📚 Year 10 AQA Statistics: Case Study Practical Exercises | Year 10 AQA 统计：案例分析实战演练
In this article, we will work through a comprehensive case study that mirrors real-world data handling and statistical analysis tasks covered in the Year 10 AQA GCSE Statistics syllabus. We will design a survey, collect and present data, calculate summary statistics, interpret graphs, and draw conclusions – all within the context of a school canteen investigation.
本文将深入一个综合案例，模拟真实世界的数据处理和统计分析任务，涵盖 Year 10 AQA GCSE 统计学课程大纲要求。我们将在学校餐厅调查的背景下，设计问卷、收集并展示数据、计算汇总统计量、解读图表并得出结论。
1. Case Study Background: School Canteen Survey | 案例背景：学校餐厅调查
A school wants to improve its canteen services. The student council decides to conduct a statistical investigation to find out students' food preferences, how much they spend weekly, and their overall satisfaction. This will involve collecting data from a sample of students, analysing the results, and presenting recommendations to the school management.
一所学校希望改善餐厅服务。学生会决定开展一项统计调查，了解学生的食物偏好、每周花费以及总体满意度。这需要从学生样本中收集数据，分析结果，并向学校管理层提出建议。
2. Designing a Questionnaire and Sampling Method | 设计问卷与抽样方法
To gather reliable data, we must design clear and unbiased questions. For example, 'What is your favourite lunch option? (Pizza / Pasta / Sandwich / Salad / Hot meal / Other)' is a closed question that is easy to analyse. We must also avoid leading questions, such as 'Don't you agree that pizza is the best?'.
为了获取可靠数据，我们必须设计清晰且无偏见的问题。例如，“您最喜欢的午餐选择是什么？（比萨饼 / 意大利面 / 三明治 / 沙拉 / 热餐 / 其他）”是一个封闭式问题，易于分析。我们还必须避免引导性问题，例如“您难道不同意比萨是最佳选择吗？”。
We need to select a sample that represents the whole student population. A stratified sample by year group ensures each year group is fairly represented. If the school has 1,000 students, with 250 in Year 10, we might sample proportionally: 10% from each year group, giving 25 Year 10 students in the sample. However, for this case study we will work with a smaller random sample of 30 students drawn from the stratified frame to keep calculations manageable.
我们需要选择一个能够代表全体学生群体的样本。按年级分层抽样可以确保每个年级都有公平的代表。如果学校有1000名学生，其中 Year 10 有250名，我们可以按比例抽样：从每个年级抽取10%，这样样本中 Year 10 学生就有25名。不过，在本案例中，我们将使用从分层框架中随机抽取的30名学生的小样本，以便于计算。
3. Collecting and Organising Data | 收集与整理数据
The sample of 30 students was surveyed. The table below shows a selection of the collected data, including student ID, year group, favourite food, weekly canteen spend (to the nearest 10p), and satisfaction rating on a scale from 1 (very dissatisfied) to 5 (very satisfied).
我们对 30 名学生样本进行了调查。下表显示了部分收集到的数据，包括学生编号、年级、最喜欢的食物、每周餐厅花费（精确到 10 便士）以及满意度评分（1 表示非常不满意，5 表示非常满意）。
| ID | Year | Favourite Food | Weekly Spend (£) | Satisfaction (1-5) |
|---|---|---|---|---|
| 1 | 9 | Pizza | 6.80 | 4 |
| 2 | 10 | Sandwich | 7.50 | 5 |
| 3 | 11 | Pasta | 9.20 | 3 |
| 4 | 9 | Hot meal | 11.00 | 2 |
| 5 | 10 | Pizza | 7.00 | 4 |
| ... (remaining 25 students follow same format) | ||||
This format allows us to handle two types of variable: categorical (favourite food, year group) and numerical (spend, satisfaction). We will use this data for all subsequent analysis.
这种格式使我们可以处理两类变量：分类变量（最喜欢的食物、年级）和数值变量（花费、满意度）。我们将使用这些数据进行所有后续分析。
4. Calculating Measures of Central Tendency | 计算集中趋势指标
For the weekly spend (£), the sum of all 30 values was found to be £246.00. We compute the mean using the formula:
对于每周花费（£），30 个数值的总和为 246.00 英镑。我们使用以下公式计算均值：
Mean x̄ = ∑xᵢ / n
Thus, mean spend = 246 ÷ 30 = £8.20. This represents the average amount a student spends in the canteen each week.
因此，平均花费 = 246 ÷ 30 = 8.20 英镑。这表示学生每周在餐厅的平均消费金额。
The median is the average of the 15th and 16th values. In our data set, the 15th value is £7.90 and the 16th is £8.10, so the median is (£7.90 + £8.10) / 2 = £8.00. The mode is the most frequently occurring spend, which is £7.50 (occurring 4 times).
为了找出中位数，我们将 30 个花费值按升序排列。中位数是第 15 个和第 16 个值的平均数。在我们的数据集中，第 15 个值是 7.90 英镑，第 16 个值是 8.10 英镑，因此中位数为 (7.90 + 8.10) / 2 = 8.00 英镑。众数是出现频率最高的花费金额，为 7.50 英镑（出现 4 次）。
For satisfaction ratings, the mean is 3.3, the median is 3, and the mode is 4. This suggests that while the most common rating is quite positive, the distribution is slightly skewed towards lower scores.
对于满意度评分，均值是 3.3，中位数是 3，众数是 4。这表明虽然最常见的评分相当正面，但分布略微偏向较低分数。
5. Measuring Spread: Range, Quartiles and Box Plot | 度量离散程度：极差、四分位数与箱线图
The range of weekly spends is the difference between the maximum (£15.00) and minimum (£4.50), giving £10.50. The range is sensitive to extreme values, so we also calculate the interquartile range (IQR).
每周花费的极差是最小值（£4.50）与最大值（£15.00）的差，为 10.50 英镑。极差容易受极端值影响，因此我们还计算四分位数间距（IQR）。
Using the sorted spend list, the lower quartile Q₁ is the median of the lower half: £6.20. The upper quartile Q₃ is £9.80. Therefore, IQR = Q₃ – Q₁ = £3.60. The five-number summary is: Min £4.50, Q₁ £6.20, Median £8.00, Q₃ £9.80, Max £15.00.
利用排序后的花费列表，下四分位数 Q₁ 是下半部分的中位数：6.20 英镑。上四分位数 Q₃ 是 9.80 英镑。因此，IQR = Q₃ – Q₁ = 3.60 英镑。五数概括为：最小值 4.50 英镑，Q₁ 6.20 英镑，中位数 8.00 英镑，Q₃ 9.80 英镑，最大值 15.00 英镑。
ers extend to the minimum and maximum that are not outliers. Since no value falls below Q₁– Q₃+1.5×IQR, the whiskers reach the actual min and max. The box plot reveals the central 50% of students spend between £6.20
Published by TutorHao | Year 10 统计 Revision Series | aleveler.com
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