📚 IGCSE Edexcel Maths: Cumulative Frequency & Box Plots (Topic 162) | IGCSE Edexcel 数学：累积频率与箱形图（主题162）
Cumulative frequency graphs and box plots are essential tools for summarising grouped data in Edexcel IGCSE Maths. They help you estimate medians, quartiles, and compare distributions visually. Mastering these topics will give you confidence in handling statistics questions, especially those involving large data sets and comparisons.
累积频率图和箱形图是 Edexcel IGCSE 数学中处理分组数据的重要工具。它们能帮你估计中位数、四分位数，并直观地比较数据分布。掌握这些内容将让你自信应对统计大题，尤其是涉及大样本和分布比较的考题。
1. What is Cumulative Frequency? | 什么是累积频率？
Cumulative frequency is the running total of the frequencies as you move through the class intervals in order. It tells you how many data values lie below the upper boundary of each class. This makes it easy to estimate the position of any percentile within the data set.
累积频率是指按组距顺序逐个累加各组的频数，得到的逐步总和。它表示截至每个组上限值以下共有多少个数据值，从而方便估计数据集中任意百分位数的位置。
For instance, if the first class has a frequency of 8 and the second class has 12, the cumulative frequency after the second class is 20. Unlike a frequency polygon, a cumulative frequency curve always goes upward or stays flat — it never decreases.
例如，第一组的频数为 8，第二组频数为 12，那么第二组结束时的累积频率就是 20。与频率多边形不同的是，累积频率曲线总是上升或持平，绝不会下降。
2. Constructing a Cumulative Frequency Table | 制作累积频率表
Begin with a grouped frequency table. Add a third column headed "Cumulative Frequency". In the first row, copy the frequency. For each subsequent row, add the current class frequency to the previous cumulative frequency. Always use the upper class boundary, not the midpoint, when plotting later.
从分组频率表开始，增加一列“累积频率”。第一行直接抄录原频数；之后每一行将当前组的频数与前一累积频率相加。注意，绘图时使用的是组上限，而不是组中值。
Consider the following example of marks in a test:
以下是一个测试成绩的例子：
| Marks (Class Interval) | Frequency | Upper Boundary | Cumulative Frequency |
|---|---|---|---|
| 0–10 | 4 | 10 | 4 |
| 10–20 | 8 | 20 | 12 |
| 20–30 | 12 | 30 | 24 |
| 30–40 | 10 | 40 | 34 |
| 40–50 | 6 | 50 | 40 |
The cumulative frequency after the last class must equal the total frequency (here 40). Always check this sum to avoid arithmetic mistakes.
最后一组结束后的累积频率必须等于总频率（这里是40）。务必检查该总和，避免计算错误。
3. Drawing a Cumulative Frequency Graph | 绘制累积频率图
include a point at (lower_boundary_of_first_class, 0). Join the points with a smooth curve — not straight line segments — to reflect the continuous nature of the data.
以(组上限, 累积频率)为坐标描出各组对应的点。由于第一个组从最低值开始，通常还需要包含一个点在第一组下限处，累积频率为0。用一条光滑曲线连接各点，而非直线段，以反映数据的连续性。
The graph's typical S-shape arises because cumulative frequency increases slowly at first, then more steeply where data are concentrated, and levels off again. Always label your axes ("Marks" and "Cumulative frequency", for example) and give the graph a title. Edexcel expects you to draw the curve carefully, using a sharp pencil and a ruler for the axes.
图形通常呈“S”形，因为累积频率起初增长缓慢，在数据集中区急剧上升，最后又趋于平缓。务必标注坐标轴（例如“成绩”和“累积频率”），并为图形加上标题。Edexcel 要求用尖铅笔仔细画出曲线，坐标轴用直尺绘制。
4. Quartiles and Percentiles | 四分位数与百分位数
From the cumulative frequency graph, you can estimate the median (Q₂), lower quartile (Q₁), and upper quartile (Q₃). If the total frequency is n, draw a horizontal line from 0.5n on the cumulative frequency axis to the curve, then drop down to the x-axis to read the median. Similarly, use 0.25n for Q₁ and 0.75n for Q₃.
根据累积频率图，你可以估计中位数(Q₂)、下四分位数(Q₁)和上四分位数(Q₃)。若总频数为n，从累积频率轴上0.5n处作水平线与曲线相交，再向下画垂线与x轴相交，即可读得中位数。类似地，用0.25n求Q₁，0.75n求Q₃。
The interquartile range (IQR) is calculated as:
四分位距(IQR)计算公式为：
IQR = Q₃ − Q₁
A smaller IQR means the middle 50% of the data are tightly packed; a larger IQR indicates greater spread. In exam questions, you may also be asked to find the 90th percentile — simply use 0.9n as the reading position.
IQR 越小，表示中间50%的数据越集中；IQR 越大则越分散。考试中也可能要求求第90百分位数——只需在累积频率轴上0.9n处读数即可。
5. Box Plots (Box-and-Whisker Diagrams) | 箱形图（盒须图）
a number line, then construct a box from Q₁ to Q₃ with a vertical line at the median. The 'whiskers' s within 1.5 × IQR from the quartiles, but Edexcel IGCSE usually uses the full data range.
箱形图展示五个数据摘要：最小值、下四分位数、中位数、上四分位数和最大值。先画一条数轴，然后在Q₁和Q₃处画矩形，并在中位数位置画一条竖线。“须”从矩形两端延伸至最小值和最大值（假设无异常值）。虽然有些课程规定须的末端为距四分位数1.5倍IQR范围内的最值，但Edexcel IGCSE通常使用全数据跨度。
You can construct a box plot directly from a cumulative frequency curve without plotting the original data. Identify the five key values, then draw the box plot above or below the cumulative frequency graph. This is a common exam requirement and needs to be neat and accurate.
你可以直接从累积频率曲线得出五个关键值，然后在累积频率图的上方或下方画出箱形图。这是常见的考试要求，画图必须整洁、准确。
6. Comparing Distributions Using Box Plots | 利用箱形图比较分布
When comparing two data sets, draw both box plots on the same scale. Comment on the median to compare average values, and on the IQR (or range) to compare spread. Use the term "skewness": if the median is closer to Q₁ the distribution is positively skewed; if closer to Q₃ it is negatively skewed.
比较两组数据时，将两幅箱形图画在同一尺度上。通过比较中位数来评价平均水平，通过比较IQR（或全距）来评价离散程度。还要用“偏度”来描述：中位数靠近Q₁则呈正偏态，靠近Q₃则呈负偏态。
Always use statistical terminology in your commentary: "The median mark for Class A is higher than Class B, suggesting better overall performance. However, Class B has a smaller IQR, indicating more consistent results." Edexcel mark schemes reward both comparative statements and numerical backing.
作答时务必使用统计术语：“A班的中位数高于B班，表明整体表现更好。但B班的IQR较小，说明成绩更稳定。”Edexcel评分标准既看重比较性陈述，也看重用数据佐证。
7. Worked Example | 例题解析
Question: The cumulative frequency table shows the lengths of 80 phone calls. Draw the cumulative frequency graph and estimate the median and IQR. Then construct a box plot.
题目：下表给出了80通电话的通话时长的累积频率。绘制累积频率图，估计中位数和IQR，再画出箱形图。
| Length (t minutes) | Cumulative Frequency |
|---|---|
| t ≤ 2 | 6 |
| t ≤ 4 | 18 |
| t ≤ 6 | 40 |
| t ≤ 8 | 60 |
| t ≤ 10 | 74 |
| t ≤ 12 | 80 |
n. Box plot: min = 0 min, Q₁ = 4.2, median = 6.0, Q₃ = 8.0, max = 12. (Draw appropriately)
描点 (0,0), (2,6), (4,18), (6,40), (8,60), (10,74), (12,80)，用光滑曲线连接。0.5×80=40，水平线与曲线交点 t≈6.0 分钟（中位数）。Q₁: 0.25×80=20 → t约 4.2 分钟。Q₃: 0.75×80=60 → t约 8.0 分钟。IQR = 8.0−4.2=3.8 分钟。箱形图：最小值0，Q₁=4.2，中位数6.0，Q₃=8.0，最大值12。（绘图略）
8. Common Mistakes | 常见错误
Mistake 1: Using class midpoints instead of upper boundaries for plotting. This leads to an incorrect curve shape and inaccurate quartile readings. Always use the upper boundary.
错误1: 用组中值而非组上限描点。这会导致曲线形状错误，四分位数读取不准。务必使用组上限。
Mistake 2: Drawing straight lines from point to point. The cumulative frequency graph must be a smooth curve. A jagged line suggests discrete data handling, which loses marks in continuous data contexts.
错误2: 用直线段连接各点。累积频率图必须是光滑曲线。锯齿状线条暗示处理成离散数据，在连续数据的题目中会被扣分。
ting to include the (lower_boundary, 0) point. If you omit it, the origin of the curve isn't anchored and median estimates may be overestimated for small values.
错误3: 漏画(最低组下限, 0)的点。如果遗漏，曲线起点未定，可能导致对小数值的中位数估计偏高。
Mistake 4: When finding quartiles, using frequencies instead of cumulative frequencies. The reading must be based on cumulative frequency axis, not the frequency axis.
错误4: 求四分位数时用频率代替累积频率。读数必须基于累积频率轴，而不是频率轴。
9. Exam Tips | 考试技巧
Tip 1: In Edexcel IGCSE, the cumulative frequency curve is often drawn on the provided grid; use a sharp pencil and check that your points are correctly plotted. Spend a moment verifying key coordinates.
技巧1: Edexcel IGCSE 考试中，累积频率曲线通常画在给定的网格纸上；应使用尖铅笔，并检查描点是否准确。花点时间验证关键坐标。
Tip 2: Write down the five-number summary clearly before drawing the box plot. Label the scale on the box plot axis — often the same axis as the cumulative frequency graph, but below or above it.
技巧2: 在画箱形图前，先清晰地写下五数摘要。箱形图的坐标轴要标注刻度——通常与累积频率图共用x轴，但画在下方或上方。
Tip 3: When comparing distributions, always quote values. Don't just say "Class A performed better"; say "Class A has a higher median (65 marks) than Class B (58 marks)". Specificity earns full marks.
技巧3: 比较分布时，务必引用数值。不要只说“A班表现更好”，而要说“A班的中位数（65分）高于B班（58分）”。具体化的表述能获得满分。
Tip 4: If asked about skewness, link it to the box plot: a longer whisker on the high side suggests positive skew, and a median closer to Q₁ supports that. Mention both visual and numerical evidence.
技巧4: 若要求讨论偏度，应联系箱形图：高端须较长表明正偏态，同时中位数靠近Q₁也支持这一判断。视觉与数值证据都要提及。
10. Summary | 总结
Cumulative frequency graphs transform grouped data into a curve from which you can read off key statistics. Box plots then condense those readings into a compact visual summary. Together, they allow you to analyse dispersion, central tendency, and skewness, and to compare multiple distributions effectively.
累积频率图将分组数据转化为一条曲线，从中可读取关键统计量。箱形图则把这些读数浓缩成简洁的视觉摘要。两者结合，使你能够分析离散程度、集中趋势和偏态，并有效比较多组数据的分布。
In Edexcel IGCSE, questions on these topics are often worth several marks and appear in the higher tier. Precision in plotting, accurate reading of the scale, and clear interpretative statements will secure top grades. Practise with past papers, paying attention to the mark schemes' demand for numerical evidence in comparisons.
在 Edexcel IGCSE 中，此类题目常占多分，且出现在提高卷里。精确绘图、准确读取刻度，以及清晰的解释性陈述将助你拿下高分。请用历年真题进行练习，特别注意评分标准中要求用数值证据进行比较。
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