📚 Case Study Practice for Year 12 CAIE Statistics | Year 12 CAIE 统计学案例分析实战演练
Welcome to this focused case study revision article for Year 12 CAIE Statistics. Using a realistic dataset from a school mathematics assessment, we will guide you through key S1 topics, including data presentation, probability, discrete random variables, the binomial distribution and the normal distribution. This hands-on approach will strengthen your problem-solving skills for the CAIE exam.
欢迎阅读这篇针对 Year 12 CAIE 统计学的案例研究复习文章。我们将利用一个真实的学校数学评估数据集，带你逐一攻克 S1 的核心主题，涵盖数据呈现、概率、离散随机变量、二项分布和正态分布。这种实战方法将强化你应对 CAIE 考试的解题能力。
1. Introducing the Dataset | 数据集介绍
A total of 50 Year 12 students sat a standard mathematics test marked out of 100. For each student, we recorded their score, gender (Male/Female) and whether they had attended extra tutoring sessions (Yes/No). A small sample of the data is shown in the table below. The full dataset is used to illustrate every statistical technique in this case study.
共有 50 名 Year 12 学生参加了一次满分 100 分的标准数学测试。我们记录了每位学生的成绩、性别（男/女）以及是否参加过课外辅导（是/否）。下表展示了数据的一小部分样本。完整的数据集将用于演示本案例研究中的每一项统计方法。
| Student | Score | Gender | Tutoring |
|---|---|---|---|
| 1 | 72 | M | Yes |
| 2 | 58 | F | No |
| 3 | 85 | M | Yes |
| 4 | 64 | F | Yes |
| 5 | 79 | F | No |
| 6 | 92 | M | Yes |
| 7 | 48 | F | No |
| 8 | 73 | M | No |
| 9 | 66 | F | Yes |
| 10 | 81 | M | No |
To compare the scores of males and females, we construct a back-to-back stem-and-leaf diagram. The stem represents the tens digit and the leaves represent the units digit, ordered from smallest to largest away from the stem. The left side of the diagram displays female scores, while the right side shows male scores.
为了比较男女生成绩，我们绘制了背靠背茎叶图。茎表示十位数，叶表示个位数，按从小到大顺序由茎向外排列。图的左侧显示女生成绩，右侧显示男生成绩。
Female Leaf | Stem | Male Leaf
8 5 2 | 4 | 3 8
9 7 4 1 | 5 | 2 5 8
8 5 5 3 0 | 6 | 0 1 4 7 9
7 4 2 | 7 | 2 2 5 5 9
3 1 | 8 | 1 3 5 5 8
| 9 | 2 4
The stem-and-leaf plot immediately reveals that the distribution of female scores is slightly more clustered in the 50s and 60s, while male scores extend further into the 90s. Both distributions are roughly unimodal with no strong skewness. Next, we compute the five-number summary to construct box plots.
从茎叶图可以立即看出，女生成绩分布略集中于50多分和60多分，而男生成绩则延伸至90分以上。两个分布均大致为单峰，且无明显偏态。接下来，我们计算五数概括以构造箱线图。
For the 25 female students, the minimum is 42, lower quartile Q1 = 55, median Q2 = 65, upper quartile Q3 = 78 and maximum is 93. For 25 males, min = 43, Q1 = 60, median = 72, Q3 = 85, max = 94. The interquartile ranges are 23 and 25 respectively, suggesting similar spread. These summaries are exactly what you need to draw accurate box plots on a CAIE exam question.
对于 25 名女生，最小值为 42，下四分位数 Q1 = 55，中位数 Q2 = 65，上四分位数 Q3 = 78，最大值为 93。对于 25 名男生，最小值为 43，Q1 = 60，中位数 = 72，Q3 = 85，最大值为 94。四分位距分别为 23 和 25，表明离散程度相似。这些概括统计数据正是你在 CAIE 考试中绘制准确箱线图所需要的。
2. Measures of Central Tendency and Spread | 集中趋势与离散程度的度量
Let us first find the overall mean score for all 50 students. Summing every score gives Σx = 3 440. Thus the sample mean is x̄ = 3 440 / 50 = 68.8. The median for the combined group, found by ordering all 50 values, is 70.5 (the average of the 25th and 26th ordered scores).
首先求出全部 50 名学生的总平均分。将所有成绩相加得 Σx = 3 440，因此样本均值 x̄ = 3 440 / 50 = 68.8。混合组的中位数，通过将所有 50 个数值排序后找到，为 70.5（即第 25 和第 26 个有序值的平均值）。
For grouped subpopulations, the male mean is 72.2 and the female mean is 65.4, confirming the visual impression from the stem plot. The standard deviation for all 50 scores is computed using the formula s = √[ Σ(x – x̄)2 / (n – 1) ]. Using calculator statistical functions we obtain s ≈ 14.1. The variance is s2 ≈ 198.8. Always remember to use the divisor n-1 for the sample variance in S1.
对于分组后的子总体，男生均值为 72.2，女生均值为 65.4，证实了从茎叶图得到的直观印象。全部 50 个成绩的标准差用公式 s = √[ Σ(x – x̄)2 / (n – 1) ] 计算。利用计算器的统计功能我们得到 s ≈ 14.1。方差为 s2 ≈ 198.8。请记住在 S1 中计算样本方差时永远使用 n-1 作为除数。
The range is 94 – 42 = 52, and the interquartile range (IQR) is 83 – 57 = 26. Because the mean is slightly less than the median, the data may have a mild negative skew, but the difference is small. In the CAIE exam, you could be asked to interpret skewness from these statistics or to suggest which measure of average is more appropriate.
全距为 94 – 42 = 52，四分位距(IQR) = 83 – 57 = 26。由于均值略低于中位数，数据可能存在轻微的负偏态，但差异很小。在 CAIE 考试中，你可能需要根据这些统计量解释偏态，或建议哪个平均数的度量更为合适。
3. Probability from the Data | 基于数据的概率计算
We now treat the dataset as a sample space and calculate probabilities directly from frequencies. Define event A: a student scores 80 marks or above. Out of 50 students, 14 achieved 80+, so P(A) = 14/50 = 0.28. Similarly, define event B: a student attended tutoring. The table shows 32 students attended tutoring, thus P(B) = 32/50 = 0.64.
现在我们将数据集视为样本空间，并直接从频数中计算概率。定义事件 A：一名学生得分不低于 80 分。在 50 名学生中，14 人达到了 80+，因此 P(A) = 14/50 = 0.28。再定义事件 B：一名学生参加了课外辅导。表格显示有 32 名学生参加了辅导，故 P(B) = 32/50 = 0.64。
Compound events are just as straightforward. Counting students who are both male and scored 80 or above gives 9 individuals. Hence P(Male ∩ A) = 9/50 = 0.18. Using the addition rule, P(Male ∪ A) = P(Male) + P(A) –
Published by TutorHao | Year 12 统计 Revision Series | aleveler.com
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