📚 Case Study Mastery: Real-World Statistics | 案例分析实战演练
Statistical reasoning comes alive when we apply it to real scenarios. In this article, we work through a series of practical case studies designed to mirror the problem‑solving style of the Year 10 AQA Statistics specification. Each case develops skills in collecting, presenting, interpreting, and evaluating data, helping you become a confident statistical thinker.
当你把统计推理应用到真实场景中时，它才真正活起来。在这篇文章中，我们通过一系列实际案例演练，模拟 Year 10 AQA 统计大纲要求的解题风格。每个案例都在训练你收集、呈现、解释和评价数据的能力，助你成为自信的统计思考者。
1. Introduction to Statistical Case Studies | 统计案例研究简介
A case study in statistics is not a single calculation; it is a story told with numbers. You will be asked to identify the problem, decide what data to collect, select appropriate diagrams and measures, perform calculations, and write a conclusion that relates back to the original context. The AQA exam often presents a short scenario followed by a series of linked questions. Being able to move smoothly from raw data to a final recommendation is the key skill.
统计学中的案例研究不是单一的计算，而是一个用数字讲述的故事。你需要明确问题、决定收集什么数据、选择合适的图表和度量、进行计算，并写出与原始背景相关的结论。AQA 考试常常给出一段简短情景，再配上一系列相互关联的问题。能够从原始数据顺畅地走向最终建议，就是核心能力。
Throughout this article we will explore eight mini‑cases covering descriptive statistics, probability, sampling, correlation, time series, and data ethics. For each one, work through the steps methodically: State what you know, plan your approach, carry out the analysis, and interpret your findings in plain English (and Chinese).
在整篇文章中，我们将探讨八个迷你案例，涵盖描述统计、概率、抽样、相关性、时间序列和数据伦理。每一个案例都请你按部就班地练习：说出已知信息、规划方法、实施分析，然后用平实的英文（和中文）解释你的发现。
2. Case Study 1: Analysing Test Scores | 案例分析1：考试成绩分析
A class of ten students sat a mathematics test. Their scores out of 80 are: 45, 52, 61, 48, 55, 72, 68, 59, 63, 50. The head of department wants a summary of the central tendency and spread. Calculate the mean, median, mode, range, and interquartile range (IQR). Then write a short report comparing the mean and median.
一个十名学生班级参加了数学测验。满分 80 分，成绩为：45, 52, 61, 48, 55, 72, 68, 59, 63, 50。学科组长需要了解数据的集中趋势和离散程度。请计算平均数、中位数、众数、极差和四分位距（IQR），然后写一份简短报告对比平均数和 median。
Step‑by‑step solution: Order the scores: 45, 48, 50, 52, 55, 59, 61, 63, 68, 72. There is no mode (all values unique).
逐步解答： 先将分数排序：45, 48, 50, 52, 55, 59, 61, 63, 68, 72。没有众数（所有值均不重复）。
Mean x̄ = (Σx)/n = (45+48+50+52+55+59+61+63+68+72)/10 = 573/10 = 57.3. Median position: (10+1)/2 = 5.5th, so median = (55+59)/2 = 57. Range = 72 − 45 = 27. To find IQR: lower half = 45,48,50,52,55 → Q1 median = 50; upper half = 59,61,63,68,72 → Q3 median = 63. IQR = Q3 − Q1 = 63 − 50 = 13.
平均数 x̄ = (Σx)/n = (45+48+50+52+55+59+61+63+68+72)/10 = 573/10 = 57.3。中位数位置：(10+1)/2 = 5.5，因此中位数 = (55+59)/2 = 57。极差 = 72 − 45 = 27。四分位距：下半部分 45,48,50,52,55 → Q1 中位数 = 50；上半部分 59,61,63,68,72 → Q3 中位数 = 63。IQR = Q3 − Q1 = 63 − 50 = 13。
The mean and median are very close (57.3 vs 57), suggesting the distribution is roughly symmetric and not pulled by extreme values. The range of 27 shows moderate spread, while the IQR of 13 indicates that the middle 50% of scores lie within 13 marks of each other.
平均数与中位数非常接近（57.3 对比 57），表明分布大致对称，没有被极端值拉偏。极差 27 说明中等程度的分散，而 IQR 为 13 则显示中间 50% 的学生分数彼此相差不超过 13 分。
3. Case Study 2: Customer Satisfaction Survey | 案例分析2：客户满意度调查
A café surveyed 100 customers about their experience. The responses: Very Satisfied (30), Satisfied (45), Neutral (15), Dissatisfied (7), Very Dissatisfied (3). The manager wants to present the data using a pie chart and a bar chart, and to calculate the proportion of customers who are at least satisfied.
一家咖啡馆调查了 100 位顾客的体验。结果：非常满意（30）、满意（45）、一般（15）、不满意（7）、非常不满意（3）。经理想用饼图和条形图展示数据，并计算至少满意的顾客比例。
Proportion at least satisfied = (30+45)/100 = 75/100 = 0.75 or 75%. For a pie chart, each category’s angle = (frequency/100) × 360°. Very Satisfied: 30% → 108°; Satisfied: 45% → 162°; Neutral: 15% → 54°; Dissatisfied: 7% → 25.2°; Very Dissatisfied: 3% → 10.8°. A bar chart can simply plot the frequencies with the categories on the horizontal axis.
至少满意的比例 = (30+45)/100 = 75/100 = 0.75，即 75%。饼图中，每类的角度 = (频数/100) × 360°。非常满意：30% → 108°；满意：45% → 162°；一般：15% → 54°；不满意：7% → 25.2°；非常不满意：3% → 10.8°。条形图只需将类别放在水平轴，频数作柱高即可。
When drawing conclusions, note that three‑quarters of customers are happy, but 10% expressed a negative opinion. The manager should investigate the reasons behind the 10 dissatisfied visitors, perhaps by reading their comments.
得出结论时，注意四分之三的顾客是满意的，但仍有 10% 表达了负面意见。经理应该调查这 10 位不满意的访客背后的原因，或许可以阅读他们的留言。
4. Case Study 3: Comparing Sports Performance | 案例分析3：体育成绩比较
Two sprinters, A and B, ran five 100 m time trials (in seconds). A: 11.2, 11.5, 10.9, 11.1, 11.8. B: 10.8, 11.3, 11.0, 10.7, 11.6. A coach wants to know who is faster on average and who is more consistent. Calculate the mean, median, range, and IQR for both athletes, and draw box plots to compare.
两名短跑运动员 A 和 B 进行了五次 100 米计时跑（单位：秒）。A：11.2, 11.5, 10.9, 11.1, 11.8。B：10.8, 11.3, 11.0, 10.7, 11.6。教练想知道谁平均更快，谁更稳定。计算两人的平均数、中位数、极差和 IQR，并绘制箱形图进行比较。
A ordered: 10.9, 11.1, 11.2, 11.5, 11.8. Mean A = (10.9+11.1+11.2+11.5+11.8)/5 = 56.5/5 = 11.30 s; median = 11.2 s; range = 11.8−10.9 = 0.9 s; Q1 = 11.0 (mid of 10.9 & 11.1), Q3 = 11.65 (mid of 11.5 & 11.8), IQR = 0.65 s.
A 排序后：10.9, 11.1, 11.2, 11.5, 11.8。A 的平均数 = (10.9+11.1+11.2+11.5+11.8)/5 = 56.5/5 = 11.30 秒；中位数 = 11.2 秒；极差 = 11.8−10.9 = 0.9 秒；Q1 = 11.0（10.9 和 11.1 的中间值），Q3 = 11.65（11.5 和 11.8 的中间值），IQR = 0.65 秒。
B ordered: 10.7, 10.8, 11.0, 11.3, 11.6. Mean B = (10.7+10.8+11.0+11.3+11.6)/5 = 55.4/5 = 11.08 s; median = 11.0 s; range = 11.6−10.7 = 0.9 s; Q1 = 10.75, Q3 = 11.45, IQR = 0.70 s.
B 排序后：10.7, 10.8, 11.0, 11.3, 11.6。B 的平均数 = (10.7+10.8+11.0+11.3+11.6)/5 = 55.4/5 = 11.08 秒；中位数 = 11.0 秒；极差 = 11.6−10.7 = 0.9 秒；Q1 = 10.75，Q3 = 11.45，IQR = 0.70 秒。
Athlete B has a lower mean (11.08 s vs 11.30 s), so B is faster on average. However, A’s IQR is slightly smaller (0.65 s vs 0.70 s), indicating A’s middle 50% of times are tighter. The range is identical. The box plots would show B’s median lower and the bulk of B’s times shifted left, but with a similar overall spread.
运动员 B 的平均数较低（11.08 秒对比 11.30 秒），所以 B 平均更快。然而，A 的 IQR 略小（0.65 秒对比 0.70 秒），表明 A 中间 50% 的成绩更紧凑。极差相同。箱形图会显示 B 的中位数更低，大部分时间偏左，但总体分散程度相似。
5. Case Study 4: Probability and Risk Assessment | 案例分析4：概率与风险评估
A game involves flipping a fair coin three times. You win £5 if you get exactly two heads, and you lose £3 otherwise. Is the game worth playing? Model the probability using a tree diagram and calculate the expected gain per game.
一个游戏需要抛一枚公平硬币三次。如果恰好出现两次正面，你赢得 5 英镑，否则输掉 3 英镑。这个游戏值得玩吗？用树状图建立概率模型，并计算每局游戏的期望收益。
The sample space has 2³ = 8 equally likely outcomes. Outcomes with exactly two heads: HHT, HTH, THH → 3 outcomes. P(exactly 2 heads) = 3/8. P(not exactly 2 heads) = 5/8. Expected gain E = (3/8)×5 + (5/8)×(−3) = (15/8) − (15/8) = 0. The expected monetary gain is zero, meaning it is a fair game. Over many plays, you would not expect to win or lose money.
样本空间共有 2³ = 8 个等可能结果。恰好两个正面的结果：HHT、HTH、THH → 3 个。P（恰好两正）= 3/8。P（非恰好两正）= 5/8。期望收益 E = (3/8)×5 + (5/8)×(−3) = (15/8) − (15/8) = 0。期望金钱收益为零，说明这是一个公平游戏。长期玩下去，不期望赢钱也不期望输钱。
If the prize were raised to £6 for two heads, the expected gain becomes (3/8)×6 + (5/8)×(−3) = 18/8 − 15/8 = 3/8 = £0.375 per game, making it favourable. Always calculate expectation to evaluate risk.
若两正的奖金提高到 6 英镑，期望收益变为 (3/8)×6 + (5/8)×(−3) = 18/8 − 15/8 = 3/8 = 每局 £0.375，游戏变得有利。评估风险时始终要计算期望值。
6. Case Study 5: Sampling Methods in Practice | 案例分析5：抽样方法实践
A school has 1200 students: 400 in Year 7, 300 in Year 8, 250 in Year 9, 150 in Year 10, and 100 in Year 11. The student council wants to survey 60 students about lunch menus. Explain how to take a stratified sample by year group, and state one advantage of this method over simple random sampling.
某校有 1200 名学生：七年级 400 人，八年级 300 人，九年级 250 人，十年级 150 人，十一年级 100 人。学生会想调查 60 名学生关于午餐菜单的意见。请说明如何按年级进行分层抽样，并说出该方法相比简单随机抽样的一个优点。
In stratified sampling, the population is divided into distinct groups (strata), and the sample size from each stratum is proportional to the stratum’s share of the population. Here, Year 7 proportion = 400/1200 = 1/3, so sample Year 7 = (1/3)×60 = 20. Year 8: 300/1200 = 1/4 → 15; Year 9: 250/1200 ≈ 0.2083 → 13 (round as needed, e.g., 13); Year 10: 150/1200 = 1/8 → 8 (round 7.5 to 8); Year 11: 100/1200 = 1/12 → 5. Then randomly select the required number within each year group.
分层抽样中，先将总体划分为互不重叠的组（层），然后从每层抽取的样本量与层在总体中的比例一致。本例中，七年级占比 = 400/1200 = 1/3，故七年级样本量 = (1/3)×60 = 20。八年级：300/1200 = 1/4 → 15；九年级：250/1200 ≈ 0.2083 → 13（根据需要四舍五入，如 13）；十年级：150/1200 = 1/8 → 8（7.5 进位为 8）；十一年级：100/1200 = 1/12 → 5。然后在各年级内随机选够所需人数。
Advantage: Stratified sampling guarantees representation from each year group, so the views of smaller cohorts (e.g., Year 11) are not overlooked, as they might be in a simple random sample that could, by chance, miss them entirely.
优点：分层抽样保证每个年级都有代表，因此较小团体（如十一年级）的意见不会被忽略；而在简单随机抽样中，有可能偶然完全漏掉他们。
7. Case Study 6: Scatter Graphs and Correlation | 案例分析6：散点图与相关性
A student recorded the number of hours spent revising (x) and the test score (y) for seven peers: (2,50), (3,55), (5,65), (4,60), (6,70), (8,78), (7,72). Plot the points, describe the correlation, draw a line of best fit, and estimate the score for someone who revised for 5.5 hours.
一名学生记录了七位同学复习小时数 (x) 与测验分数 (y)：(2,50), (3,55), (5,65), (4,60), (6,70), (8,78), (7,72)。请描点，描述相关性，画出最佳拟合线，并估计复习 5.5 小时的同学可能得的分数。
Plotting reveals a strong positive correlation: as revision hours increase, test scores rise fairly linearly. The line of best fit should pass near the middle of the points. Using the points (2,50) and (8,78), gradient m ≈ (78−50)/(8−2) = 28/6 ≈ 4.67. Equation roughly: y − 50 = 4.67(x − 2) → y = 4.67x + 40.7. For x = 5.5 hours, y ≈ 4.67×5.5 + 40.7 = 25.7 + 40.7 = 66.4, so about 66 marks.
描点后显示强正相关：复习时间越长，分数呈线性上升。最佳拟合线应接近点的中心。利用 (2,50) 与 (8,78) 两点，斜率 m ≈ (78−50)/(8−2) = 28/6 ≈ 4.67。大致方程为：y − 50 = 4.67(x − 2) → y = 4.67x + 40.7。对于 x = 5.5 小时，y ≈ 4.67×5.5 + 40.7 = 25.7 + 40.7 = 66.4，约 66 分。
Caution: Estimating within the data range (interpolation) is reasonably reliable; predicting beyond the existing data (extrapolation), e.g., for 12 hours, would be risky because the linear pattern may not continue.
注意：在数据范围内估计（内插）相当可靠；超出已有数据进行预测（外推），例如 12 小时，就有风险，因为线性趋势可能不会保持。
8. Case Study 7: Time Series and Forecasting | 案例分析7：时间序列与预测
Quarterly sales figures (£000s) for a small shop: Q1=120, Q2=150, Q3=130, Q4=180, Q1=140, Q2=170. Compute a four‑point moving average to smooth the series, and comment on the trend and seasonal pattern.
某小店的季度销售额（千英镑）：Q1=120, Q2=150, Q3=130, Q4=180, Q1=140, Q2=170。计算四点移动平均以修匀序列，并评论趋势和季节模式。
Four‑point moving averages: (120+150+130+180)/4 = 580/4 = 145 (centred at between Q2 and Q3 of first year); (150+130+180+140)/4 = 600/4 = 150; (130+180+140+170)/4 = 620/4 = 155. The moving averages are rising: 145, 150, 155, indicating an upward trend. The original data show higher sales in Q4 and Q2, suggesting a seasonal peak in those quarters. Forecasting could use the trend, but seasonal adjustments would be needed to obtain quarter‑specific predictions.
四点移动平均：(120+150+130+180)/4 = 580/4 = 145（定位在第一年 Q2 与 Q3 之间）；(150+130+180+140)/4 = 600/4 = 150；(130+180+140+170)/4 = 620/4 = 155。移动平均数列上升：145, 150, 155，表明呈上升趋势。原始数据显示 Q4 和 Q2 销售额较高，提示这些季度存在季节性高峰。预测时可以借助趋势，但要进行季节性调整才能得到具体季度的预测值。
Always plot both the original data and the moving average on the same graph to visualise how the smooth line reveals the underlying trend.
始终将原始数据和移动平均画在同一张图上，这样可以直观地看出平滑线条如何揭示潜在趋势。
9. Case Study 8: Interpreting Data Ethically | 案例分析8：数据解读的伦理
A newspaper reported: ‘Crime up 200% in our town!’ The actual figures showed incidents rose from 2 to 6. Another graph truncated the vertical axis to start at 50, making a change from 80 to 82 look dramatic. Discuss how these presentations mislead and how they could be corrected.
一家报纸报道：“本镇犯罪率上升 200%！”实际数字显示事件从 2 起增加到 6 起。另一张图将纵轴截断从 50 开始，使得从 80 到 82 的变化显得很剧烈。讨论这些呈现方式如何误导，以及如何纠正。
Using a percentage increase on a very small base exaggerates the perceived danger. The rise from 2 to 6 is indeed a 200% relative increase, but the absolute increase is only 4 incidents, which may be random fluctuation. A more honest statement would give both absolute numbers and note the baseline is tiny. For the graph, truncating the axis is acceptable only if clearly labelled with a break symbol and if the intention is not to deceive; otherwise, the full axis starting at zero should be shown.
在很小的基数上使用百分比增长会夸大感知危险。从 2 到 6 确实是 200% 的相对增长，但绝对增加值仅为 4 起事件，可能是随机波动。更诚实的表述应同时给出绝对数字并指出基数很小。对于图表，截断纵轴只有在清楚标出中断符号且不以误导为目的时才可接受；否则应展示从零开始的完整坐标轴。
As statistical consumers, always ask: What is the baseline? Is the axis scaled fairly? Are percentages hiding a tiny absolute change? Ethical data presentation builds trust.
作为统计信息的消费者，要始终追问：基数是多少？坐标轴刻度是否公平？百分数是否掩盖了微小的绝对变化？符合伦理的数据呈现才能建立信任。
10. Conclusion: Key Skills from the Case Studies | 结论：案例研究中的关键技能
These eight cases have taken you through the core Year 10 AQA Statistics toolkit: summarising data with averages and spread, visualising categorical and numerical data, comparing distributions, modelling chance with tree diagrams, designing unbiased samples, exploring relationships with scatter graphs, smoothing time series, and spotting misleading statistics. In the exam, read the context carefully, choose the right mathematical tool, show your workings clearly, and always write a sentence that links your numbers back to the real‑world question.
这八个案例带你走过了 Year 10 AQA 统计的核心工具箱：用平均数和离散量概括数据、可视化分类与数值数据、比较多组分布、用树状图模拟机会、设计无偏样本、用散点图探索关系、修匀时间序列以及识别误导性统计。在考试中，请仔细阅读背景，选择合适的数学工具，清晰展示计算过程，并且始终用一句话把你的数字与现实世界的问题联系起来。
Practice these case studies several times until you can switch confidently between calculation and interpretation. Statistics is not just about getting the numbers right; it is about telling the truth with data.
多次练习这些案例，直到你能够自信地在计算和解释之间切换。统计不仅仅是把数字算对，更是要用数据讲述真相。
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