量的极差也值得计算：最高（42）− 最低（24）= 18 包。这告诉经理需求量波动较大，因此需要灵活应对，尤其是在像周五这样的繁忙日子。
7. Combining Two Data Sets: Comparing Class Heights | 合并两组数据：比较两个班级的身高
Let's add data from another class of 15 students, whose heights (in cm) are: 145, 150, 142, 148, 152, 139, 147, 151, 146, 149, 143, 148, 150, 145, 147. We want to compare the two classes' heights using averages and range to see which class is taller on average and which is more consistent.
让我们加入另一个班级 15 名学生的数据，他们的身高（cm）为：145, 150, 142, 148, 152, 139, 147, 151, 146, 149, 143, 148, 150, 145, 147。我们想用平均数和极差来比较两个班级的身高，看看哪个班级平均身高更高，哪个班级更整齐。
Step 1: Order Class B' 150, 151, 152. There are 15 values, so the median is the 8th value, which is 147 cm. Mode? 8+148+149+150+150+151+152 = 2202. Divide by 15: 2202 ÷ 15 = 146.8 cm. Range: 152 − 139 = 13 cm.
步骤 1：将 B 班数据排序：139, 142, 143, 145, 145, 146, 147, 147, 148, 148, 149, 150, 150, 151, 152。共有 15 个值，中位数是第 8 个值，即 147 cm。众数呢？145, 147, 148, 150 各出现两次，因此有多个众数（双峰/多峰）。平均数：总和 = 139+142+143+145+145+146+147+147+148+148+149+150+150+151+152 = 2202。除以 15：2202 ÷ 15 = 146.8 cm。极差：152 − 139 = 13 cm。
Now compare with Class A from Case Study 2: Mean 141.7 cm, Median 142 cm, Range 11 cm. Class B has a higher mean (146.8 cm vs 141.7 cm) and a higher median (147 cm vs 142 cm), so Class B is generally taller. However, Class B's range is larger (13 cm vs 11 cm), meaning their heights are slightly more spread out. Class A is more consistent.
现在与案例二中的 A 班比较：平均数 141.7 cm，中位数 142 cm，极差 11 cm。B 班的平均数更高（146.8 cm vs 141.7 cm），中位数也更高（147 cm vs 142 cm），因此 B 班总体上更高。然而，B 班的极差更大（13 cm vs 11 cm），这意味着他们的身高分布稍微更分散。A 班则更整齐。
Always use both an average and the range for a fair comparison. An average alone can hide important information about spread.
进行公平比较时一定要同时使用平均数和极差。单独的平均数可能隐藏重要的分散程度信息。
8. Spotting Misleading Graphs and Errors | 识别误导性图表和错误
Statistics can be misused if graphs are drawn poorly. Suppose someone draws the bar chart for favourite colours but starts the vertical axis from 8 instead of 0. The bar for Red (11) might then look more than twice as tall as the bar for Other (1), exaggerating the difference. Always check that axes start at zero unless there is a very good reason and it is clearly labelled with a break symbol.
如果图表绘制不当，统计数据可能被误用。假设有人在绘制喜爱颜色条形图时，将纵轴起点设为 8 而不是 0。那么代表红色的条（11）可能看起来比代表"其他"的条（1）高出两倍多，从而夸大了差异。务必检查坐标轴是否从零开始，除非有非常充分的理由且用断裂符号明确标示。
Common mistakes students make include: forgetting to order data before finding the median; dividing by the wrong number when finding the mean (e.g. dividing by 5 for 6 numbers); and mixing up the x-axis and y-axis labels. Always take a moment to re-read the question and check your steps. In a frequency table, a quick sum of frequencies can catch counting errors.
学生们常犯的错误包括：在求中位数之前忘记排序；求平均数时除以了错误的数字（例如 6 个数却除以 5）；混淆了 x 轴和 y 轴的标签。务必花点时间重新读题并检查步骤。在频数表中，快速求和频数可以发现计数错误。
9. Practice Questions with Full Solutions | 实战练习题及详解
Question 1: The shoe sizes of 10 students are: 4, 5, 3, 4, 6, 4, 5, 4, 3, 5. Find the mode, median, mean, and range.
问题 1：10 名学生的鞋码为：4, 5, 3, 4, 6, 4, 5, 4, 3, 5。求出众数、中位数、平均数和极差。
Solution: median = 4. Mean: sum = 3+3+4+4+4+4+5+5+5+6 = 43, divided by 10 = 4.3. Range = 6 − 3 = 3.
解答：排序数据：3, 3, 4, 4, 4, 4, 5, 5, 5, 6。众数 = 4（出现四次）。中位数：有 10 个值，中位数是第 5 和第 6 个值的平均数。两者都是 4，因此中位数 = 4。平均数：总和 = 3+3+4+4+4+4+5+5+5+6 = 43，除以 10 = 4.3。极差 = 6 − 3 = 3。
Question 2: A fruit seller records the number of oranges sold over 6 days: Mon 15, Tue 22, Wed 18, Thu 25, Fri 30, Sat 40. Draw a line graph and state the trend. On which day was the increase the greatest?
问题 2：某水果商记录了 6 天售出的橙子数量：周一 15，周二 22，周三 18，周四 25，周五 30，周六 40。绘制线形图并描述趋势。哪一天的增幅最大？
Solution: Plot days on x-axis, sales on y-axis. The line rises overall, with a slight dip on Wednesday. Increase from Tue to Wed is negative (−4), so greatest increase is from Fri to Sat (30 to 40, +10). The trend is generally upward, especially towards the weekend.
解答：在 x 轴上标出天数，y 轴上标出销量。线条总体上升，周三略有下降。周二到周三的增幅为负（−4），因此最大增幅是从周五到周六（从 30 到 40，+10）。趋势总体上升，尤其是临近周末时。
10. Drawing Conclusions from Data | 从数据中得出结论
Statistics is not just about calculating numbers; it is about telling a story. In our colour survey, we concluded that red and blue are the clear favourites, so a school uniform supplier might order more items in those colours. For the class heights, we might advise the PE teacher that Class B tends to be taller, which could influence team selections. For the tuck shop, the manager could prepare extra stock on Fridays and investigate why Thursday sales dip.
统计学不仅仅是计算数字；它是在讲述故事。在我们喜爱的颜色调查中，我们得出结论红色和蓝色明显最受欢迎，因此校服供应商可能会多订购这些颜色的商品。对于班级身高，我们可以建议体育老师 B 班学生普遍更高，这可能影响球队选拔。对于小吃店，经理可以在周五多备货，并调查周四销量下降的原因。
Always write a short conclusion in your own words. Use phrases like "The data suggests that...", "On average, ...", "There is a clear trend showing ...". Support your statement with numbers, e.g. "The mean increased from 141.7 cm to 146.8 cm, confirming Class B is taller."
务必用自己的话写一小段结论。使用诸如"数据表明……"，"平均而言，……"，"有一个明显的趋势显示……"之类的表达。用数字支撑你的陈述，例如"平均数从 141.7 cm 增加到 146.8 cm，证实 B 班更高。"
11. Tips for Recording and Organising Your Own Data | 记录和整理你个人数据的技巧
When you conduct your own investigation, always design a clear data collection sheet before you start. For example, if you are surveying favourite fruits, make columns for Fruit, Tally, and Frequency. Use tallies correctly (four vertical lines and a diagonal for five). Collect data without bias: don't ask leading questions like "Isn't mango the best?" Instead, ask neutrally, "Which fruit do you prefer?"
当你自己进行调查时，一定要在开始前设计一张清晰的数据收集表。例如，如果你在调查最喜欢的水果，可以设置水果、划记和频数这几列。正确使用划记法（四条竖线和一条对角线代表五）。无偏见地收集数据：不要问诱导性问题，比如"芒果难道不是最好的吗？"而应中性提问："你更喜欢哪种水果？"
After gathering data, always double-check your counts. An error in tallying can make all your later calculations wrong. Store your raw data safely in case you need to re-check.
收集数据后，务必仔细核对你点的数。划记中的错误可能会让你后面所有的计算都出错。妥善保存原始数据，以备需要重新核对时使用。
12. Summary of Key Skills and Final Practice | 关键技能总结和最后的练习
Let's quickly recap: You should be able to collect categorical and numerical data, make frequency tables, draw and interpret bar charts, pictograms, and line graphs. You must be able to calculate the mean, median, mode, and range from a list or a table. You also need to compare two data sets using these measures and write a simple conclusion.
让我们快速回顾一下：你应该能够收集分类数据和数值数据，制作频数表，绘制并解读条形图、象形图和线形图。你必须能够从列表或表格中计算平均数、中位数、众数和极差。你还需要能够使用这些度量来比较两组数据并写出简单的结论。
d range. Also, suggest one way the library could use this information. (Answer: ordered: 2,3,3,4,5,5,6,7. Mode=3 and 5 (bimodal), median=4.5, mean=4.375, range=5. The library could use the average to plan book purchases.)
来试试这个最后的挑战：图书馆记录了 8 名学生一个月的借书量：3, 5, 2, 7, 4, 5, 3, 6。求平均数、中位数、众数和极差。另外，提出一种图书馆可以利用这一信息的方式。（答案：排序：2,3,3,4,5,5,6,7。众数=3 和 5（双峰），
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