📚 Topic 159: Cumulative Frequency and Box Plots | 主题159：累积频率与箱线图
Cumulative frequency and box plots are essential tools for summarising and comparing data distributions in IGCSE Mathematics. They allow you to quickly see the spread, central tendency, and skewness of a dataset without having to examine every individual value. This article explains how to construct cumulative frequency tables and curves, how to find the median and quartiles from grouped data, and how to draw and interpret box plots – all tailored to the Edexcel IGCSE syllabus.
累积频率和箱线图是IGCSE数学中总结和比较数据分布的重要工具。它们能让你迅速了解数据集的分散程度、集中趋势和偏态分布，而无需逐一检查每个数据值。本文讲解了如何构建累积频率表和累积频率曲线，如何从分组数据中找出中位数和四分位数，以及如何绘制和解读箱线图——所有内容都紧扣Edexcel IGCSE考试大纲。
1. What is Cumulative Frequency? | 什么是累积频率？
Cumulative frequency is the running total of frequencies up to a certain point in a data set. For grouped data, it tells you how many observations fall below the upper boundary of each class interval. This running total helps you see how the data accumulates and makes it easier to estimate positional measures like the median.
累积频率是指数据集中截至某个点的频率累计总和。对于分组数据，它表示有多少观测值落在每个组区间上界以下。这种累计和有助于观察数据的累积方式，并便于估算中位数等位置量数。
2. Constructing a Cumulative Frequency Table | 构建累积频率表
Start with a grouped frequency table showing class intervals and their frequencies. Add a third column labelled 'Cumulative Frequency'. For the first row, the cumulative frequency equals the frequency of that interval. For each subsequent row, add the frequency of the current interval to the previous cumulative frequency. The final cumulative frequency must equal the total number of data values.
从一张显示组区间和频率的分组频率表开始。添加第三列，标注为“累积频率”。第一行的累积频率等于该区间的频率；每一后续行都取当前区间的频率加上前一行累积频率。最终的累积频率必须等于数据总数。
Example: Test scores of 40 students.
示例：40名学生的考试成绩。
| Marks (x) | Frequency | Cumulative Frequency |
|---|---|---|
| 0 ≤ x < 10 | 2 | 2 |
| 10 ≤ x < 20 | 5 | 7 |
| 20 ≤ x < 30 | 8 | 15 |
| 30 ≤ x < 40 | 12 | 27 |
| 40 ≤ x < 50 | 10 | 37 |
| 50 ≤ x < 60 | 3 | 40 |
3. Drawing a Cumulative Frequency Curve | 绘制累积频率曲线
Plot points using the upper class boundary of each interval on the horizontal axis and the corresponding cumulative frequency on the vertical axis. Join the points with a smooth curve – never use a ruler to join them with straight lines. The curve should always start at the lower boundary of the first class with a cumulative frequency of zero (or at a sensible origin depending on the data).
以每个区间的组上界为横坐标，以对应的累积频率为纵坐标描点。用平滑曲线连接各点——切勿用直尺画成折线。曲线应从第一个区间的下界处、累积频率为零的位置开始（或根据数据从合理的原点开始）。
4. Estimating Median and Quartiles | 估计中位数和四分位数
To find the median, locate half of the total frequency on the vertical axis (e.g. 20 for total 40) and draw a horizontal line to the curve, then drop a vertical line to read the corresponding value on the horizontal axis. The lower quartile Q₁ is found at one‑quarter of the total frequency; the upper quartile Q₃ is at three‑quarters of the total frequency. Always show construction lines on your diagram.
求中位数时，先在纵轴上找到总频率的一半（如总数为40，一半为20），画水平线交于曲线，再向下引垂直线读出横坐标。下四分位数 Q₁ 在总频率的四分之一处找到；上四分位数 Q₃ 在总频率的四分之三处。务必在图中保留作图辅助线。
Median position = n/2, Q₁ position = n/4, Q₃ position = 3n/4
中位数位置 = n/2，Q₁ 位置 = n/4，Q₃ 位置 = 3n/4
5. Interquartile Range (IQR) | 四分位距（IQR）
The interquartile range is defined as IQR = Q₃ – Q₁. It measures the spread of the middle 50% central half of the data is closely packed, while a larger IQR suggests more variability.
四分位距定义为 IQR = Q₃ – Q₁。它衡量数据中间50%的分散程度，且不受极端值影响。IQR 越小，表示数据的中央一半越集中；IQR 越大，表示变异性越强。
From our example, using estimated values Q₃ = 44.5 and Q₁ = 24:
IQR = 44.5 – 24 = 20.5
我们的示例中，根据估计值 Q₃ = 44.5 和 Q₁ = 24：
IQR = 44.5 – 24 = 20.5
6. Drawing a Box Plot (Box-and-Whisker) | 绘制箱线图（盒须图）
e: plot whiskers to the most extreme values that are within 1.5×IQR of the quartiles, and show outliers as crosses (this is examined in some higher-tier questions).
箱线图需要五个关键值：最小值、Q₁、中位数、Q₃和最大值。画一条水平数轴，从 Q₁ 到 Q₃ 画一个矩形箱体，并在中位数位置画一条垂直线；然后从箱体两端画出须线延伸到最小值和最大值。如果有异常值，Edexcel IGCSE 较高层次题目中通常采用简单规则：须线延伸到距离四分位数 1.5×IQR 以内的最远值，并用叉号标出异常值。
The five‑number summary for our example:
Minimum = 2, Q₁ = 24, Median = 35.4, Q₃ = 44.5, Maximum = 58
我们示例的五数概括如下：
最小值 = 2，Q₁ = 24，中位数 = 35.4，Q₃ = 44.5，最大值 = 58
7. Interpreting Box Plots | 解读箱线图
The box plot reveals the symmetry or skewness of the distribution. If the median line is closer to the left edge of the box, the data is positively skewed (right‑skewed). If it is closer to the right edge, the data is negatively skewed (left‑skewed). The length of the box and whiskers also indicates the spread. A short whisker combined with a long box can suggest a concentration of values near that end of the distribution.
箱线图能揭示分布的对称性或偏态。如果中位线靠近箱体左边缘，数据呈正偏态（右偏）；如果靠近右边缘，则呈负偏态（左偏）。箱体和须线的长度同样指示分散程度。短的须线加长的箱体可能意味着数值集中在分布的那一端。
8. Comparing Distributions | 比较分布
When two or more box plots are drawn on the same scale, you can directly compare medians, IQRs, and overall ranges. In Edexcel IGCSE exam questions, you are often asked to make two specific comparisons, e.g. 'Class B has a higher median mark and a larger interquartile range, indicating that on average they performed better but were less consistent.' Always support comparisons with statistical values from the box plots.
当两个或多个箱线图绘制在同一标尺上时，你可以直接比较中位数、IQR 和全距。在 Edexcel IGCSE 考试题中，通常要求写出两个具体的比较点，例如：“B班的中位数更高且四分位距更大，表明他们平均表现更好但一致性较差。”始终用从箱线图中读出的统计值来支撑比较。
9. Common Mistakes and Tips | 常见错误与技巧
Students often confuse cumulative frequency with frequency when plotting points. Remember: always use the upper class boundary for the x‑coordinate. Another common error is drawing straight lines instead of a smooth curve. When estimating quartiles, read the scale carefully and be precise. Practice tracing horizontal and vertical lines accurately. Lastly, when drawing box plots, ensure the scale is linear and the box is clearly labelled.
学生常把累积频率与频率搞混，绘图时一定要用组上界作为 x 坐标。另一个常见错误是画成折线而非平滑曲线。估算四分位数时，仔细读数并力求精确。练习准确地引出水平和垂直辅助线。此外，绘制箱线图时，确保数轴是均匀刻度，且箱体标注清晰。
10. Using Technology to Check Your Work | 使用技术检查你的作业
Modern calculators and spreadsheet software can quickly generate cumulative frequency tables and box plots. While you won't have access to these in your Edexcel IGCSE exam, using them while studying helps you confirm your hand‑drawn graphs and strengthens your understanding. Get comfortable with plotting by hand, as this is what will be tested.
现代计算器和电子表格软件可以快速生成累积频率表和箱线图。虽然在 Edexcel IGCSE 考试中无法使用这些工具，但在学习时使用它们能帮助你验证手绘图形并加深理解。要熟练手工绘图，因为这才是考试所考查的内容。
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