📚 Year 8 WJEC Statistics: Case Study Practical Exercises | Year 8 WJEC 统计：案例分析实战演练
In this article, we will work through a complete statistical investigation from start to finish. This case study is designed for Year 8 students following the WJEC curriculum. You will see how to plan a survey, collect data, organise it into tables and charts, calculate averages and spread, and finally interpret your findings. Learning by doing is the best way to master statistics.
在本文中，我们将从头到尾完成一个完整的统计调查案例。这个案例专为学习 WJEC 课程的 Year 8 学生设计。你将看到如何规划调查、收集数据、将数据整理成表格和图表、计算平均数和离散程度，并最终解释你的发现。边做边学是掌握统计学的最好方法。
1. Setting the Scene: The Research Question | 设定场景：研究问题
Every statistical investigation begins with a clear question. For our case study, a Year 8 student named Mia wants to find out: ‘How many hours per week do Year 8 students spend reading for pleasure?’ She is curious whether boys and girls have different reading habits. This question is specific, measurable, and relevant to her school community.
每项统计调查都从一个明确的问题开始。在本案例研究中，一位名叫 Mia 的 Year 8 学生想要弄清楚：“Year 8 学生每周花多少小时进行课外阅读？”她很好奇男生和女生的阅读习惯是否不同。这个问题具体、可衡量，并且与她的学校社区相关。
Mia decides to conduct a survey using a simple questionnaire. She plans to ask a random sample of 30 students from her year group: 15 boys and 15 girls. This is a stratified sampling approach to ensure both genders are fairly represented.
Mia 决定用一份简单的问卷进行调查。她计划从年级中随机抽取 30 名学生：15 名男生和 15 名女生。这是一种分层抽样方法，确保男女都有公平的代表性。
2. Designing the Data Collection Sheet | 设计数据收集表
A well-designed data collection sheet makes recording responses easy and reduces errors. Mia prepares a simple tally chart with two columns: ‘Gender’ and ‘Hours per week (to the nearest hour)’. She includes a space for tally marks and a frequency count for each interval.
一个设计良好的数据收集表能让记录回答变得简单并减少错误。Mia 准备了一个简单的计数表，包含两栏：“性别”和“每周小时数（四舍五入到最接近的小时）”。她为每个区间留出了画记号和记录频数的空间。
Before collecting data, Mia defines her response categories. Since reading hours can vary from 0 to more than 10, she groups the data into class intervals: 0–1 hours, 2–3 hours, 4–5 hours, 6–7 hours, and 8 or more hours. This will make the data easier to handle and display.
在收集数据之前，Mia 定义了回答类别。由于阅读时间可能从 0 到 10 小时以上不等，她将数据分组为区间：0–1 小时、2–3 小时、4–5 小时、6–7 小时和 8 小时及以上。这将使数据更易于处理和展示。
3. Collecting the Raw Data | 收集原始数据
Mia approaches 30 students during break time and records their answers anonymously. Here are the raw data she collects, with hours rounded to the nearest whole number:
Mia 在课间休息时间询问了 30 名学生，并匿名记录了他们的回答。以下是她收集的原始数据，小时数四舍五入为最接近的整数：
| Boys’ hours: 2, 0, 1, 3, 5, 1, 0, 2, 4, 6, 1, 2, 0, 3, 7 |
| Girls’ hours: 4, 2, 5, 3, 6, 4, 5, 7, 3, 8, 4, 6, 5, 2, 4 |
Notice that the raw data are just lists of numbers. Without organising them, it is hard to see any patterns. The next step is to sort and group these numbers into frequency tables.
注意，原始数据仅仅是一串数字。不加以整理，很难看出任何模式。下一步是将这些数字分类并分组成频数表。
4. Creating Tally Charts and Frequency Tables | 创建计数表和频数表
Mia uses the class intervals she defined earlier to build a tally chart for boys and another for girls. She goes through each response and places a tally mark in the correct interval. After every four tally marks, the fifth mark crosses the previous four to make a gate of five, which speeds up counting.
Mia 用她之前定义的区间为男生和女生分别制作计数表。她逐一查看每个回答，并在正确的区间内画记号。每四个记号之后，第五个记号横穿前四个形成“五”字栅栏，这能加快计数速度。
Here is the completed frequency table for boys:
以下是完成的男生频数表：
| Hours | Tally | Frequency |
|---|---|---|
| 0–1 | |||| || | 7 |
| 2–3 | |||| | 4 |
| 4–5 | || | 2 |
| 6–7 | || | 2 |
| 8+ | | | 0 |
And the frequency table for girls:
以下是女生的频数表：
| Hours | Tally | Frequency |
|---|---|---|
| 0–1 | | | 0 |
| 2–3 | |||| | 4 |
| 4–5 | |||| || | 7 |
| 6–7 | ||| | 3 |
| 8+ | | | 1 |
Immediately, a difference is visible: more boys are concentrated in the lower intervals, while girls tend to be in higher intervals. Now we can move on to visual representations.
差异立刻显现出来：更多男生集中在较低的区间，而女生则倾向于较高的区间。现在我们可以继续进行可视化表示。
5. Drawing Comparative Bar Charts | 绘制对比条形图
A comparative bar chart is an excellent way to display two sets of data side by side. Mia decides to draw a dual bar chart with the class intervals on the horizontal axis and frequency on the vertical axis. She uses one colour for boys and another for girls, and includes a clear key.
对比条形图是并排展示两组数据的绝佳方式。Mia 决定绘制双条形图，横轴为区间，纵轴为频数。她用一种颜色代表男生，另一种颜色代表女生，并附上清晰的图例。
To construct the chart accurately, she draws axes with a ruler, labels them, and chooses a scale where 1 cm represents 1 student on the frequency axis. Bars for the same interval are drawn touching each other but with a small gap between different intervals. The title of the chart is ‘Weekly Reading Hours of Year 8 Boys and Girls’.
为了准确绘制图表，她用尺子画轴，标注标签，并选择比例尺，在频数轴上 1 厘米代表 1 名学生。同一区间的条形紧挨着绘制，但不同区间之间留有微小间隙。图表的标题为“Year 8 男女每周阅读小时数”。
From the bar chart, it is easy to see that the modal class interval for boys is 0–1 hours, while for girls it is 4–5 hours. The shape of the distributions suggests that girls read more hours per week than boys in this sample.
从条形图中很容易看出，男生的众数区间是 0–1 小时，而女生的是 4–5 小时。分布的形状表明，在这个样本中女生每周阅读的时间比男生长。
6. Pie Charts for Proportional Comparison | 用饼图进行比例比较
A pie chart shows proportions of a whole. Mia wants to create separate pie charts for boys and girls to compare the relative frequencies. To draw a pie chart, she needs to calculate the angle for each sector using the formula:
饼图显示各部分占整体的比例。Mia 想分别为男生和女生制作饼图，以比较相对频数。要绘制饼图，她需要用公式计算每个扇区的角度：
Sector angle = (Frequency ÷ Total frequency) × 360°
For boys, total frequency is 15. The sector angle for 0–1 hours: (7 ÷ 15) × 360° = 168°. For 2–3 hours: (4 ÷ 15) × 360° = 96°. For 4–5 hours: (2 ÷ 15) × 360° = 48°. For 6–7 hours: (2 ÷ 15) × 360° = 48°. The 8+ interval has zero frequency, so no sector is drawn.
对于男生，总频数为 15。0–1 小时的扇区角度：(7 ÷ 15) × 360° = 168°。2–3 小时：(4 ÷ 15) × 360° = 96°。4–5 小时：(2 ÷ 15) × 360° = 48°。6–7 小时：(2 ÷ 15) × 360° = 48°。8+ 区间的频数为零，因此不绘制扇区。
For girls, total frequency is also 15. The angles are: 2–3 hours: (4 ÷ 15) × 360° = 96°; 4–5 hours: (7 ÷ 15) × 360° = 168°; 6–7 hours: (3 ÷ 15) × 360° = 72°; 8+ hours: (1 ÷ 15) × 360° = 24°. Comparing the two pie charts clearly shows the shift in reading habits between genders.
对于女生，总频数也是 15。角度为：2–3 小时：(4 ÷ 15) × 360° = 96°；4–5 小时：(7 ÷ 15) × 360° = 168°；6–7 小时：(3 ÷ 15) × 360° = 72°；8+ 小时：(1 ÷ 15) × 360° = 24°。比较两个饼图可以清楚地看出不同性别之间阅读习惯的差异。
7. Calculating the Mode and Modal Class | 计算众数和众数区间
The mode is the value or category that occurs most often. From the raw data, Mia can find the mode for each gender. For boys, the individual hours reported were: 2, 0, 1, 3, 5, 1, 0, 2, 4, 6, 1, 2, 0, 3, 7. The value 1 appears three times, as does 0 and 2. So there are multiple modes: 0, 1, and 2 hours. In grouped data, we refer to the modal class interval, which for boys is 0–1 hours with a frequency of 7.
众数是出现次数最多的数值或类别。Mia 可以从原始数据中找出每种性别的众数。男生的具体小时数为：2, 0, 1, 3, 5, 1, 0, 2, 4, 6, 1, 2, 0, 3, 7。数值 1 出现了三次，0 和 2 也是如此。因此存在多个众数：0、1 和 2 小时。在分组数据中，我们指的是众数区间，对于男生，众数区间为 0–1 小时，频数为 7。
For girls, the raw data: 4, 2, 5, 3, 6, 4, 5, 7, 3, 8, 4, 6, 5, 2, 4. The value 4 appears four times, making it the mode. The modal class interval is 4–5 hours. Modes give a quick snapshot of the most typical reading behaviour in each group.
女生的原始数据：4, 2, 5, 3, 6, 4, 5, 7, 3, 8, 4, 6, 5, 2, 4。数值 4 出现了四次，是众数。众数区间是 4–5 小时。众数可以快速反映每组中最典型的阅读行为。
8. Finding the Median and Quartiles | 求中位数和四分位数
The median is the middle value when data are ordered. For boys, ordered data: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7. With 15 values, the median is the 8th value: 2 hours. The lower quartile (Q₁) is the median of the first 7 values: the 4th value is 1 hour. The upper quartile (Q₃) is the median of the last 7 values: the 12th value is 4 hours.
中位数是数据排序后位于中间的数值。男生排序后的数据：0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7。共有 15 个值，中位数是第 8 个值：2 小时。下四分位数（Q₁）是前 7 个值的中位数：第 4 个值为 1 小时。上四分位数（Q₃）是后 7 个值的中位数：第 12 个值为 4 小时。
For girls, ordered data: 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8. The median is the 8th value: 4 hours. Q₁ is the 4th value: 3 hours. Q₃ is the 12th value: 6 hours. Medians confirm that girls typically read 2 hours more per week than boys in this sample.
女生排序后的数据：2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8。中位数是第 8 个值：4 小时。Q₁ 是第 4 个值：3 小时。Q₃ 是第 12 个值：6 小时。中位数证实，在这个样本中女生通常每周比男生多阅读 2 小时。
9. Calculating the Mean | 计算平均数
The mean gives the arithmetic average. For boys, sum of hours = 0+0+0+1+1+1+2+2+2+3+3+4+5+6+7 = 37. Mean = 37 ÷ 15 ≈ 2.47 hours. For girls, sum = 2+2+3+3+4+4+4+4+5+5+5+6+6+7+8 = 68. Mean = 68 ÷ 15 ≈ 4.53 hours.
平均数给出算术平均值。男生的总小时数 = 0+0+0+1+1+1+2+2+2+3+3+4+5+6+7 = 37。平均数 = 37 ÷ 15 ≈ 2.47 小时。女生的总和 = 2+2+3+3+4+4+4+4+5+5+5+6+6+7+8 = 68。平均数 = 68 ÷ 15 ≈ 4.53 小时。
The mean is influenced by extreme values. For boys, the 7-hour reading week pulls the mean up slightly compared to the median. For girls, the distribution is more symmetric, so the mean and median are closer. The mean confirms that girls read on average over 2 hours more per week than boys.
平均数受极端值影响。对于男生，每周阅读 7 小时的值使平均数略高于中位数。对于女生，分布更对称，因此平均数与中位数更接近。平均数证实女生平均每周比男生多阅读超过 2 小时。
10. Measuring Spread with the Range and Interquartile Range | 用极差和四分位距衡量离散程度
Spread describes how spread out the data are. The range is the difference between the maximum and minimum values. For boys: range = 7 – 0 = 7 hours. For girls: range = 8 – 2 = 6 hours. These ranges are quite similar, but range alone can be misleading because of outliers.
离散程度描述数据的分散情况。极差是最大值与最小值之差。男生：极差 = 7 – 0 = 7 小时。女生：极差 = 8 – 2 = 6 小时。这些极差非常相似，但仅看极差可能会因异常值而产生误导。
The interquartile range (IQR) is a more robust measure of spread, focusing on the middle 50% of data. IQR = Q₃ – Q₁. For boys: IQR = 4 – 1 = 3 hours. For girls: IQR = 6 – 3 = 3 hours. The identical IQR suggests that although girls read more, the consistency of reading hours in the middle half is similar for both genders.
四分位距（IQR）是更稳健的离散程度度量，关注中间 50% 的数据。IQR = Q₃ – Q₁。男生：IQR = 4 – 1 = 3 小时。女生：IQR = 6 – 3 = 3 小时。相同的 IQR 表明，虽然女生阅读更多，但两组中间一半学生的阅读时间一致性相似。
11. Drawing a Box Plot for Each Gender | 为每种性别绘制箱线图
A box plot (box-and-whisker diagram) uses the five-number summary: minimum, Q₁, median, Q₃, and maximum. For boys: Min = 0, Q₁ = 1, Median = 2, Q₃ = 4, Max = 7. For girls: Min = 2, Q₁ = 3, Median = 4, Q₃ = 6, Max = 8.
箱线图（盒须图）使用五数概括：最小值、Q₁、中位数、Q₃ 和最大值。男生：最小值 = 0，Q₁ = 1，中位数 = 2，Q₃ = 4，最大值 = 7。女生：最小值 = 2，Q₁ = 3，中位数 = 4，Q₃ = 6，最大值 = 8。
Mia draws a horizontal scale from 0 to 9 hours and places the boxes side by side. The box spans from Q₁ to Q₃ with a line inside at the median. Whiskers extend to the minimum and maximum. The girls’ box is shifted noticeably to the right, showing higher reading hours overall. The overlap of the boxes is small, which visually suggests a real difference between the groups.
Mia 绘制了从 0 到 9 小时的水平刻度，并将箱线图并排放置。箱体从 Q₁ 延伸到 Q₃，箱内一条线标示中位数。须线延伸到最小值和最大值。女生的箱体明显右移，显示总体阅读时间更长。箱体的重叠很小，这在视觉上表明两组之间存在真实差异。
12. Interpreting Findings and Drawing Conclusions | 解释发现并得出结论
Now Mia reviews all her findings. Both the mean and median reading hours for girls (4.53 and 4) are clearly higher than for boys (2.47 and 2). The modal class interval for boys is 0–1 hours, while for girls it is 4–5 hours. The pie charts and bar charts visually reinforce that boys are more likely to read very little, whereas girls are more likely to read for several hours a week.
现在 Mia 回顾所有发现。女生的平均阅读小时数（4.53 和 4）无论是平均数还是中位数都明显高于男生（2.47 和 2）。男生的众数区间为 0–1 小时，而女生的众数区间为 4–5 小时。饼图和条形图在视觉上加强了这样一种印象：男生更可能阅读很少，而女生更可能每周阅读数小时。
Mia writes her conclusion carefully, stating that in her sample of 30 Year 8 students, girls tended to read for pleasure significantly more than boys. She acknowledges the limitation that this is a small sample from one school, so the results may not apply to all Year 8 students. She suggests further research with a bigger sample and perhaps investigating reasons for the difference.
Mia 仔细地写下结论，说明在她的 30 名 Year 8 学生样本中，女生的课外阅读时间明显多于男生。她承认这个样本较小且仅来自一所学校，因此结果可能不适用于所有 Year 8 学生。她建议用更大的样本做进一步研究，并且可以调查差异背后的原因。
This case study has taken you through every stage of the statistical enquiry cycle: posing a question, collecting data, processing and presenting data, and interpreting results. Practising such case studies is the best way to build confidence for your WJEC statistics assessments.
这个案例研究带你经历了统计探究周期的每个阶段：提出问题、收集数据、处理和呈现数据，以及解释结果。练习这样的案例分析是为你的 WJEC 统计评估建立信心的最佳方式。
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