📚 Year 11 AQA Statistics: Case Study Practice | Year 11 AQA 统计：案例分析实战演练
Welcome to this GCSE Statistics case study walkthrough, designed to help you apply key statistical techniques to a realistic data investigation. We will follow one complete analysis – from data collection to hypothesis testing – using the relationship between revision hours and maths exam scores among Year 11 students. This article will sharpen your skills in handling data, calculating summary statistics, drawing graphs, measuring correlation, building regression models, and making inferences, all within the AQA specification.
欢迎来到这份 GCSE 统计案例研究实战演练，旨在帮助你运用关键的统计方法完成一次真实的数据调查。我们将全程跟踪一个完整的分析过程——从数据收集到假设检验——围绕 Year 11 学生的复习时间与数学考试成绩之间的关系展开。本文将通过这一实例，强化你在数据整理、概括统计量计算、图形绘制、相关性度量、回归建模以及统计推断方面的能力，所有内容均紧扣 AQA 考纲要求。
1. Introducing the Case Study | 案例背景介绍
In this case study, a school's statistics class suspects that the number of hours a Year 11 student spends revising maths per week is positively associated with their end-of-year exam score. To investigate this, they decide to survey 20 students at random and record two variables: revision hours per week (x) and percentage score in the maths exam (y). The goal is to describe the relationship, measure its strength, build a model for prediction, and test whether the average revision time differs from a national benchmark of 25 hours.
在这项案例研究中，某校统计班的同学猜测 Year 11 学生每周用于数学复习的小时数与期末考试成绩之间存在正相关。为了验证这一想法，他们随机调查了 20 名学生，并记录了两个变量：每周复习小时数（x）和数学考试百分制得分（y）。分析的目标是描述两者关系、度量关系强度、建立预测模型，并检验平均复习时间是否与国家基准 25 小时存在差异。
2. Data Collection and Sampling | 数据收集与抽样方法
The class uses a simple random sampling method: each Year 11 student is assigned a number, and 20 numbers are drawn using a random number generator. This approach avoids bias and gives every student an equal chance of being selected. The survey asks two clear questions: 'How many hours did you spend revising maths in an average week this term?' and 'What was your percentage score in the latest maths exam?' The raw data are recorded in a table.
班上同学采用了简单随机抽样方法：先给每位 Year 11 学生一个编号，再用随机数生成器抽取 20 个号码。这种方法可以避免偏差，确保每个学生被选中的概率相同。调查提出了两个明确的问题：“本学期你平均每周花多少小时复习数学？”以及“最近一次数学考试的百分制成绩是多少？”所有原始数据均记录在表格中。
The sample size of 20 is manageable but small enough to introduce sampling variability. Students are reminded that the results only apply to this particular cohort and should not be generalised without caution. Ethical considerations, such as anonymity and informed consent, are also discussed before collecting the data.
样本容量 20 虽然便于处理，但偏小，会引入一定的抽样变异性。同学们认识到，分析结果只适用于这一特定年级群体，不能轻易推广到其他学生。在收集数据之前，他们还讨论了匿名性和知情同意等伦理要求。
| Student | Revision hours (x) | Maths score (y) |
|---|---|---|
| 1 | 4 | 45 |
| 2 | 6 | 52 |
| 3 | 8 | 55 |
| 4 | 10 | 62 |
| 5 | 12 | 65 |
| 6 | 15 | 70 |
| 7 | 18 | 72 |
| 8 | 20 | 78 |
| 9 | 23 | 80 |
| 10 | 26 | 82 |
| 11 | 29 | 84 |
| 12 | 32 | 87 |
| 13 | 35 | 90 |
| 14 | 38 | 91 |
| 15 | 40 | 93 |
| 16 | 43 | 94 |
| 17 | 46 | 95 |
| 18 | 48 | 96 |
| 19 | 50 | 98 |
| 20 | 55 | 99 |
3. Descriptive Statistics: Central Tendency and Spread | 描述性统计：集中趋势与离散程度
For revision hours, the mean is calculated by summing all 20 values and dividing by 20: x̄ y in this continuous-like data, but the modal class could be considered. The range is 55 - 4 = 51 hours, showing considerable variability.
在寻找关系之前，有必要先用集中趋势和离散程度的指标分别对每个变量进行概括。对于复习小时数，平均值为所有 20 个数据的总和除以 20：x̄ = 558 ÷ 20 = 27.9 小时。中位数是排序后第 10 和第 11 个值的平均数，即 (26 + 29) ÷ 2 = 27.5 小时。这组数据总体上没有明显的众数，但可以考虑众数区间。极差为 55 - 4 = 51 小时，体现出较大的变异性。
(20+1)/4 ≈ 5.25th, which corresponds to a value of 12.5 hours (interpolated between 12 and 15). The upper quartile Q3 is the value at the 15.75th position, giving 40 + 0.75×(43-40) = 42.25 hours. Hence IQR = 42.25 - 12.5 = 29.75 hours. The standard deviation for this sample is approximately 15.2 hours, but the IQR is more robust to outliers.
为了更可靠地描述离散程度，我们计算四分位距 (IQR)。将数据排序后，下四分位数 Q1 位于第 (20+1)/4 ≈ 5.25 个位置，对应的数值为 12.5 小时（在 12 与 15 之间插值）。上四分位数 Q3 位于第 15.75 个位置，得到 40 + 0.75×(43-40) = 42.25 小时。因此 IQR = 42.25 - 12.5 = 29.75 小时。该样本的标准差大约为 15.2 小时，但 IQR 对异常值更具抗干扰性。
For the maths scores, the mean is ȳ = 1588 ÷ 20 = 79.4%, with a median of (82+84) ÷ 2 = 83%. The scores range from 45% to 99%. The lower and upper quartiles of y are 65% and 94%, giving an IQR of 29% res suggests the distribution might be approximately normal, a property we will use later.
对于数学成绩，平均分为 ȳ = 1588 ÷ 20 = 79.4%，中位数为 (82+84) ÷ 2 = 83%。成绩范围从 45% 到 99%。成绩的下四分位数和上四分位数分别为 65% 和 94%，因而 IQR 为 29%。成绩分布看上去大致对称，这提示数据可能近似正态分布，后面我们会用到这一性质。
4. Visualising the Data: Box Plots and Histograms | 数据可视化：箱线图与直方图
Visual summaries help detect patterns quickly. A box plot for revision hours shows a slightly right-skewed distribution, as the distance from Q3 to the maximum is larger than from Q1 to the minimum. The median sits near the middle of the box. For maths scores, the box plot indicates a fairly symmetric distribution with a slightly longer left tail (minimum 45 is farther from Q1 than the maximum 99 is from Q3).
图形化的概括能帮助快速发现数据模式。复习小时数的箱线图呈现出轻微右偏的分布，因为从 Q3 到最大值的距离大于从 Q1 到最小值的距离。中位数大致位于箱体中间。数学成绩的箱线图则显示大致对称的分布，但左尾稍长（最小值 45 离 Q1 比最大值 99 离 Q3 更远）。
A histogram of revision hours with bins of width 5 hours would reveal a peak in the 20-30 hour range and a gradual tail towards higher values. The scores histogram would be roughly bell-shaped, centred around 80%. These plots confirm the absence of extreme outliers and support the use of correlation and normal-based methods.
若将复习小时数以 5 小时为区间宽度绘制直方图，会看到在 20–30 小时区间出现峰值，并向高值方向逐渐拖尾。成绩数据的直方图则大致呈钟形，中心位于 80% 左右。这些图形证实数据中没有极端异常值，支持后续使用相关分析和基于正态分布的方法。
5. Scatter Diagram and Direction of Association | 散点图与相关方向
Plotting revision hours on the horizontal axis and maths score on the vertical axis reveals a clear pattern: as revision hours increase, the exam scores tend to rise as well. The points form an elliptical cloud stretching from the bottom-left to the top-right, indicating a strong positive association. There are no obvious outliers or clusters, and the relationship appears linear rather than curved.
以复习小时数为横轴、数学成绩为纵轴绘制散点图后，可以清晰地看到一种模式：随着复习时间的增加，考试成绩总体上也趋于提高。这些点构成一个从左下向右上延伸的椭圆形云团，表明两者之间存在强烈的正相关。图中没有明显的异常点或分群现象，关系也表现为线性而非曲线形状。
This visual inspection justifies proceeding to calculate a correlation coefficient and fitting a linear regression line. The scatter diagram also helps us consider whether a causal claim is reasonable: more revision is likely to lead to higher scores, but other factors such as prior ability, sleep, and teaching quality may also influence the outcome.
这种肉眼观察为下一步计算相关系数并拟合线性回归直线提供了依据。散点图还促使我们思考因果推断是否
Published by TutorHao | Year 11 统计 Revision Series | aleveler.com
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