Year 10 CIE Statistics: In-Depth Past Paper Analysis | 十年级 CIE 统计历年真题深度解析

📚 Year 10 CIE Statistics: In-Depth Past Paper Analysis | 十年级 CIE 统计历年真题深度解析

Past papers are an essential revision tool for Year 10 CIE IGCSE Statistics. They reveal common question types, mark allocation, and examiner expectations. This article provides a deep dive into typical past paper questions, offering detailed solutions and strategic insights to help you achieve top grades.

历年真题是十年级 CIE IGCSE 统计复习的重要工具。它们显示了常见的题型、分值分配和考官的期望。本文将深入分析典型的真题题目，提供详细的解题方法和策略见解，帮助你在考试中获得高分。

1. Calculating Mean and Standard Deviation | 计算平均数与标准差

A typical exam question provides a small data set and asks for the mean and standard deviation. For example: ‘The number of books read by 8 students are: 5, 7, 4, 6, 8, 5, 3, 10. Calculate the mean and the standard deviation.’ Always show clear working to secure method marks.

典型的考题会给出一个小型数据集，要求计算平均数和标准差。例如：”8 名学生阅读的书籍数量为：5, 7, 4, 6, 8, 5, 3, 10。请计算平均数和标准差。” 必须展示清晰的解题步骤来获得过程分。

First, find the mean x̄ using the formula:

首先，使用公式求出平均数 x̄：

x̄ = Σx / n

Sum of values = 5 + 7 + 4 + 6 + 8 + 5 + 3 + 10 = 48. Number of values n = 8, so x̄ = 48 / 8 = 6.

数值之和 = 5 + 7 + 4 + 6 + 8 + 5 + 3 + 10 = 48。数据个数 n = 8，因此 x̄ = 48 / 8 = 6。

Next, compute the standard deviation using the formula for a sample:

接下来，使用样本标准差公式计算：

s = √[ Σ(x – x̄)² / (n – 1) ]

Calculate each deviation, square it, and sum them: (5-6)²=1, (7-6)²=1, (4-6)²=4, (6-6)²=0, (8-6)²=4, (5-6)²=1, (3-6)²=9, (10-6)²=16. Sum = 36. Divide by n-1 = 7 → 36/7 ≈ 5.1429. Finally, take the square root: s ≈ √5.1429 ≈ 2.27 (3 s.f.).

计算每个偏差，平方后求和：(5-6)²=1, (7-6)²=1, (4-6)²=4, (6-6)²=0, (8-6)²=4, (5-6)²=1, (3-6)²=9, (10-6)²=16。总和 = 36。除以 n-1 = 7 → 36/7 ≈ 5.1429。最后开平方：s ≈ √5.1429 ≈ 2.27（保留三位有效数字）。

Many candidates lose marks by using n instead of n-1. CIE usually expects the sample standard deviation unless stated otherwise. Always check the question wording carefully.

很多考生因为用了 n 而非 n-1 而丢分。CIE 通常要求计算样本标准差，除非题目另有说明。务必仔细审题。

2. Cumulative Frequency and Quartiles | 累积频率图与四分位数

Questions on cumulative frequency often require you to complete a table, draw a smooth curve, and then read off the median, quartiles, and interquartile range. For instance, a table gives class intervals and frequencies; you must add a cumulative frequency column and plot the upper class boundary against cumulative frequency.

累积频率的题目通常要求填写表格、绘制平滑曲线，然后读出中位数、四分位数和四分位距。例如，表格给出组距和频数；你需要加一列累积频率，并以组距上限为横坐标、累积频率为纵坐标描点画图。

To find the median, locate the value on the cumulative frequency axis at half the total frequency. For a total of 80, the median is at 40 on the vertical axis; draw a horizontal line to the curve and then a vertical line down to read the data value. The lower quartile is at 1/4 of total frequency (20), and upper quartile at 3/4 (60).

求中位数时，在累积频率轴上找到总频数一半的位置。如果总频数为 80，在纵轴 40 处画水平线与曲线相交，再向下画垂直线读取数据值。下四分位数位于总频数的 1/4 处（20），上四分位数位于 3/4 处（60）。

Interquartile range (IQR) = Upper quartile − Lower quartile. Always state the units in your answer. A common pitfall is using the midpoints or lower boundaries instead of upper boundaries; remember cumulative frequency is plotted at the upper boundary of each class.

四分位距 (IQR) = 上四分位数 − 下四分位数。答案务必注明单位。常见错误是使用了组中值或下限而不是上限；记住累积频率必须在每个组的上限处描点。

3. Probability and Tree Diagrams | 概率与树状图

CIE frequently tests probability with tree diagrams, especially when events are without replacement. A question might read: ‘A bag contains 4 red and 6 blue marbles. Two marbles are drawn at random without replacement. Find the probability that both are the same color.’

CIE 常以树状图考查概率，尤其是不放回的题目。典型问题是：”袋中有 4 个红球和 6 个蓝球。随机抽取两个球且不放回。求两个球颜色相同的概率。”

Draw a tree diagram with branches for the first pick (P(Red)=4/10, P(Blue)=6/10). For the second pick, adjust the probabilities: if red was taken first, then 3 red and 6 blue remain, giving P(Red|Red)=3/9. Similarly, compute other branches and multiply along paths. Both same color means (Red and Red) or (Blue and Blue).

画出树状图，第一抽的分支为 P(红)=4/10，P(蓝)=6/10。第二抽的概率需要调整：若第一抽为红，则剩余 3 红 6 蓝，P(红|红)=3/9。同样计算其他分支，并沿着路径相乘。同色意味着（红且红）或（蓝且蓝）。

P(same color) = P(RR) + P(BB) = (4/10 × 3/9) + (6/10 × 5/9) = 12/90 + 30/90 = 42/90 = 7/15

Always simplify fractions and show the probability as a fraction, decimal, or percentage as requested. Check that the sum of all final outcomes equals 1.

始终化简分数，并按题目要求表示为分数、小数或百分数。检查所有最终结果的概率之和是否为 1。

4. Scatter Diagrams and Correlation | 散点图与相关

A past paper task may give paired data and ask you to plot a scatter diagram, describe the correlation, and draw a line of best fit. For example, data showing study hours (x) and test score (y) for 10 students. You must plot points accurately, then position a straight line passing through the mean point (x̄, ȳ) as closely as possible to all points.

历年真题可能给出成对数据，要求绘制散点图、描述相关关系并画出最佳拟合线。例如，给出 10 名学生学习时间 (x) 与测试成绩 (y) 的数据。你必须准确描点，然后画一条最贴近所有点的直线，且该直线应通过均值点 (x̄, ȳ)。

Correlation description: positive, negative, or none. Strength: strong, moderate, weak. Avoid vague terms like ‘good’ or ‘bad’. Once the line of best fit is drawn, you can use it to estimate y given x (interpolation) or predict beyond the data range (extrapolation). Such estimates should be clearly shown by dashed lines on the graph.

描述相关性：正相关、负相关或无相关。强度：强、中等、弱。避免使用”好”或”坏”等模糊词汇。画出最佳拟合线后，可以利用它根据 x 估算 y（内插法），或在数据范围外预测（外推法）。必须在图上用虚线标出估算过程。

5. Sampling Techniques | 抽样方法

Exam questions often ask you to describe random, stratified, systematic, or quota sampling and to assess their advantages. A typical command word is ‘Describe how to obtain a stratified sample of 50 students from a school of 500, given that 300 are juniors and 200 are seniors.’

考试常要求描述随机抽样、分层抽样、系统抽样或配额抽样等方法并评估其优势。典型的指令语是：”描述如何从一所 500 名学生的学校中抽取 50 人的分层样本，已知其中 300 名为低年级学生，200 名为高年级学生。”

For stratified sampling, calculate the number from each stratum proportionally: Juniors = (300/500)×50 = 30, Seniors = (200/500)×50 = 20. Then use simple random sampling within each group (e.g., number them and use a random number generator). This ensures the sample accurately reflects the population structure.

对于分层抽样，按比例计算每层应抽取的人数：低年级 = (300/500)×50 = 30，高年级 = (200/500)×50 = 20。然后在各层内使用简单随机抽样（例如，编号并使用随机数生成器）。这能确保样本准确反映总体结构。

Always mention that the sample is representative and reduces bias. Avoid confusion with quota sampling, which is non-random but also sets quotas proportionally.

务必指出样本具有代表性且能减少偏差。注意与配额抽样的区别，后者虽也按比例设定配额，但并不是随机抽取。

6. Histograms and Frequency Density | 直方图与频率密度

Histograms are tested with unequal class widths. You must understand frequency density: frequency density = frequency ÷ class width. A question may ask you to complete a histogram from a table, or to extract frequencies from a given histogram.

直方图常用于组距不等的情况。必须理解频率密度：频率密度 = 频数 ÷ 组距。题目可能要求根据表格补全直方图，或根据给定的直方图提取频数。

Frequency density = Frequency / Class width

When drawing, the vertical axis represents frequency density, and the area of each bar is proportional to the frequency. Thus, area = class width × frequency density = frequency. Always use a ruler, label axes, and ensure bars touch.

绘图时，纵轴代表频率密度，每个柱形的面积与频数成比例。因此，面积 = 组距 × 频率密度 = 频数。必须使用直尺、标记坐标轴，并确保柱形互相接触。

7. Box Plots and Comparing Data Sets | 箱线图与数据比较

A typical exam question provides two sets of data and asks you to draw box-and-whisker plots, then compare the distributions. For instance, compare the test scores of two classes using median, IQR, and range.

典型的考试题会给出两组数据，要求绘制箱线图，然后比较分布情况。例如，用中位数、四分位距和极差比较两个班级的测试成绩。

To draw a box plot, you need the five-number summary: minimum, lower quartile, median, upper quartile, maximum. Draw a scale and mark these points, then construct the box between Q1 and Q3 with the median line inside. Whiskers extend to min and max (or to the nearest points within 1.5 IQR if outliers are considered, though Year 10 usually uses the full range).

绘制箱线图需要五数综合：最小值、下四分位数、中位数、上四分位数、最大值。画出刻度并标出这些点，在 Q1 和 Q3 之间画箱体，内部标上中位数线。须线延伸至最小和最大值（若考虑异常值则延伸到 1.5 IQR 以内的最远点，但十年级通常使用全距）。

When comparing, state which data set has a higher median and which has a greater spread (IQR and/or range). Support each statement with values. Avoid saying one is ‘better’ without context.

比较时，要说明哪组数据中位数更高，哪组分布更离散（IQR 和/或极差），用具体数值支撑陈述。不要在没有背景的情况下说哪个”更好”。

8. Stem-and-Leaf Diagrams | 茎叶图

CIE often includes stem-and-leaf questions that require you to draw an ordered diagram and then find the median, mode, and range. For data: 23, 25, 28, 31, 32, 32, 35, 40, the stem is the tens digit, leaf is the units. Always include a key and order the leaves.

CIE 茎叶图题目通常要求绘制有序茎叶图，然后求中位数、众数和极差。对于数据：23, 25, 28, 31, 32, 32, 35, 40，茎为十位数，叶为个位数。务必包括图例并对叶子排序。

Stem 2: leaves 3, 5, 8; Stem 3: leaves 1, 2, 2, 5; Stem 4: leaf 0. Key: 2|3 = 23. Median position = (n+1)/2. With 8 values, median = (28+31)/2 = 29.5. Mode = 32.

茎 2：叶 3, 5, 8；茎 3：叶 1, 2, 2, 5；茎 4：叶 0。图例：2|3 = 23。中位数位置 = (n+1)/2。8 个值的中位数 = (28+31)/2 = 29.5。众数为 32。

Stem-and-leaf diagrams keep the original data visible and make it easy to identify the shape of the distribution. Sometimes a back-to-back stem diagram is used to compare two data sets.

茎叶图能保留原始数据，并易于识别分布形态。有时会用背靠背茎叶图比较两组数据。

9. Pie Charts and Bar Charts | 饼图与条形图

Exam questions may ask you to construct a pie chart from frequency data. Convert each frequency to an angle using the formula: angle = (frequency / total frequency) × 360°. For example, if 90 students chose science out of 200, angle = (90/200)×360 = 162°.

考题可能要求根据频数数据绘制饼图。使用公式将每个频数转换为角度：角度 = (频数 / 总频数) × 360°。例如，200 名学生中有 90 人选择科学，则角度 = (90/200)×360 = 162°。

Use a protractor and label each sector clearly. For bar charts, ensure bars are of equal width when data are categorical, and there are equal gaps between bars. An examiner might ask why a pie chart is suitable – an answer is to show proportions of a whole.

使用量角器并清晰标注每个扇形。条形图在处理分类数据时须保证条形等宽，且条形之间留出相等空隙。考官可能询问饼图适用的原因——答案是可展示整体的比例构成。

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

In regression analysis, past papers ask you to draw a line of best fit on a scatter diagram and then use it for estimation. The line should pass through the mean point (x̄, ȳ) and balance points above and below. When making predictions, distinguish between interpolation (within the data range) and extrapolation (outside), noting the latter is less reliable.

在回归分析中，真题要求在散点图上画出最佳拟合线并用它进行估算。该直线应通过均值点 (x̄, ȳ)，并使两侧的点数大致平衡。进行预测时，要区分内插法（数据范围内）和外推法（范围外），并指出外推法可靠性较低。

To find the equation of the line in the form y = mx + c, select two points on the line, calculate gradient m = (y₂ − y₁)/(x₂ − x₁), and then determine c by substituting one point. Show all working clearly.

若要写成直线方程 y = mx + c，在直线上选取两点，计算斜率 m = (y₂ − y₁)/(x₂ − x₁)，再代入一点求出截距 c。过程要详细展示。

11. Exam Technique and Common Mistakes | 考试技巧与常见错误

Always read the question carefully, highlighting key instructions like ‘give your answer to 3 significant figures’ or ‘state the units’. Show all steps — even if the final answer is wrong, method marks are awarded. Use a ruler and compass for drawings, and label axes on graphs.

认真审题，圈出关键指令，如”答案保留三位有效数字”或”注明单位”。展示所有解题步骤——即使最终答案错误，也能获得过程分。绘图要使用直尺和圆规，并为图形坐标轴添加标签。

Common mistakes: confusing frequency with frequency density in histograms; plotting cumulative frequency at midpoints; using n instead of n-1 for standard deviation; misreading scales; and omitting keys or titles on statistical diagrams. Time management is crucial — allocate about 1 minute per mark.

常见错误：混淆直方图中的频数和频率密度；在组中值处描累积频率点；求标准差时分母误用 n 而非 n-1；误读坐标轴刻度；统计图表缺少图例或标题。时间管理至关重要——按每分 1 分钟左右分配答题时间。

Regular practice of past papers under timed conditions will build your confidence and speed. Review the mark schemes to understand exactly what examiners require for full marks.

定期进行定时真题训练可以提升信心和解题速度。仔细研读评分标准，准确理解考官对满分答案的要求。

Published by TutorHao | Statistics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Year 10 CIE Statistics: In-Depth Past Paper Analysis | 十年级 CIE 统计历年真题深度解析