📚 Year 11 CAIE Statistics: In-depth Analysis of Past Exam Questions | Year 11 CAIE 统计：历年真题深度解析
Statistics is a key component of the CAIE Year 11 curriculum, consistently appearing in both core and extended papers. In this article, we offer a thorough breakdown of the most common question types seen in past exams, from histograms and cumulative frequency diagrams to probability trees and scatter graphs. By understanding the underlying methods and examiner expectations, you can confidently tackle these questions and secure high marks.
统计学是CAIE Year 11课程的核心模块，在核心卷和扩展卷中反复出现。本文将深度解析历年真题中最常见的题型，包括直方图、累积频率图、概率树以及散点图等。通过掌握这些方法并了解考官评分要点，你将能够自信应对这类题目并确保高分。
1. Understanding Frequency Density in Histograms | 理解直方图中的频率密度
sity = frequency ÷ class width. Always check whether the vertical axis is already labelled 'Frequency density'; if not, you must compute it before drawing bars.
许多CAIE真题要求你补全直方图或从直方图计算频数。核心公式是频率密度 = 频数 ÷ 组距。一定要先确认纵轴是否已标明 “频率密度”；如果没有，在绘图前需要先计算出该值。
When using a histogram to find frequencies, multiply the frequency density by the class width. In exams, watch out for unequal class widths – a common trap. If a bar has frequency density 1.5 and class width 10, the frequency is 15.
当利用直方图求频数时，用频率密度乘以组距即可。考试中要警惕不相等的组距——这是常见陷阱。如果某柱频率密度为1.5，组距为10，则频数为15。
A classic past paper question gives a partially completed histogram and a table of frequencies for some intervals. Your job is to use the given bars to find the correct frequency density scale, then fill in the missing bars or frequencies.
经典的真题会给出一张未完成的直方图以及部分区间的频数表。你需要利用已知的柱子确定频率密度的比例尺，然后补全缺失的柱状或频数。
2. Cumulative Frequency and Quartiles | 累积频率与四分位数
CAIE exams love cumulative frequency curves. You plot the upper boundary of each class against the cumulative frequency, then draw a smooth curve. To locate the median, find the value on the x-axis that corresponds to half of the total frequency (n/2). The lower quartile is at n/4 and the upper quartile at 3n/4.
CAIE考试非常偏爱累积频率曲线。你需要将每个组的上限与对应的累积频数绘出，然后画出平滑曲线。求中位数时，从纵轴总频数的一半 (n/2) 引水平线与曲线相交，再读横轴值。下四分位数在 n/4 处，上四分位数在 3n/4 处。
ot from the results. Always use a ruler and guide lines on your graph to show working; examiners award marks for clear method.
后续题目常常要求计算四分位距 (IQR = Q₃ − Q₁) 或根据结果绘制箱形图。务必在图上用直尺画出辅助线并展示过程；清晰的作图方法能帮助你获得方法分。
A recent extended paper asked students to compare two populations using cumulative frequency diagrams. By reading the medians and IQRs from both curves, you can comment on central tendency and spread, making precise comparisons rather than vague statements.
近期一份扩展卷真题要求学生利用两条累积频率曲线比较两个总体的差异。通过分别读取中位数和四分位距，你可以对集中趋势和离散程度做出精确比较，而不是泛泛而谈。
3. Box-and-Whisker Plots: Comparing Distributions | 箱形图：比较分布
Box plots are a staple in CAIE statistics questions. You must identify the minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum from a dataset or a cumulative frequency diagram, then draw the box plot with an appropriate scale.
箱形图是CAIE统计题中的常客。你需要从数据集或累积频率图中找出最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值，然后以合适的比例尺绘制箱形图。
When comparing two box plots, comment on both the median (which distribution has a higher centre) and the interquartile range or overall range (which is more spread out). Always support your statement with numerical values, e.g., 'The median for group A is 35 while for group B it is 42, so group B generally scores higher.'
比较两组箱形图时，既要评论中位数（哪个分布的中心更高），也要讨论四分位距或全距（哪个更分散）。务必用具体数值支撑你的观点，例如：“A组中位数为35，而B组为42，因此B组总体得分更高。”
Past papers often provide an incomplete box plot and ask you to construct the box plot for a second dataset on the same axis. Pay close attention to the scale of the graph and label the axis clearly.
真题常常给出一个不完整的箱形图，要求你在同一坐标轴上绘制第二个数据集的箱形图。要特别注意横轴的比例尺并清晰标注数轴。
4. Mean, Median and Mode from Frequency Tables | 从频数表求均值、中位数和众数
Frequency tables appear in nearly every CAIE statistics paper. To find the mean of grouped data, multiply each class midpoint (x) by its frequency (f), sum the fx values, then divide by total frequency Σf. The mode is the class with the highest frequency; for the modal class, simply state the interval.
频数表几乎出现在每份CAIE统计试卷中。计算分组数据的平均数时，用每个组的中点值 (x) 乘以频数 (f)，求出 Σfx，再除以总频数 Σf。众数所在组就是频数最高的那个区间；只需说出该区间即可。
The median from a grouped frequency table is estimated by locating the (n/2)th value. Some questions provide a cumulative frequency column to help. If not, you may need to add frequencies until you cross the halfway point and then state the corresponding interval.
分组频数表的中位数可以通过定位第 n/2 个数据来估算。有些题目会给出累积频数列；如果没有，你需要累加频数直到超过一半的位置，然后指出对应的区间。
A common exam error is using the class boundaries incorrectly when calculating the midpoint. For example, a class '20–29' includes all values from 20 to 29, so the midpoint is 24.5, not 24.
一个常见考试错误是在计算中点时错误使用组界。例如，“20–29”这一组包含了20到29的所有值，因此中点是24.5，而不是24。
5. Probability Tree Diagrams | 概率树图
Tree diagrams are heavily tested in CAIE exams, especially when events occur without replacement. You start with branches showing the probability of each outcome at the first stage, then extend branches for the second event, adjusting the probabilities if the first outcome has changed the situation.
树图在CAIE考试中出现频率很高，尤其是涉及无放回的情境。你首先画出表示第一阶段每种结果概率的分支，然后延伸出第二阶段事件的分支，并根据第一阶段结果调整概率。
Always label the branches with the probabilities as fractions or decimals, and multiply along the required path to find a combined probability. For 'at least one' type questions, it is often quicker to use the complementary probability: 1 − P(none).
务必在分支上标注分数或小数的概率，并沿所需路径相乘求出组合概率。对于“至少有一个”这类问题，利用互补概率 1 − P(一个都没有) 通常会更快。
In recent papers, candidates have been asked to complete a partially drawn tree diagram and then use it to answer several probability questions. Write probabilities in their simplest form to avoid losing accuracy marks.
在近期试卷中，考生需补全部分树图并用它回答若干概率问题。概率要写成最简分数，以免因未化简而丢分。
6. Conditional Probability with Venn Diagrams | 使用韦恩图的条件概率
e overlapping events and are often used alongside conditional probability. The formula P(A|B) = P(A ∩ appears frequently. You must be able to read sets from a diagram and substitute correctly.
韦恩图有助于形象化重叠事件，并常与条件概率搭配考查。公式 P(A|B) = P(A ∩ B) / P(B) 频繁出现。你必须能从图中读取集合信息并正确代入。
In a typical exam question, you are given the number of elements in each region of a Venn diagram and asked to find probabilities like 'a randomly chosen person likes only tea'. Remember to work with the total sample space as the denominator.
典型的考题会给出韦恩图各区域的人数，然后要求计算诸如“随机选一个人只喜欢茶”的概率。要记住将总样本空间作为分母进行计算。
Be careful with phrasing: 'A and B' means the intersection, while 'A or B' means the union. Some diagrams also show the complement, so learning to interpret shaded regions accurately is essential for conditional probability items.
要小心措辞：“A 且 B” 指交集，而 “A 或 B” 指并集。有些图还展示了补集，因此准确解读阴影区域对于条件概率题至关重要。
7. Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs test your ability to plot bivariate data and determine the type of correlation – positive, negative, or none. CAIE questions often provide a table of values and expect you to plot points accurately on a grid, then describe correlation in context.
散点图考查你绘制双变量数据并判断相关类型（正相关、负相关或无相关）的能力。CAIE题目通常会给出数值表格，要求你在网格上精确描点，然后结合实际情境描述相关性。
After plotting, you may be asked to draw a line of best fit by eye. The line should pass through the centre of the points, with roughly equal numbers of points above and below. Avoid forcing it through the origin unless the context justifies it.
描点后，可能会要求你通过目测画出最佳拟合线。这条线应穿过点的中心，上下点数大致相等。除非实际情境支持，否则不要强制通过原点。
Correlation alone does not imply causation. In extended papers, examiners expect you to recognise this and comment sensibly, e.g., 'Although there is a strong positive correlation between hours spent studying and test scores, other factors like prior knowledge also play a role.'
相关性本身并不代表因果关系。在扩展卷中，考官期望你能意识到这一点并做出合理评论，例如：“尽管学习时间与考试成绩之间存在强正相关，但如先前知识等其他因素也有影响。”
8. Line of Best Fit and Predictions | 最佳拟合线与预测
Once a line of best fit is drawn, you can use it to make predictions. Interpolation (predicting within the range of data) is generally reliable, while extrapolation (predicting beyond the data range) should be treated with caution and always noted as such.
画出最佳拟合线后，你可以用它进行预测。内插（在已有数据范围内预测）通常比较可靠，而外推（超出数据范围预测）需谨慎对待，并应明确说明这一点。
Past paper questions often ask: 'Use your line of best fit to estimate the test score for a student who studied for 5 hours.' Clearly draw lines on the graph to show your reading, and state the estimated value with units.
真题常会问：“利用你的最佳拟合线估计学习5小时的学生考试成绩。”务必在图上用线条展示读图过程，并明确写出带单位的估计值。
pers. Determine the gradient and intercept, and write the equation in the form y = mx + c.
在更高级别的试卷中，还可能考查利用趋势线上两个间距较大的点求出直线方程。计算出斜率和截距，并以 y = mx + c 的形式写出方程。
9. Stem-and-Leaf Plots and Back-to-Back Comparisons | 茎叶图及背靠背比较
Stem-and-leaf diagrams are a quick and effective way to display small sets of data. CAIE exam questions often ask you to complete a stem-and-leaf plot from a list, ensuring the leaves are in order and a key is provided.
茎叶图是展示小量数据集的快捷有效方法。CAIE考题常会要求你根据列表补全茎叶图，并确保叶的数值有序且提供图例。
Back-to-back stem-and-leaf plots allow direct comparison of two distributions sharing the same stem. Use these to identify modes, medians, and ranges, then write a short comparative sentence focusing on both central tendency and spread.
背靠背茎叶图可以让两组数据共享同一茎部，直接进行比较。利用它找出众数、中位数和全距，然后写简短比较句，同时关注集中趋势和离散程度。
A common examiner tip: when finding the median from a back-to-back plot, count the total number of leaves on each side separately. For an even number of leaves, average the two middle values – and show your working to gain full method marks.
考官常见提示：从背靠背图中找中位数时，要分别数出两侧叶子个数。若叶子数为偶数，取中间两个值的平均——并展示计算过程以获得完整方法分。
10. Interpreting Statistical Diagrams in Context | 在情境中解读统计图表
Simply producing a graph is not enough; CAIE places heavy emphasis on interpretation. You must be ready to answer questions like 'What does the interquartile range tell you about the consistency of the scores?' or 'Explain why the median is a better measure than the mean in this scenario.'
仅仅画出图表还不够，CAIE十分重视解读能力。你必须准备好回答诸如“四分位距说明了分数的稳定性怎样？”或“解释为何在此情境下中位数比平均数更合适测量”之类的问题。
When a distribution is skewed, the median is often preferred because it is not affected by extreme values. For symmetric distributions, the mean and median are close, and the mean may provide a more useful overall measure.
当分布有偏态时，通常首选中位数，因为它不受极端值影响。对于对称分布，均值与中位数接近，此时均值可提供更有用的综合衡量。
Also, be ready to criticise a given statistical claim. For example, a question might state, 'Company X says its average salary is £40,000,' but you notice the median is only £25,000; this suggests a few high earners are pulling up the mean, making it less representative.
同样，要能批判给出的统计声明。例如，题目可能说“X公司称其平均薪资为40000英镑”，但你发现中位数只有25000英镑；这表明少数高薪者拉高了均值，使得均值缺乏代表性。
11. Common Mistakes and Examiner Tips | 常见错误与考官提示
Examiner reports consistently highlight several avoidable errors. One is misreading scales on histograms or frequency polygons, particularly when there is an uneven scale. Always check the first significant mark and count divisions carefully.
考官报告反复强调几种可避免的错误。其一是误读直方图或频数多边形的坐标尺度，特别是不均匀分度时。务必检查第一个主刻度并仔细数清格数。
Another frequent mistake is confusing frequency density with frequency when drawing histograms. Students often plot the frequency directly, which is only correct if all class widths are equal. When in doubt, calculate frequency density for each bar first.
另一个常见错误是在画直方图时混淆频率密度与频数。学生常常直接绘制频数，只有在所有组距相等时这才是正确的。若有疑问，先计算出每个柱的频率密度。
In probability questions, forgetting to consider that items are not replaced leads to using the same denominator for both stages. Always reduce the denominator for the second event when sampling without replacement.
在概率题中，忘记考虑不放回情境会导致两个阶段使用相同的分母。无放回抽样时，第二阶段的分母必须相应减小。
12. Practice with a Real Exam-Style Question | 真题实战（附解析）
Let's put it all together with a question typical of a CAIE past paper: 'The table shows the masses, m kg, of 50 parcels. Draw a histogram, find an estimate for the mean mass, and construct a cumulative frequency diagram to estimate the median and interquartile range.'
让我们用一道典型的CAIE真题来综合演练：“下表显示了50个包裹的质量 m kg。绘制直方图，估算平均质量，并构建累积频率图以估计中位数和四分位距。”
y 3.33; 12 < m ≤ 16, frequency 8, density 2. Draw the histogram with mass on the horizontal axis and frequency density on the vertical axis, using a consistent scale.
首先计算每个组的频率密度：0 < m ≤ 2，频数8，密度4；2 < m ≤ 6，频数14，密度3.5；6 < m ≤ 12，频数20，密度3.33；12 < m ≤ 16，频数8，密度2。绘制直方图，横轴为质量，纵轴为频率密度，使用统一比例尺。
20×9 + 8×14 = 8 + 56 + 180 + 112 = 356. Divide by 50 to get an estimated mean of 7.12 kg.
求均值时，找出每组中点：1, 4, 9, 14。乘以频数并求和：8×1 + 14×4 + 20×9 + 8×14 = 8 + 56 + 180 + 112 = 356。除以50得到估计均值为7.12 kg。
= 25 lies between 6 and 12; reading the graph gives roughly 7.8 kg. The lower quartile (12.5) is about 3.5 kg, upper quartile (37.5) about 11 kg, so IQR ≈ 7.5 kg.
累积频数为：8, 22, 42, 50。在图中绘出上界值2, 6, 12, 16及其累积频数。绘制平滑曲线。中位数在 n/2=25 处，介于6到12之间，读图约得7.8 kg。下四分位数(12.5)约3.5 kg，上四分位数(37.5)约11 kg，因此 IQR ≈ 7.5 kg。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图