Year 9 AQA Statistics: Case Study Practical Work | 九年级AQA统计：案例分析实战演练

📚 Year 9 AQA Statistics: Case Study Practical Work | 九年级AQA统计：案例分析实战演练

In AQA GCSE Statistics (Year 9), case study exercises bring the subject to life by asking you to apply data handling, representation, analysis and interpretation in real-world contexts. This article walks through a complete practical example using a dataset that links revision hours to test scores. You will see how to organise raw numbers, choose the correct graphs, calculate summary statistics, explore correlation, fit a line of best fit and communicate findings like a true statistician. Every step mirrors the style of exam-style investigation tasks and helps build the fluency you need for Paper 1 and Paper 2.

在AQA GCSE统计（九年级）中，案例研究练习通过将数据处理、数据呈现、分析和解释应用于真实情境，让这门学科变得鲜活起来。本文将通过一个将复习时间与考试成绩联系起来的完整实例，带你一步步掌握如何整理原始数据、选择正确的图表、计算汇总统计量、探究相关性、拟合最佳拟合线，并像一名真正的统计学家那样交流你的发现。每个步骤都模拟了调查型考题的风格，帮助你建立Paper 1和Paper 2所需的熟练度。

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1. Understanding the Case Study Context | 理解案例背景

A local school wants to investigate whether the number of hours Year 9 pupils spend on independent revision each week has any relationship with their end-of-year mathematics percentage score. The school surveyed 20 randomly selected pupils and collected two variables: weekly revision hours (to the nearest half hour) and mathematics test score (%). The aim is to describe the distribution of both variables, check for any association, and if appropriate, build a simple model to predict scores from revision time.

当地一所学校想调查九年级学生每周用于自主复习的小时数与他们的年终数学百分比成绩之间是否存在某种关系。学校随机抽取了20名学生，收集了两个变量：每周复习小时数（精确到半小时）和数学测试成绩（%）。目标是描述这两个变量的分布、检查是否存在关联，并在适当的情况下建立一个根据复习时间预测成绩的简单模型。

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2. Data Collection and Organisation | 数据收集与整理

The raw data are shown in the table below. Before any analysis can begin, the data must be checked for errors, sorted and grouped where necessary. The table includes an ID column for reference but statistical work focuses on the two numerical columns.

原始数据如下表所示。在开始任何分析之前，必须先检查数据错误、进行排序并在必要时进行分组。表中包含一列ID以供参考，但统计分析的重点是两列数值。

Pupil ID	Revision hours (h)	Test score (%)
1	0.5	32
2	1.0	41
3	1.5	45
4	2.0	48
5	2.0	55
6	2.5	52
7	3.0	60
8	3.0	62
9	3.5	58
10	4.0	70
11	4.0	68
12	4.5	75
13	5.0	78
14	5.0	82
15	5.5	85
16	6.0	88
17	6.5	84
18	7.0	92
19	8.0	95
20	9.0	98

For the revision hours, grouping into equal class intervals helps to construct a frequency table. A sensible choice is 0–2, 2–4, 4–6, 6–8, 8–10 hours. Similarly, test scores can be grouped into 30–49, 50–69, 70–89, 90–100 for broader insight.

对于复习小时数，将其划分为等宽的组距有助于构建频数分布表。合理的选择是0–2、2–4、4–6、6–8、8–10小时。同样，测试成绩可以划分为30–49、50–69、70–89、90–100，以便进行更宏观的观察。

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3. Visualising the Data: Graphs and Charts | 数据可视化：图表

For univariate data, the revision hours can be displayed as a histogram (with frequency density if intervals are unequal, but here equal widths mean frequency is fine). A box plot is also valuable to show the five-number summary. For test scores, a cumulative frequency curve can estimate the median and quartiles. The bivariate relationship is best shown with a scatter graph – plotting revision hours on the horizontal axis and test score on the vertical axis. Always label axes clearly: ‘Revision time (hours)’ and ‘Mathematics score (%)’.

对于单变量数据，复习小时数可以用直方图展示（如果组距不等需要使用频率密度，但此处组距相等，所以直接使用频数即可）。箱形图也能有效地显示五数概括。对于测试成绩，累积频率曲线可以估计中位数和四分位数。双变量关系最好用散点图来展示——将复习小时数放在横轴，测试成绩放在纵轴。务必清晰地标注坐标轴：“复习时间（小时）”和“数学成绩（%）”。

From the scatter graph, the points follow an upward trend, indicating a positive association. A line of best fit will be discussed later. In an exam, you must plot points as small crosses and avoid joining them dot-to-dot unless it is a time series.

从散点图看，数据点呈上升趋势，表明存在正相关。最佳拟合线将在后文讨论。在考试中，你必须用小叉号绘制数据点，除非是时间序列图，否则不要将点连成线。

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4. Measures of Central Tendency | 集中趋势量数

For the 20 revision hours: mean x̄ = (0.5+1.0+1.5+2.0+2.0+2.5+3.0+3.0+3.5+4.0+4.0+4.5+5.0+5.0+5.5+6.0+6.5+7.0+8.0+9.0) / 20 = 84.0/20 = 4.2 hours. Median: with n=20, the median lies between the 10th and 11th ordered values. Ordered hours: 0.5,1.0,1.5,2.0,2.0,2.5,3.0,3.0,3.5,4.0,4.0,4.5,5.0,5.0,5.5,6.0,6.5,7.0,8.0,9.0. The 10th value is 4.0, 11th is 4.0, so median = 4.0 hours. Mode: 4.0 and 5.0 both appear twice, so bimodal (4.0, 5.0). Because the mean (4.2) is slightly above the median (4.0), the distribution has a slight positive skew.

对于20个复习小时数：均值 x̄ = (0.5+1.0+1.5+2.0+2.0+2.5+3.0+3.0+3.5+4.0+4.0+4.5+5.0+5.0+5.5+6.0+6.5+7.0+8.0+9.0) / 20 = 84.0/20 = 4.2 小时。中位数：n=20，中位数位于第10个和第11个有序值之间。排序后的小时数：0.5,1.0,1.5,2.0,2.0,2.5,3.0,3.0,3.5,4.0,4.0,4.5,5.0,5.0,5.5,6.0,6.5,7.0,8.0,9.0。第10个值是4.0，第11个值是4.0，因此中位数为4.0小时。众数：4.0和5.0都出现了两次，因此是双峰（4.0, 5.0）。由于均值（4.2）略高于中位数（4.0），分布呈轻微正偏态。

For test scores: mean ≈ 69.4% (calculated similarly). Median: 10th and 11th ordered scores are 70 and 75? Let’s list scores: 32,41,45,48,52,55,58,60,62,68,70,75,78,82,84,85,88,92,95,98. 10th=68, 11th=70, median = 69%. The mode is none unique. The closeness of mean and median suggests fairly symmetric distribution.

对于测试成绩：均值约为69.4%（计算方法类似）。中位数：排序后第10个68%，第11个70%，中位数 = 69%。没有唯一的众数。均值与中位数接近表明分布大致对称。

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5. Measures of Spread | 离散程度量数

The range is the simplest measure. For revision hours: 9.0 – 0.5 = 8.5 hours. For test scores: 98 – 32 = 66%. However, the range is heavily affected by outliers. The interquartile range (IQR) is more robust. For revision hours, Q₁ is the median of the lower half (first 10 values): the 5.5th position gives (2.0+2.5)/2 = 2.25. Q₃ is from upper half: 15.5th position = (6.0+6.5)/2 = 6.25. IQR = 6.25 – 2.25 = 4.0 hours. Standard deviation (s) is also required: using the formula s = √[ Σ(x – x̄)² / (n–1) ]. Calculating step-by-step, we find s ≈ 2.31 hours. Always show your working in an investigation.

极差是最简单的离散度量。对复习小时数：9.0 – 0.5 = 8.5小时。对测试成绩：98 – 32 = 66%。然而，极差受异常值影响很大。四分位距（IQR）更具稳健性。对于复习小时数，Q₁是低半组（前10个值）的中位数：第5.5个位置为 (2.0+2.5)/2 = 2.25。Q₃来自高半组：第15.5个位置 = (6.0+6.5)/2 = 6.25。IQR = 6.25 – 2.25 = 4.0小时。标准差（s）也是必需的：使用公式 s = √[ Σ(x – x̄)² / (n–1) ]。逐步计算后，得到s ≈ 2.31小时。在调查中要始终展示你的计算过程。

For test scores IQR: Q₁=55%, Q₃=85%, IQR=30%. This indicates the middle 50% of scores are spread over 30 percentage points. Knowing both central tendency and spread allows you to compare groups. For instance, if another class had a similar mean but smaller IQR, you would conclude their performance is more consistent.

测试成绩的IQR：Q₁=55%，Q₃=85%，IQR=30%。这表明中间50%的成绩分布在30个百分点的范围内。同时了解集中趋势和离散程度可以比较不同的组别。例如，如果另一班级均值相近但IQR较小，你就可以得出其成绩更稳定的结论。

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6. Scatter Graphs and Correlation | 散点图与相关性

Plotting revision hours (x) against test score (y) reveals a clear positive trend. The correlation coefficient r quantifies the strength and direction. Using a calculator, we can compute r for this dataset. The procedure uses the sums: Σx, Σy, Σx², Σy², Σxy. Here, Σx=84, Σy=1388, Σx²=446.5, Σy²=104678, Σxy=6732. Substituting into the PMCC formula yields r ≈ 0.976. A value very close to +1 indicates a very strong positive linear correlation.

将复习小时（x）与测试成绩（y）绘制成散点图后，可以观察到明显的正趋势。相关系数r可以量化相关的强度和方向。使用计算器可以计算出本数据集的r。计算过程用到以下总和：Σx、Σy、Σx²、Σy²、Σxy。此处，Σx=84，Σy=1388，Σx²=446.5，Σy²=104678，Σxy=6732。代入积差相关系数公式得到r ≈ 0.976。非常接近+1的值表明存在极强的正线性相关。

It is vital to remember that correlation does not imply causation. Pupils who revise more tend to score higher, but other factors (prior attainment, quality of revision, attendance) also play a part. In your report, always discuss possible lurking variables and avoid making causal claims without experimental evidence.

关键是要记住：相关关系并不意味着因果关系。复习更多的学生往往得分更高，但其他因素（原有基础、复习质量、出勤率）也在起作用。在报告中，请务必讨论可能的潜在变量，在没有实验证据的情况下不要做出因果断言。

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7. Line of Best Fit and Predictions | 最佳拟合线与预测

With such a strong correlation, it is reasonable to fit a linear model of the form y = a + bx. The slope b is calculated as b = (nΣxy – ΣxΣy) / (nΣx² – (Σx)²). Plugging in numbers: b = (20×6732 – 84×1388) / (20×446.5 – 84²) = (134640 – 116592) / (8930 – 7056) = 18048 / 1874 ≈ 9.63 (to 2 d.p.). The intercept a = (Σy – bΣx)/n = (1388 – 9.63×84) / 20 = (1388 – 808.92) / 20 = 579.08/20 ≈ 28.95. Thus the regression equation is: Score ≈ 29.0 + 9.6 × (revision hours).

在相关性如此强的情况下，拟合一个形如 y = a + bx 的线性模型是合理的。先计算斜率b：b = (nΣxy – ΣxΣy) / (nΣx² – (Σx)²)。代入数字：b = (20×6732 – 84×1388) / (20×446.5 – 84²) = (134640 – 116592) / (8930 – 7056) = 18048 / 1874 ≈ 9.63（保留两位小数）。截距a = (Σy – bΣx)/n = (1388 – 9.63×84) / 20 = (1388 – 808.92) / 20 = 579.08/20 ≈ 28.95。因此回归方程为：成绩 ≈ 29.0 + 9.6 ×（复习小时数）。

Equation: y = 29.0 + 9.6x

This means a pupil who does 3 hours of revision is predicted to score 29 + 9.6×3 = 57.8%, whereas 6 hours predicts 29 + 9.6×6 = 86.6%. Interpolation (within the data range 0.5–9 h) is acceptable, but extrapolation beyond 9 hours is unreliable because we do not know if the linear trend continues.

这意味着复习3小时的学生预测成绩为29 + 9.6×3 = 57.8%，而复习6小时则预测为29 + 9.6×6 = 86.6%。在数据范围（0.5–9小时）内的插值是可以接受的，但超出9小时的外推则不可靠，因为我们不知道线性趋势是否会延续。

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8. Probability from Data | 从数据中估计概率

Statistical case studies often ask you to estimate probabilities using relative frequency. For example, ‘What is the probability that a randomly chosen pupil revises more than 5 hours per week?’ From the data, 9 out of 20 pupils exceed 5 hours (values: 5.5,6.0,6.5,7.0,8.0,9.0 plus the boundary 5.0? We count strictly >5: 5.5,6.0,6.5,7.0,8.0,9.0 = 6 pupils). So estimated P(more than 5 h) = 6/20 = 0.3. You can also construct a two-way table by grouping both variables, then calculate conditional probabilities. For instance, P(score > 70 | revision > 5 h) = (6 pupils with revision >5 and score >70) / 6 = 6/6 = 1, based on our data, showing a perfect conditional relative frequency in this sample. Always comment on reliability: small sample sizes give imprecise estimates.

统计案例研究经常要求你利用相对频率来估计概率。例如，“随机选择一名学生，他每周复习超过5小时的概率是多少？”根据数据，20人中有9人超过5小时？严格大于5：5.5,6.0,6.5,7.0,8.0,9.0共6人。所以估计P(超过5小时) = 6/20 = 0.3。你还可以通过将两个变量分组构造双向表，然后计算条件概率。例如：P(成绩>70 | 复习>5小时) = （复习>5且成绩>70的6人）/ 6 = 6/6 = 1，在这个样本中显示出完美的条件相对频率。请始终评价可靠性：小样本量会给出不精确的估计。

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9. Writing a Statistical Report | 撰写统计报告

A well-structured report should include: (1) Introduction – state the problem, the population and the sample; (2) Data presentation – tables and graphs with clear titles; (3) Numerical analysis – averages, spread, correlation; (4) Interpretation – comment on trends, possible reasons, any unusual observations; (5) Conclusion – summarise findings, mention limitations and suggest improvements. Use the PPDAC cycle (Problem, Plan, Data, Analysis, Conclusion) to organise your work. Always refer to your graphs and calculations explicitly in the text.

一份结构良好的报告应包含以下部分：（1）引言——陈述问题、总体和样本；（2）数据呈现——带有清晰标题的表格和图表；（3）数值分析——平均数、离散程度、相关性；（4）解释——评论趋势、可能的原因、任何异常观察值；（5）结论——总结发现、提及局限性并提出改进建议。使用PPDAC循环（问题、计划、数据、分析、结论）来组织你的工作。务必在正文中明确引用你的图表和计算结果。

For this case, a sample conclusion could be: ‘There is a very strong positive linear relationship between weekly revision time and test score (r = 0.976). On average, each extra hour of revision is associated with an increase of approximately 9.6 percentage points. However, the sample of 20 is small and from one school, so results may not generalise to all Year 9 pupils.’ Remember to use non-technical language in the summary for a general audience.

对于本例，一个示例性的结论可以是：“每周复习时间与测试成绩之间存在极强的正线性关系（r = 0.976）。平均而言，每增加一小时复习时间，成绩大约提高9.6个百分点。然而，样本量仅为20人，且来自同一所学校，因此结果可能无法推广到所有九年级学生。”记得在面向普通读者的总结中使用非技术性语言。

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10. Common Mistakes to Avoid | 避免常见错误

Mistake 1: Using the wrong graph – a bar chart for continuous data instead of a histogram, or a line graph for unconnected points. Continuous data needs a histogram or frequency polygon. Mistake 2: Confusing correlation with causation – never state ‘revising more causes higher scores’ without a controlled experiment. Mistake 3: Extrapolating without caution – predicting for 20 hours of revision when the maximum data point is 9 hours is inappropriate. Mistake 4: Ignoring outliers or not discussing them – an extremely low score with high revision could be an outlier; always investigate. Mistake 5: Calculation slips – misplacing decimal points, forgetting to square values, or using n instead of n–1 for sample standard deviation. Double-check with a calculator and write intermediate steps.

错误1：使用了错误的图表——用条形图表示连续数据而不是直方图，或者对不连续的点使用折线图。连续数据需要使用直方图或频数折线图。错误2：混淆相关关系与因果关系——在没有对照实验的情况下，永远不要说“复习更多导致分数更高”。错误3：不加小心地进行外推——当数据点最大值为9小时，却预测复习20小时的成绩是不恰当的。错误4：忽略异常值或不讨论它们——复习时间很多却得分极低可能是一个异常值；一定要进行调查。错误5：计算错误——点错小数点、忘记平方、或在计算样本标准差时用了n而不是n–1。使用计算器复核并写出中间步骤。

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11. Practice Exercise | 实战练习

Now test yourself. A new dataset gives the number of hours 15 Year 9 pupils spent on a maths app and their scores: (2 h, 45%), (3 h, 50%), (4 h, 60%), (4 h, 58%), (5 h, 62%), (5 h, 65%), (6 h, 70%), (6 h, 68%), (7 h, 74%), (7 h, 72%), (8 h, 80%), (8 h, 78%), (9 h, 85%), (10 h, 88%), (10 h, 90%). Tasks: a) Draw a scatter graph. b) Comment on the correlation. c) Calculate the equation of the line of best fit. d) Predict the score for 5.5 hours. e) Explain why predicting at 20 hours is unreliable. Write a short paragraph summarising limitations.

现在来测试一下自己。一个新数据集给出了15名九年级学生使用数学APP的小时数及其分数：(2 h, 45%), (3 h, 50%), (4 h, 60%), (4 h, 58%), (5 h, 62%), (5 h, 65%), (6 h, 70%), (6 h, 68%), (7 h, 74%), (7 h, 72%), (8 h, 80%), (8 h, 78%), (9 h, 85%), (10 h, 88%), (10 h, 90%)。任务：a) 绘制散点图。b) 对相关性进行评论。c) 计算最佳拟合线方程。d) 预测学习5.5小时的成绩。e) 解释为什么预测20小时不可靠。写一小段关于局限性的总结。

Answers for checking: b) Strong positive correlation. c) Using similar calculations, b ≈ 4.86, a ≈ 37.6, so y = 37.6 + 4.86x. d) ≈ 64.3%. e) Far beyond the data range, so relationship may not hold. The sample size is also small.

答案供核对：b) 强正相关。c) 利用相似的计算过程，b ≈ 4.86，a ≈ 37.6，因此 y = 37.6 + 4.86x。d) ≈ 64.3%。e) 远远超出数据范围，因此关系可能不成立。样本量也偏小。

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12. Summary and Key Takeaway | 总结与关键要点

AQA GCSE Statistics case studies reward methodical, well-communicated analysis. Always begin by understanding the context, organise your data, choose the right visual and numerical summaries, then interpret with caution. Remember the investigation cycle: Problem, Plan, Data, Analysis, Conclusion (PPDAC). Practise with different datasets until the steps become automatic. When writing, clarity is more important than complex vocabulary. Show your working clearly, use units, and never forget to comment on the reliability of your findings.

AQA GCSE统计的案例研究奖励有条不紊、表达清晰的分析。始终从理解背景开始，整理数据，选择合适的图形和数值汇总，然后谨慎地解释。记住调查循环：问题、计划、数据、分析、结论（PPDAC）。用不同的数据集反复练习，直到这些步骤变得自然而然。在写作时，清晰比复杂的词汇更重要。清晰地展示计算过程，使用正确的单位，并且永远不要忘记评论你研究结果的可靠性。

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