📚 GCSE Statistics Case Study: Gaming Time vs. Maths Scores | GCSE 统计案例分析：游戏时间与数学成绩
In GCSE Statistics, applying your knowledge to real-world case studies is essential for deep understanding and exam success. This practical walk‑through takes you through a complete statistical investigation: exploring the relationship between weekly gaming hours and maths test scores among Year 11 students. You will design the study, collect and present data, calculate summary statistics, create visual displays, analyse correlation and regression, and finally evaluate the whole process. By the end, you will have a model answer for any similar 'investigative task' question.
在GCSE统计中，将知识应用于真实案例是深入理解和考试成功的关键。本次实战演练带你完整经历一项统计调查：探究11年级学生每周游戏时间与数学考试成绩之间的关系。你将设计研究方案、收集并呈现数据、计算概括统计量、绘制图表、分析相关与回归，并最终对整个流程进行评估。学完以后，你将拥有一份应对任何类似'调查任务'题目的标准范例。
1. Setting Up the Investigation | 设定调查问题
Every statistical investigation begins with a clear question and a hypothesis. Our aim is to find out whether there is an association between the time a student spends playing video games each week and their performance in a maths test. We treat gaming time as the independent variable (explanatory) and maths score as the dependent variable (response). The null hypothesis H₀ states that there is no linear correlation between the two variables, while the alternative hypothesis H₁ suggests a negative correlation: as gaming hours increase, maths scores tend to decrease.
每一项统计调查都始于明确的问题和假设。我们的目标是探究学生每周玩电子游戏的时间与数学测试成绩之间是否存在联系。我们将游戏时间视为自变量（解释变量），数学分数视为因变量（响应变量）。原假设 H₀ 表示两个变量之间没有线性相关，而备择假设 H₁ 则表明存在负相关：随着游戏时间增加，数学成绩趋于下降。
2. Designing the Data Collection | 设计数据收集方法
To gather reliable data, we decide to ask a sample of students to report their average weekly gaming hours (to the nearest hour) and grant permission to access their most recent standardised maths test score (out of 100). Both variables are quantitative and continuous. We design a short questionnaire that collects gaming time and student ID, then later we match the ID with the maths score obtained from the school's database. This reduces self‑reporting bias for the test result.
为了收集可靠的数据，我们决定请一组学生报告他们每周平均的游戏时间（精确到小时），并授权我们获取他们最近一次标准化数学测试的分数（满分100分）。这两个变量都是定量连续型变量。我们设计了一份简短问卷来收集游戏时间和学生编号，随后通过学校数据库将编号与数学成绩匹配起来。这可以减少测试成绩的自我报告偏差。
3. Sampling Strategy | 抽样策略
We aim for a simple random sample of 10 Year 11 students from a large secondary school. Using a random number generator on the school roll, we select participants to minimise selection bias. A sample size of 10 is small but sufficient for this introductory case study; in a real investigation we would aim for at least 30 to meet the conditions for more reliable inference. We ensure that participation is voluntary and anonymous.
我们的目标是从一所大型中学里抽取10名11年级学生组成简单随机样本。利用随机数生成器在全校名册上选取参与者，以尽量减少选择偏差。虽然10个样本量较小，但对于这个入门案例已经足够；在实际调查中，我们会争取至少30个样本以便满足更可靠推断的条件。我们确保参与是自愿且匿名的。
4. Collecting and Recording Data | 收集与记录数据
The raw data is recorded in a structured table. Each student is given a label A–J to preserve anonymity. Gaming time is recorded in hours per week, and the maths score is the percentage achieved in the test. Here is the collected dataset:
原始数据被记录在一个结构化的表格中。每位学生用字母 A–J 标记以保护隐私。游戏时间记录为每周小时数，数学分数为测试取得的百分数。以下为收集到的数据集：
| Student | Gaming time (hours) | Maths score (%) |
|---|---|---|
| A | 5 | 85 |
| B | 10 | 70 |
| C | 8 | 75 |
| D | 2 | 90 |
| E | 12 | 65 |
| F | 6 | 80 |
| G | 9 | 72 |
| H | 3 | 88 |
| I | 7 | 78 |
| J | 11 | 68 |
A quick scan suggests that students who game less tend to have higher scores, but we need formal analysis.
快速浏览后发现，游戏时间较短的学生分数往往更高，但我们需要正式的分析来验证。
5. Organising and Tabulating Data | 整理与制表
Before analysis we can sort the data by gaming time to make patterns clearer. We also check for any outliers or errors. The gaming times range from 2 to 12 hours; maths scores range from 65% to 90%. Both variables show reasonable spread. For later calculations we will work with the unsorted table, but it is useful to note the paired values remain linked.
在分析之前，我们可以按游戏时间对数据排序，让规律更明显。同时检查是否有离群值或错误。游戏时间范围从2到12小时；数学分数从65%到90%。两个变量都展现出合理的分散程度。为后续计算我们会使用未排序的表格，但要特别注意成对数据的联结关系。
6. Graphical Presentation: Scatter Plot | 图形呈现：散点图
The most appropriate graph for bivariate quantitative data is a scatter diagram. We plot gaming hours on the x‑axis and maths score on the y‑axis. Each point represents one student. The resulting graph shows a clear downward pattern, suggesting a strong negative correlation.
对于双变量定量数据，最合适的图表是散点图。我们将游戏时间画在 x 轴，数学分数画在 y 轴。每个点代表一名学生。得出的图形呈现明显的下降模式，表明存在很强的负相关。
- Try this yourself: On graph paper, label the x‑axis from 0 to 14 hours and the y‑axis from 60% to 100%. Plot the ten points. You should see a steep slope downwards.
- 自己动手试: 在坐标纸上，将 x 轴标为 0 到 14 小时，y 轴标为 60% 到 100%。描出这十个点。你会看到一条陡峭的下行趋势。
7. Measures of Central Tendency and Spread | 集中趋势与离散程度
Summarising each variable individually gives us a fuller picture. We calculate the mean, median, range, and interquartile range (IQR).
分别对每个变量进行概括能给我们更全面的信息。我们计算均值、中位数、极差和四分位距 (IQR)。
Gaming time (x)
Data: 5, 10, 8, 2, 12, 6, 9, 3, 7, 11. Sum = 73, so mean x̄ or quartiles, the lower half is 2, 3, 5, 6, 7 giving Q₁ = 5; the upper half 8, 9, 10, 11, 12 gives Q₃ = 10; IQR = 10 − 5 = 5 hours.
数据：5, 10, 8, 2, 12, 6, 9, 3, 7, 11。总和 = 73，均值 x̄ = 73 ÷ 10 = 7.3 小时。求中位数时，排序：2, 3, 5, 6, 7, 8, 9, 10, 11, 12。中位数 = (7 + 8) ÷ 2 = 7.5 小时。众数：无。极差 = 12 − 2 = 10 小时。四分位数中，下半组 2, 3, 5, 6, 7 得 Q₁ = 5；上半组 8, 9, 10, 11, 12 得 Q₃ = 10；IQR = 10 − 5 = 5 小时。
Maths score (y)
Data: 85, 70, 75, 90, 65, 80, 72, 88, 78, 68. Sum = 771, mean ȳ 5, 68, 70, 72, 75 → Q₁ = 70. Q₃ of the upper five: 78, 80, 85, 88, 90 → Q₃ = 85. IQR = 15.
数据：85, 70, 75, 90, 65, 80, 72, 88, 78, 68。总和 = 771，均值 ȳ = 77.1。排序：65, 68, 70, 72, 75, 78, 80, 85, 88, 90。中位数 = (75 + 78) ÷ 2 = 76.5。极差 = 90 − 65 = 25。下半组五个的中位数为 Q₁ = 70；上半组中位数 Q₃ = 85；IQR = 15。
On their own, the summaries tell us about the average and spread of each variable, but they do not measure the relationship. That is the job of correlation.
这些统计量本身告诉我们每个变量的均值和分布，但并不能衡量两者之间的关系。这就要靠相关性分析来完成。
8. Correlation Analysis | 相关性分析
Pearson's product‑moment correlation coefficient r quantifies the strength and direction of a linear relationship. The formula is:
皮尔逊积矩相关系数 r 可以量化线性关系的强度和方向。其公式为：
r = Σ(x − x̄)(y − ȳ) / √[ Σ(x − x̄)² Σ(y − ȳ)² ]
We construct a working table to find the components. Use the means x̄ = 7.3 and ȳ = 77.1.
我们绘制一张计算表来求出各个组成部分。使用均值 x̄ = 7.3 和 ȳ = 77.1。
| x | y | x − x̄ | y − ȳ | (x−x̄)(y−ȳ) | (x−x̄)² | (y−ȳ)² |
|---|---|---|---|---|---|---|
| 5 | 85 | −2.3 | 7.9 | −18.17 | 5.29 | 62.41 |
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