📚 Case Study in Action: IGCSE CCEA Statistics | IGCSE CCEA 统计：案例分析实战演练
This case study follows a realistic school canteen survey and shows how to apply CCEA IGCSE Statistics skills from data collection to interpretation. The same survey of 200 students is used throughout, so you can see how methods connect in one investigation.
本案例研究跟踪一项真实的学校食堂问卷调查，展示如何应用 CCEA IGCSE 统计技能，从数据收集一直到结果解释。整个案例始终使用同一项针对 200 名学生的调查，因此你可以看到各种方法如何在一个调查中相互联系。
1. Setting Up the Case Study | 案例设定
The case study uses a survey of 200 students at Northfield Academy about the school canteen. The aim is to investigate spending habits, satisfaction and factors affecting choice.
本案例以 Northfield Academy 对 200 名学生开展的学校食堂问卷调查为基础，旨在研究消费习惯、满意度以及影响选择的因素。
Variables collected include year group, daily spend, satisfaction score from 1 to 5, whether the student buys a meal deal, and queue time in minutes.
收集的变量包括年级、每日消费、1 至 5 分的满意度评分、是否购买套餐以及排队时间（分钟）。
This case connects descriptive statistics, probability and bivariate analysis to one realistic context, just like CCEA exam questions.
本案例将描述统计、概率和双变量分析与一个真实情境联系起来，与 CCEA 考试题类似。
2. Data Collection and Types | 数据收集与数据类型
Data were collected using a paper questionnaire handed out during form time. Each student answered ten closed questions, which produce clean numerical or categorical data.
数据通过班主任时间发放的纸质问卷收集。每名学生回答了十个封闭式问题，这些问题能产生清晰的数值型或分类型数据。
Year group is categorical nominal data because it is a label, while daily spend and queue time are continuous numerical data.
年级是名义分类数据，因为它是标签；而每日消费和排队时间是连续数值数据。
Satisfaction score is discrete numerical data because it can only take whole-number values from 1 to 5.
满意度评分是离散数值数据，因为它只能取 1 到 5 的整数值。
Identifying the correct data type is essential before choosing a chart or summary statistic.
在选择图表或汇总统计量之前，正确识别数据类型至关重要。
3. Sampling Methods | 抽样方法
Since interviewing all 1,000 students would take too long, the team selected a sample of 200.
由于采访全部 1000 名学生耗时太长，小组选取了 200 人作为样本。
A stratified sample by year group is suitable here: the proportion in each year matches the whole school population.
按年级分层抽样在这里很合适：每个年级的比例与全校总体比例一致。
For example, if Year 11 has 25% of the school, then 50 Year 11 students should be sampled.
例如，如果 11 年级占全校 25%，那么样本中应抽取 50 名 11 年级学生。
A random sample within each year removes bias and gives every student an equal chance of selection.
每个年级内随机抽样可以消除偏差，使每名学生被选中的机会相等。
4. Organising Data with Tables | 用表格整理数据
The raw data were first entered into a frequency table showing daily spend in class intervals of £0–£2, £2–£4, £4–£6 and £6–£8.
原始数据首先录入频数表，显示每日消费的组距为 0–2、2–4、4–6 和 6–8 英镑。
The table includes tally marks, frequency, and cumulative frequency to make calculations easier later.
该表包括记数符号、频数和累计频数，以便后续计算。
| Daily spend (£) | Frequency | Cumulative frequency |
|---|---|---|
| 0 ≤ x < 2 | 30 | 30 |
| 2 ≤ x < 4 | 85 | 115 |
| 4 ≤ x < 6 | 60 | 175 |
| 6 ≤ x < 8 | 25 | 200 |
Class width is constant at £2, which allows accurate histogram and frequency density work.
组距统一为 2 英镑，这有助于准确绘制直方图和计算频数密度。
5. Visualising Data: Charts and Graphs | 数据可视化：图表
A bar chart was used for the categorical variable ‘meal deal purchased? Yes or No’ because the categories are separate.
对于“是否购买套餐？是或否”这一分类变量，使用条形图，因为各类别相互独立。
A histogram displays the continuous daily spend data; frequency density equals frequency divided by class width.
直方图用于显示连续型每日消费数据；频数密度等于频数除以组距。
Frequency density = Frequency ÷ Class width
A pie chart could show the proportion of students in each year group, but the total angles must add to 360°.
饼图可以显示各年级学生比例，但各扇形角度总和必须为 360°。
Choosing the correct chart avoids misleading the reader; line graphs should not be used for unrelated categories.
选择正确的图表可以避免误导读者；折线图不应用于无关联的类别。
6. Averages and Measures of Central Tendency | 平均数与集中趋势
For the satisfaction scores, the mean is found by summing all scores and dividing by 200.
对于满意度评分，平均数是将所有评分相加再除以 200。
Mean = Σx ÷ n
The median satisfaction score was 4, meaning half the students gave 4 or less and half gave 4 or more.
满意度评分的中位数为 4，这意味着半数学生给出 4 分或以下，半数给出 4 分或以上。
The mode was 4 because it occurred most often, with 85 students choosing this score.
众数为 4，因为它出现次数最多，有 85 名学生选择了这个分数。
In this case the mean, median and mode are close, suggesting the satisfaction data are roughly symmetric.
本案例中平均数、中位数和众数接近，表明满意度数据大致对称。
If an extreme value is present, the median is more reliable than the mean because it is not affected by outliers.
如果存在极端值，中位数比平均数更可靠，因为它不受异常值影响。
7. Spread: Range, Quartiles and Standard Deviation | 离散程度：极差、四分位数与标准差
The range of daily spend is highest value minus lowest value, for example £7.80 – £0.50 = £7.30.
每日消费的极差为最大值减最小值，例如 7.80 – 0.50 = 7.30 英镑。
The interquartile range (IQR) is the upper quartile minus the lower quartile: IQR = Q₃ – Q₁.
四分位距（IQR）是上四分位数减去下四分位数：IQR = Q₃ – Q₁。
For spend data, Q₁ was £1.90 and Q₃ was £4.60, so the IQR was £2.70. This shows the spread of the middle 50%.
消费数据中，Q₁ 为 1.90 英镑，Q₃ 为 4.60 英镑，因此四分位距为 2.70 英镑，表示中间 50% 数据的离散程度。
Standard deviation measures how far values are from the mean. If the mean spend is £3.20 with a standard deviation of £1.45, most students spend within £1.75 and £4.65.
标准差衡量数值与平均数的距离。如果平均消费为 3.20 英镑，标准差为 1.45 英镑，则大多数学生的消费在 1.75 至 4.65 英镑之间。
s = √[ Σ(x − x̄)² ÷ (n − 1) ]
A smaller standard deviation means the data are tightly packed around the mean, while a larger one indicates greater variation.
标准差越小，数据越集中在平均数附近；标准差越大，变异程度越大。
8. Probability from Case Data | 案例数据中的概率
Probability can be estimated from relative frequency in the case data.
概率可以通过案例数据中的相对频数来估计。
If 120 out of 200 students bought a meal deal, the estimated probability that a random student buys a meal deal is 120/200 = 0.6.
如果 200 名学生中有 120 人购买了套餐，那么随机抽取一名学生购买套餐的概率估计为 120/200 = 0.6。
If 30 students in Year 10 and 40 in Year 11 bought a meal deal, the probability of selecting a meal-deal buyer from Year 10 is 30/200 = 0.15.
如果 10 年级有 30 人、11 年级有 40 人购买套餐，则从 10 年级中抽到套餐购买者的概率为 30/200 = 0.15。
For independent events, multiply probabilities: the chance that two randomly chosen students both bought a meal deal is 0.6 × 0.6 = 0.36.
对于独立事件，应将概率相乘：随机选出的两名学生都购买套餐的概率为 0.6 × 0.6 = 0.36。
Biased estimates occur when the sample is not representative, so probability statements must always mention the sample basis.
当样本不具代表性时，估计值会有偏差，因此概率表述必须始终说明样本基础。
9. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph was drawn with daily spend on the x-axis and satisfaction score on the y-axis for each student.
以每日消费为 x 轴、满意度评分为 y 轴，为每名学生绘制散点图。
The points showed a weak positive correlation, meaning students who spent more were slightly more satisfied, but the relationship was not strong.
散点显示弱正相关，说明消费更多的学生满意度略高，但关系并不强。
Correlation does not imply causation: higher spend may be linked to buying hot food, not directly to satisfaction.
相关并不意味着因果：较高消费可能与购买热食有关，而非直接导致满意度高。
A line of best fit can be drawn through the points to predict satisfaction for a given spend.
可以通过散点绘制最佳拟合线，用来预测给定消费下的满意度。
An outlier was a student spending £7.60 but rating satisfaction 1; this point should be checked before drawing the line of best fit.
有一个异常点：某学生消费 7.60 英镑但满意度评分为 1；在绘制最佳拟合线之前应核查该点。
10. Time Series and Forecasting | 时间序列与预测
The canteen recorded the number of meal deals sold each day over three weeks to form a time series.
食堂记录了三周内每天售出的套餐数量，形成时间序列。
A moving average smooths out daily fluctuations and reveals the trend.
移动平均可以平滑每日波动并揭示趋势。
If the three-day moving average increased from 70 to 85 to 92, the trend was upward.
如果三天移动平均从 70 升至 85 再升至 92，则趋势是上升的。
The seasonal pattern shows higher sales on Fridays and lower sales on Mondays.
季节性模式显示周五销量较高，周一销量较低。
Forecasts from time series should only be short-term and must state that they assume the trend continues.
时间序列预测应仅限短期，并须说明假设趋势持续。
11. Drawing Conclusions and Limitations | 结论与局限性
The case study found that most students spend £2–£4 daily and satisfaction is centred at 4, but queue time is still an issue.
本案例发现大多数学生每日消费为 2–4 英镑，满意度集中在 4 分，但排队时间仍然是一个问题。
The main limitations are a sample of only 200 students, possible non-response bias, and answers that may not be fully honest.
主要局限包括样本仅 200 名学生、可能存在无应答偏差，以及回答可能不完全真实。
Future work could collect data for a longer period and compare different year groups using a two-way table.
未来研究可以收集更长时间的数据，并利用双向表比较不同年级。
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