Year 10 CAIE Statistics: Mock Test Analysis | Year 10 CAIE 统计：单元测试模拟卷解析

📚 Year 10 CAIE Statistics: Mock Test Analysis | Year 10 CAIE 统计：单元测试模拟卷解析

This article provides a detailed walkthrough of a typical Year 10 CAIE Statistics unit test, highlighting key question types, step-by-step solutions, and common pitfalls. By working through this mock test analysis, students can strengthen their understanding of fundamental statistical concepts and improve their exam technique.

本文详细解析一份典型的 Year 10 CAIE 统计单元测试，突出关键题型、分步解题方法和常见错误。通过这份模拟卷解析，学生可以巩固基本统计概念的理解，并提高考试技巧。

1. Calculating the Mean from Grouped Data | 计算分组数据的平均数

In the mock test, Question 1 presented a frequency table of student heights. The task was to estimate the mean height. 在模拟测试中，第1题给出了一组学生身高的频数表，要求估计平均身高。

To solve, follow these steps: 解法步骤如下：

English Step	中文步骤
1. Find the midpoint of each class (e.g., 150 ≤ h < 160 → midpoint = 155).	1. 找出每个组距的中点（如 150 ≤ h < 160 → 中点 = 155）。
2. Multiply each midpoint by its frequency to get fx.	2. 将每个中点乘以其频数，得到 fx。
3. Sum all fx values and divide by total frequency (Σfx / Σf).	3. 将所有 fx 求和，除以总频数（Σfx / Σf）。

The result is an estimate because we assume all data in a class lies at the midpoint. 结果是估计值，因为我们假设该组所有数据都位于中点。

A common mistake is using class boundaries instead of midpoints, or forgetting to multiply by frequencies. 常见错误是使用组界而不是中点，或忘记乘以频数。

The formula for the estimated mean is: 估计均值的公式为：

Mean ≈ Σ(f × midpoint) / Σf

Always check that your total frequency is correct before dividing. 在相除之前，务必检查总频数是否正确。

2. Finding the Median and Mode from a Frequency Table | 从频数表找中位数和众数

The test included a discrete frequency table showing the number of books read by students in a month. 测试中包含一个离散频数表，显示学生一个月内读书的本数。

For the mode, simply identify the data value with the highest frequency. 求众数时，只需找出频数最高的数据值。

For the median, first calculate the cumulative frequency. Then locate the position using (n+1)/2 if the data is ungrouped, or n/2 for grouped data. 对于中位数，先计算累积频数。若数据未分组，用 (n+1)/2 确定位置；若分组，则用 n/2。

If the position falls within a class interval for grouped data, use linear interpolation: 如果位置落在分组数据的某个组距内，则使用线性插值：

Median = L + [ (n/2 – CF₍ₚᵣₑᵥ₎) / f₍ₘₑₔ₎ ] × w

In the mock question, the cumulative frequency just exceeded half the total at a data value of 4, so the median was 4 books. 在模拟题中，累积频数刚好在数据值4处超过总数的一半，因此中位数为4本书。

Remember: the median is a measure of central tendency resistant to outliers, whereas the mean is pulled by extreme values. 记住：中位数是一种不受异常值影响的集中趋势度量，而平均数会被极端值拉偏。

3. Drawing and Interpreting Stem-and-Leaf Diagrams | 绘制并解读茎叶图

One question required students to organise raw marks from a class test into an ordered stem-and-leaf diagram. 有一题要求学生将班级测验的原始分数整理成有序的茎叶图。

A stem-and-leaf diagram retains the original data while showing the shape of the distribution. 茎叶图保留原始数据，同时展示分布的形状。

Key steps: 关键步骤：

Choose the stem as the tens digit and the leaf as the units digit. 选择十位数作为茎，个位数作为叶。
Order the leaves in ascending order. 将叶按升序排列。
Always include a key (e.g., 5 | 2 means 52). 务必包含图例（如 5 | 2 表示 52）。

From the diagram, you can easily find the median (the middle value) and the mode. 从图中，你可以轻松找到中位数（中间值）和众数。

In the mock test, students then had to find the median, which involved counting to the ((n+1)/2)th leaf. 模拟测试中，学生接着需要找到中位数，这需要数到第 ((n+1)/2) 片叶。

Common pitfalls: forgetting to align stems, omitting a stem with no leaves, or misordering leaves. 常见陷阱：忘记对齐茎、遗漏没有叶的茎、或叶排序错误。

4. Constructing and Using Cumulative Frequency Graphs | 绘制并利用累积频率图

In this part, a grouped frequency table of reaction times was given, and candidates had to draw a cumulative frequency curve. 本题中，给出了反应时间的分组频数表，考生需要绘制累积频率曲线。

First, complete the cumulative frequency column by adding each frequency to the previous total. 首先，通过将每个频数与上一累积频数相加，完成累积频数栏。

Then plot the upper boundary of each class interval against its cumulative frequency. 然后以每个组距的上限为横坐标，对应的累积频数为纵坐标，描点。

Join the points with a smooth curve, not straight line segments. 用平滑曲线连接各点，不要用直线段。

From the graph you can estimate the median, quartiles, and percentiles. 根据图表，可以估计中位数、四分位数和百分位数。

The mock test asked for the median and the interquartile range. Locate the value at 50% of the total frequency for the median. 模拟测试要求求中位数和四分位距。在总频数的50%处定位中位数。

Always read the scale accurately and show construction lines on the graph. 务必准确读取刻度，并在图上画出构造线。

5. Understanding Histograms with Unequal Class Widths | 理解不等宽组距的直方图

A challenging question involved a histogram representing the distances travelled by cyclists, where class widths were not equal. 一道较难的题目涉及表示骑车者行驶距离的直方图，其中组距宽度不相等。

In a histogram, the area of each bar is proportional to frequency. When classes have different widths, we must use frequency density. 在直方图中，每个长条的面积与频数成正比。当组距宽度不同时，必须使用频率密度。

Frequency density = Frequency / Class width. 频率密度 = 频数 / 组距宽度。

In the question, a table needed completing with frequency densities, and then the bars were drawn. 在那道题中，需要完成一张频率密度表，然后画出长条。

Always label axes clearly: the horizontal axis with the continuous variable, and the vertical axis as ‘Frequency density’. 务必将坐标轴标注清楚：横轴是连续变量，纵轴是“频率密度”。

A common error is plotting frequency instead of frequency density on the vertical axis, which distorts the representation. 常见错误是在纵轴上绘制频数而非频率密度，这会扭曲图形表达。

6. Calculating Range and Interquartile Range | 计算极差和四分位距

Both raw data and a box-and-whisker plot were used in the mock test to assess measures of spread. 模拟测试中使用了原始数据和箱线图来评估离散程度。

The range is simply the difference between the maximum and minimum values. 极差就是最大值与最小值之差。

The interquartile range (IQR) is the difference between the upper quartile (Q3) and the lower quartile (Q1). 四分位距 (IQR) 是上四分位数 (Q₃) 与下四分位数 (Q₁) 之差。

To find Q1 and Q3 from a list of numbers, first order them, then find the medians of the lower and upper halves. 要从一列数据中找出 Q₁ 和 Q₃，先排序，再找出下半部分和上半部分的中位数。

In the question using the box plot, students had to read Q1 and Q3 directly from the graph and subtract. 在利用箱线图的题目中，学生需要直接从图上读取 Q₁ 和 Q₃，然后相减。

The IQR tells us the spread of the middle 50% of data, making it robust to outliers. 四分位距告诉我们中间50%数据的分布范围，使其对异常值具有稳健性。

Remember: always subtract (Q3 – Q1), not the reverse. 记住：总是用上四分位数减下四分位数（Q₃ – Q₁），不可颠倒。

7. Solving Basic Probability Problems | 求解基本概率问题

The probability section began with simple single-event and combined-event problems. 概率部分以简单的单一事件和组合事件问题开始。

Write down the sample space clearly. For example, when rolling a fair die, the sample space is {1,2,3,4,5,6}. 清楚地写出样本空间。例如，掷一枚均匀骰子时，样本空间是 {1,2,3,4,5,6}。

Probability of an event = Number of favourable outcomes / Total number of possible outcomes. 事件概率 = 有利结果的数量 / 所有可能结果的总数。

One question asked for the probability of picking a red card or a king from a standard deck. Use the addition rule: 一题要求从一副标准扑克牌中抽到一张红牌或一张K的概率。使用加法法则：

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Be careful not to double-count the king of hearts and king of diamonds when adding probabilities of red cards and kings. 当把红牌和K的概率相加时，小心不要重复计算红桃K和方块K。

Another common mistake is writing probabilities greater than 1 or as ratios without simplification. 另一个常见错误是写出大于1的概率，或比率未化简。

8. Using Tree Diagrams for Dependent and Independent Events | 利用树状图处理相关和独立事件

The mock test featured a tree diagram question where students had to find the probability of drawing counters without replacement. 模拟测试中有一道树状图问题，要求学生求无放回抽取筹码的概率。

Draw branches for each stage, labelling them with probabilities that sum to 1 at each node. 为每个阶段画出分支，并在每个节点标注概率，这些概率之和为1。

For independent events, the probabilities on the second set of branches are identical to the first. 对于独立事件，第二组分枝上的概率与第一组相同。

For dependent events, the probabilities on the second stage change depending on the first outcome. 对于相关事件，第二阶段的分支概率会根据第一阶段的结果而变化。

To find the combined probability of a path, multiply along the branches. 要计算一条路径的联合概率，沿分支相乘。

In the ‘without replacement’ question, after removing one counter, the denominator decreased by one. 在“无放回”的题目中，取出一个筹码后，分母减一。

A typical error is forgetting to adjust the probabilities for the second draw, which results in an incorrect tree. 一个典型错误是忘记调整第二次抽取的概率，导致树状图错误。

When the question asks for ‘at least one red’, it is often easier to work out 1 – P(no reds). 当题目要求“至少一个红球”时，通常计算 1 – P(无红球) 更容易。

9. Interpreting Scatter Graphs and Correlation | 解读散点图与相关性

A scatter graph displayed data on hours of revision and test scores. 一张散点图显示了复习小时数与测验分数之间的数据。

Describe the correlation: positive, negative, or no correlation. Also comment on its strength (strong, moderate, weak). 描述相关性：正相关、负相关或无相关。并评价其强度（强、中等、弱）。

In the mock test, the correlation was strong positive, meaning that as revision time increased, test scores tended to increase. 在模拟测试中，呈强正相关，意味着复习时间增加，测验分数也倾向于增加。

Students were then asked to draw a line of best fit by eye, balancing points above and below the line. 然后要求学生通过目测画一条最佳拟合线，平衡线上和线下的点。

The line should pass through the mean point (x̄, ȳ) if calculated, but for an estimate a reasonable straight line is sufficient. 如果已计算，该线应通过均值点 (x̄, ȳ)，但作为估计，一条合理的直线即可。

Using the line to estimate values inside the data range is called interpolation; estimating outside is extrapolation, which is less reliable. 利用该线估计数据范围内的值称为内插；估计范围外的值称为外推，后者可靠性较低。

When stating estimates from a line of best fit, always show how you read from the graph and include units. 当从最佳拟合线给出估计值时，务必展示如何从图上读取，并注明单位。

10. Choosing and Evaluating Statistical Diagrams | 选择和评价统计图表

The final question asked students to compare two different charts (a pie chart and a bar chart) presenting the same categorical data. 最后一题要求学生比较呈现相同分类数据的两种不同图表（饼图和条形图）。

A pie chart easily shows proportions of a whole, but it is difficult to compare exact frequencies. 饼图易于显示整体中的比例，但难以比较精确的频数。

A bar chart makes it easy to read individual frequencies and compare categories, but does not show the part-to-whole relationship directly. 条形图便于读取个别频数并比较类别，但不能直接显示部分与整体的关系。

When writing an evaluation, mention suitability, clarity, ease of comparison, and any misleading aspects. 撰写评价时，要提及适用性、清晰度、比较的便利性以及任何误导之处。

In the mock test, students had to justify which diagram was better for identifying which category had the mode. The bar chart was more suitable because the mode stands out immediately by bar height. 在模拟测试中，学生需要判断哪张图更适合确定哪个类别为众数。条形图更合适，因为通过条形高度可以立即看出众数。

Remember: there is no single ‘best’ diagram; it depends on what you want to show. 记住：没有唯一的“最佳”图表；这取决于你想要展示什么。

Always label all parts of a diagram—axes, bars, sectors, and include a title and key. 务必将图表的各部分标注清楚——坐标轴、条形、扇形，并包含标题和图例。

Published by TutorHao | Statistics Revision Series | aleveler.com

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Year 10 CAIE Statistics: Mock Test Analysis | Year 10 CAIE 统计：单元测试模拟卷解析