📚 Year 10 WJEC Statistics: Case Studies in Practice | 英国 WJEC 十年级统计：案例分析实战演练
Statistics is not just about numbers; it is about telling a story with data. For Year 10 students following the WJEC specification, applying statistical methods to real-world scenarios is the key to deep understanding and exam success. This article walks you through a complete case study from question design to final conclusion, modelling the investigative cycle expected in your course. Along the way, we highlight typical pitfalls and show how each technique links to the assessment objectives.
统计学不仅关乎数字，更是用数据讲故事的过程。对于学习 WJEC 大纲的十年级学生来说，将统计方法应用到真实情境中是深入理解并取得考试成功的关键。本文将通过一个完整的案例，带你经历从问题设计到最终结论的全过程，模拟课程所要求的统计调查周期。同时，我们还将指出常见误区，并说明每项技巧与考核目标的联系。
1. Setting the Scene: A School Canteen Survey | 场景设定：学校食堂调查
Imagine your school wants to improve lunch options. The headteacher asks your statistics class to find out which new meal students would most like to see on the menu. This is a classic WJEC-style investigation where you start with a clear hypothesis: “Students prefer hot meals over cold sandwiches.” You must plan, collect, analyse, and interpret data to either support or reject this hypothesis.
想象一下，你的学校希望改善午餐选择。校长请统计课的同学调查学生最希望在菜单上看到哪种新餐食。这是一个典型的 WJEC 风格调查，你需要从一个明确的假设出发：“相比冷三明治，学生更偏爱热餐。”你必须规划、收集、分析并解读数据，以支持或否定这一假设。
The first step is to define the population and decide how to sample. The population is all students in Years 7–11, but it may be impractical to ask everyone. A stratified sample by year group could work, ensuring proportional representation. Alternatively, a simple random sample using a random number generator on the school roll would be easier and is worth discussing in your write-up.
第一步是界定总体并决定抽样方法。总体是所有 7 至 11 年级的学生，但询问每一个人可能不现实。按年级分层抽样可以确保比例代表性。或者，利用随机数生成器从全校名单中抽取简单随机样本更简便，也值得在报告中加以讨论。
2. Designing an Unbiased Questionnaire | 设计无偏见的问卷
Your data collection tool must avoid leading questions. Instead of “Don’t you think hot meals are better?”, ask “Which type of meal do you prefer for lunch?” with options: Hot meal, Cold sandwich, Salad, Other. A pilot study with five students can test whether questions are clear and response categories cover all possibilities. WJEC examiners love seeing this in a case study.
数据收集工具必须避免诱导性问题。不要问“你不觉得热餐更好吗？”，而应问“你更喜欢哪种类型的午餐？”，选项为：热餐、冷三明治、沙拉、其他。通过 5 名学生进行预调查，可以检验问题是否清晰、回答选项是否包含所有可能。WJEC 考官非常喜欢在案例分析中看到这一点。
We also need to decide on a response format. A closed question with tick boxes gives categorical data, which is easy to summarise in frequency tables and charts. If we ask “How many times per week would you choose a hot meal?”, that yields discrete numerical data, allowing us to calculate means and ranges. Combining both types adds depth to the analysis.
我们还需要决定回答格式。带勾选框的封闭式问题可以产生分类数据，便于用频数表和图表进行汇总。如果我们问“你每周会选择多少次热餐？”，则会产生离散数值数据，从而可以计算平均值和极差。结合两种数据类型可以增加分析的深度。
3. Sampling Methods in Action | 抽样方法的实际应用
For our canteen survey, let’s take a stratified sample of 60 students across five year groups. We find the total number in each year from the school office: Year 7 (200), Year 8 (180), Year 9 (160), Year 10 (150), Year 11 (110). Total = 800. The number sampled from Year 7 = (200 ÷ 800) × 60 = 15. We calculate similarly for others: Year 8 (13.5 → 14), Year 9 (12), Year 10 (11.25 → 11), Year 11 (8.25 → 8). Rounding ensures whole numbers, but note that the sum should remain 60—adjust if necessary.
在我们的食堂调查中，我们从五个年级中抽取 60 名学生的分层样本。我们从学校办公室找到各年级的总人数：7 年级（200），8 年级（180），9 年级（160），10 年级（150），11 年级（110）。总计 800。从 7 年级抽取的人数 = (200 ÷ 800) × 60 = 15。依此类推：8 年级（13.5 → 14），9 年级（12），10 年级（11.25 → 11），11 年级（8.25 → 8）。取整后确保总数为 60，必要时进行调整。
Stratified sampling reduces bias and guarantees that each year group is fairly represented. An alternative systematic sample (e.g., every 10th student entering the canteen) might be quicker but could over-represent the years that have lunch at that time. Justify your choice in the report.
分层抽样可以减少偏差，并保证每个年级都能公平地被代表。备选的系统抽样（例如，每第 10 个进入食堂的学生）可能更快捷，但可能会过多地代表在那个时间段吃午餐的年级。在报告中需要为你的选择提供理由。
4. Collecting and Recording Raw Data | 收集并记录原始数据
Data is collected during Tuesday lunchtime to avoid Monday or Friday anomalies. Each selected student is given a paper slip with the question and tick boxes. All responses are anonymous. The raw data looks like: Hot, Cold, Hot, Salad, Hot, Other, Hot, Cold, … We record 60 responses in a tally chart. For the numerical question “How many times per week?”, we obtain numbers like 5, 3, 0, 4, 2, …
数据收集安排在周二午餐时间，以避免周一或周五的异常情况。每位被选中的学生都会收到一张写有问题和勾选框的纸片。所有回答都是匿名的。原始数据看起来像是：热餐、冷餐、热餐、沙拉、热餐、其他、热餐、冷餐……我们将 60 个回答记录在划记表中。对于数值型问题“每周多少次？”，我们得到诸如 5、3、0、4、2 等数字。
Organising data at this stage is crucial. A tally chart for the meal preference includes columns for Tally, Frequency, and perhaps Relative Frequency. The numerical data can be ordered: 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5,… ready for a stem-and-leaf diagram or a frequency table with groups.
在这个阶段整理数据至关重要。餐食偏好的划记表包括画记、频数，或许还有相对频数等列。数值数据可以排序：0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5……为茎叶图或分组频数表做好准备。
5. Presenting Categorical Data with Charts | 用图表展示分类数据
From the tally, suppose the frequencies are: Hot meal = 28, Cold sandwich = 15, Salad = 10, Other = 7. We can construct a bar chart where each bar height represents frequency. The bars should be separate, with equal width, and labelled clearly. A pie chart is also appropriate: the angle for Hot = (28/60) × 360° = 168°. Cold = 90°, Salad = 60°, Other = 42°. Always check the angles sum to 360°.
根据划记结果，假设频数为：热餐 = 28，冷三明治 = 15，沙拉 = 10，其他 = 7。我们可以绘制条形图，每根条形的高度代表频数。条形之间应分开、宽度相等，并标注清楚。饼图同样适用：热餐对应的圆心角 = (28/60) × 360° = 168°；冷餐 90°，沙拉 60°，其他 42°。务必检查角度之和是否为 360°。
Which chart is better? A bar chart makes it easier to compare actual frequencies, while a pie chart emphasises the proportion of the whole. WJEC often asks you to justify your choice. For this case, because we want to highlight that nearly half the sample chose hot meals, a pie chart combined with percentages could be very effective.
哪种图更好？条形图更容易比较实际频数，而饼图则强调各部分在整体中的占比。WJEC 经常要求你说明选择的理由。在本案例中，由于我们希望突出近一半的样本选择了热餐，带有百分比的饼图可能非常有效。
6. Summarising Numerical Data: Averages and Spread | 汇总数值数据：平均数与离散程度
For the “times per week” data, we calculate the mean: sum of all values divided by 60. Let’s say the total is 180, so the mean = 3.0 times per week. The mode is 5 (most frequent). The median is the 30.5th value in order: between 3 and 3, so median = 3. The range = max – min = 5 – 0 = 5. The interquartile range (IQR) can be found by locating Q1 (15.25th → 2) and Q3 (45.75th → 4), giving IQR = 4 – 2 = 2.
对于“每周次数”数据，我们计算均值：所有数值之和除以 60。假设总和为 180，则均值 = 每周 3.0 次。众数为 5（出现频率最高）。中位数是排序后第 30.5 个值：位于 3 和 3 之间，所以中位数 = 3。极差 = 最大值 – 最小值 = 5 – 0 = 5。通过定位 Q1（第 15.25 个 → 2）和 Q3（第 45.75 个 → 4），可求得四分位距 IQR = 4 – 2 = 2。
These statistics tell us that the typical student would choose a hot meal about 3 times a week, but there is variation. The relatively small IQR (2) compared to the range (5) suggests that the middle 50% of students are quite consistent, while a few never choose hot meals and a few always do.
这些统计量告诉我们，典型的学生每周大约会选择 3 次热餐，但存在个体差异。与极差（5）相比，相对较小的 IQR（2）意味着中间 50% 的学生选择频率相当一致，而少数学生从不选热餐，少数总是选热餐。
7. Comparing Subgroups with Dual Bar Charts and Box Plots | 通过复式条形图和箱线图比较子群体
We can break down the meal preference by gender. Suppose among 30 boys, 18 chose hot meals; among 30 girls, 10 chose hot meals. A dual bar chart can display these side by side, making it visually clear that boys have a stronger preference for hot meals. For the numerical data, side-by-side box plots are excellent: they show the median, quartiles, and range for boys’ and girls’ weekly hot-meal choices at a glance.
我们可以按性别细分餐食偏好。假设在 30 名男生中，18 名选择了热餐；在 30 名女生中，10 名选择了热餐。复式条形图可以将这些数据并列显示，让人直观地看到男生对热餐的偏好更强。对于数值数据，并排箱线图则非常出色：它们一目了然地展示出男生和女生每周选择热餐次数的中位数、四分位数和极差。
In our box plot comparison, boys might have a higher median (4) and girls a lower median (2). The overlapping interquartile ranges still indicate some similarity, but the shift in median supports a possible difference. This is the kind of comparative analysis WJEC expects in the “interpret” part of the statistical enquiry cycle.
在我们的箱线图比较中，男生的中位数可能更高（4），女生的中位数较低（2）。虽然重叠的四分位距仍表明一定程度的相似性，但中位数的偏移支持了可能存在差异的推断。这正是 WJEC 在统计调查周期中“解读”部分所期望的比较分析。
8. Introducing Probability Concepts from the Data | 从数据中引入概率概念
From our sample, we can estimate probabilities. The relative frequency of a student preferring hot meals is 28/60 ≈ 0.467. If we select one student at random from the sample, P(Hot) = 0.467. This is an experimental probability, which can be compared to a theoretical probability if we had one. We can also ask: “If two students are chosen at random, what is the probability both prefer cold sandwiches?” Assuming independence and using P(Cold) = 15/60 = 0.25, then P(Both Cold) ≈ 0.25 × 0.25 = 0.0625.
从样本中，我们可以估计概率。学生偏好热餐的相对频数为 28/60 ≈ 0.467。如果从样本中随机选择一名学生，P(热餐) = 0.467。这是一个实验概率，如果有理论概率，可以将两者进行比较。我们还可以问：“如果随机选择两名学生，两人都喜欢冷三明治的概率是多少？”假设独立性，利用 P(冷餐) = 15/60 = 0.25，则 P(两人均选冷餐) ≈ 0.25 × 0.25 = 0.0625。
WJEC Year 10 questions often link back to expected frequency: in a year group of 200 students, how many would we expect to choose salad? Expected = (10/60) × 200 = 33.3, roughly 33 students. Such predictions come with the caveat that they are based on a sample and subject to sampling variability.
WJEC 十年级试题常常联系到期望频数：在一个 200 人的年级中，我们预期有多少人会选择沙拉？期望值 = (10/60) × 200 = 33.3，约 33 名学生。这种预测需要附带说明，即它们基于样本，并受抽样变异的影响。
9. Drawing Conclusions and Evaluating the Investigation | 得出结论并评估调查过程
Our hypothesis was “Students prefer hot meals over cold sandwiches.” The data shows 28 out of 60 chose hot meals, compared with 15 for cold. The mean weekly frequency of hot meal choice was 3.0. This supports the hypothesis but does not prove it absolutely—28 is less than half (30) of the sample, so it is not an overwhelming majority. The margin is enough to suggest a preference, but we must discuss sampling error and bias.
我们的假设是“相比冷三明治，学生更偏爱热餐”。数据显示 60 人中有 28 人选择了热餐，而冷餐仅 15 人。每周选择热餐的平均次数为 3.0。这支持了假设，但并未绝对证明——28 不足样本数（30）的一半，因此并非压倒性多数。这一差距足以表明一种偏好，但我们必须讨论抽样误差和偏差。
Evaluating the study: the sample size (60) is adequate for a school of 800, but could be larger. Stratified sampling ensured representation, but non-response might have occurred if some selected students were absent or refused. The questionnaire was tested with a pilot, reducing response error. However, the choice of “Other” could hide strong opinions like “pasta” that might have influenced results.
评估本次调查：对于 800 人的学校，样本量（60）是足够的，但还可以更大。分层抽样确保了代表性，但如果某些被选中的学生缺席或拒绝回答，则可能产生无回答误差。问卷经过预调查测试，减少了回答误差。然而，“其他”选项可能隐藏了像“意面”这样可能影响结果的强烈意见。
10. Common Mistakes and Examiner Tips | 常见错误与考官提示
One frequent mistake is constructing a bar chart for continuous data – bar charts are for categorical or discrete data, while histograms are used for continuous grouped data. Another is calculating the mean from a frequency table incorrectly by forgetting to multiply each value by its frequency before summing. Also, when finding the median from a stem-and-leaf diagram, students often forget to count to the (n+1)/2th position.
一个常见错误是为连续数据绘制条形图——条形图适用于分类或离散数据，而直方图则用于连续的分组数据。另一个错误是在频数表中计算均值时，忘记先将每个值与其频数相乘再求和。此外，在茎叶图中求中位数时，学生常常忘记要数到第 (n+1)/2 个位置。
When interpreting box plots, do not just state the median; compare the interquartile range and the overall spread. For probability, always express answers as fractions or decimals between 0 and 1, and simplify where possible. Finally, in the evaluation section, always refer back to the original hypothesis and mention how the study could be improved—WJEC awards marks for critical reflection.
解读箱线图时，不要只陈述中位数；还要比较四分位距和整体散布情况。处理概率时，始终将答案表示为 0 到 1 之间的分数或小数，并尽可能化简。最后，在评估部分，一定要回扣原始假设，并提及研究可以如何改进——WJEC 对批判性反思给予评分。
11. Putting It All Together: The Statistical Enquiry Cycle | 融会贯通：统计调查周期
The WJEC specification emphasises a cyclical process: Pose a question or hypothesis, Plan data collection, Collect data, Process and represent data, Interpret and discuss, and finally Evaluate. Our canteen case study followed this cycle exactly. Whenever you tackle a case study in class or in the exam, mentally tick off each stage to ensure you haven’t missed an opportunity to demonstrate statistical thinking.
WJEC 大纲强调一个循环过程：提出问题或假设、规划数据收集、收集数据、处理并展示数据、解读并讨论，最后进行评估。我们的食堂案例研究正是严格遵循了这一周期。每当你在课堂或考试中处理案例分析时，可以心中逐一勾销每个阶段，确保没有遗漏展示统计思维的机会。
Remember that real data is messy. Outliers may appear, patterns may be weak, and conclusions tentative. That is perfectly fine. The skill WJEC wants to see is your ability to work systematically, choose appropriate techniques, and comment insightfully on the findings, not to discover dramatic results.
请记住，真实数据往往是杂乱的。可能出现异常值，模式可能微弱，结论也可能是尝试性的。这完全没有问题。WJEC 希望看到的技能，是你能够系统地进行工作、选择合适的技巧，并对发现给出深刻的评论，而非一定要得出戏剧性的结果。
12. Practice Case Study: Sports Club Membership | 实践案例：体育俱乐部会员资格
Now try your own. Investigate the question: “Is there an association between year group and the type of sport chosen for after‑school clubs?” Design a questionnaire, sample 50 students across Years 7, 9 and 11, record data in a two‑way table, draw a compound bar chart, and calculate row percentages. Use a comparative pie chart or stacked bar chart to look for patterns. Write up your findings following the cycle above.
现在请自己尝试一个案例。调查问题：“年级与课后俱乐部选择的运动类型之间是否存在关联？”设计一份问卷，抽取 7、9、11 年级共 50 名学生作为样本，用双向表记录数据，绘制复合条形图，并计算行百分比。利用比较饼图或堆叠条形图寻找模式。按照上述周期撰写你的调查结果。
This exercise will consolidate all the skills covered: sampling, data representation, probability, average and spread calculations, and critical evaluation. Share your work with a classmate for peer review—another tip from experienced WJEC teachers!
这个练习将巩固所有已学技能：抽样、数据展示、概率、平均数和离散程度的计算，以及批判性评估。把你的作业与同学分享，进行同伴互评——这也是经验丰富的 WJEC 教师给出的另一个建议！
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