📚 Cumulative Frequency and Box Plots | 累积频率与箱形图
Cumulative frequency diagrams and box plots are essential tools for summarising and comparing grouped data in Edexcel IGCSE Mathematics. A cumulative frequency curve shows the running total of frequencies, while a box plot displays a visual five‑number summary of the data. Mastering these topics will help you interpret distributions, find medians and quartiles, and draw accurate statistical diagrams in your exam.
累积频率图和箱形图是爱德思 IGCSE 数学中总结和比较分组数据的基本工具。累积频率曲线展示频率的累加和，而箱形图则用图形化的五数概括呈现数据。掌握这些内容能帮助你在考试中解读分布、求出中位数和四分位数，并绘制准确的统计图。
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 classes in a frequency table. It tells you how many data values lie below the upper boundary of each class.
累积频率是在频率表中逐组累加得到的频率总和。它告诉你位于每组上界以下的数据值有多少个。
= 19, meaning 19 values are less than or equal to the upper boundary of the second class.
例如，若第一组的频数为 7，则其累积频率为 7。若第二组的频数为 12，则累积频率变为 7 + 12 = 19，表示有 19 个数据值小于或等于第二组的上界。
Cumulative frequency is always plotted against the upper class boundary, never the midpoint. This is crucial for drawing the graph correctly.
累积频率总是相对于组的上界绘制的，而不是组中点。正确绘制图表时必须牢记这一点。
2. Constructing a Cumulative Frequency Table | 构建累积频率表
Start with a grouped frequency table that includes class intervals and frequencies. Add a third column for cumulative frequency and fill it in by successive addition.
首先准备一张包含组区间和频数的分组频率表。增加第三列“累积频率”，并通过逐次相加填入数值。
Here is an example using test scores:
以下是一个关于测验分数的例子：
| Score, x | Frequency | Cumulative frequency |
|---|---|---|
| 0 < x ≤ 10 | 4 | 4 |
| 10 < x ≤ 20 | 7 | 11 |
| 20 < x ≤ 30 | 10 | 21 |
| 30 < x ≤ 40 | 8 | 29 |
| 40 < x ≤ 50 | 3 | 32 |
Always check that the final cumulative frequency equals the total frequency (here 32). This confirms your table is correct.
务必检查最终的累积频率是否等于总频数（此处为 32），这能确认表格的正确性。
3. Drawing a Cumulative Frequency Curve | 绘制累积频率曲线
Plot the cumulative frequency against the upper boundary of each class. Use the points (10, 4), (20, 11), (30, 21), (40, 29), (50, 32) from the table above. The graph must also start at the lower boundary of the first class with a cumulative frequency of 0, so include (0, 0).
将累积频率相对于每一组的上界描点。使用上表中的点 (10, 4), (20, 11), (30, 21), (40, 29), (50, 32)。图形还必须从第一个组的下界、累积频率为 0 开始，因此需包含 (0, 0)。
Join the points with a smooth, continuously increasing curve — never use straight line segments. The curve should be S‑shaped for most data sets, rising gently at both ends and more steeply in the middle.
用光滑、持续上升的曲线连接各点 —— 切勿使用直线段。对于大多数数据集，曲线应呈 S 形，两端平缓上升，中间较陡。
4. Finding the Median and Quartiles | 求中位数和四分位数
To estimate the median, lower quartile (LQ) and upper quartile (UQ) from the graph, first find the positions on the cumulative frequency axis:
要从图中估算中位数、下四分位数 (LQ) 和上四分位数 (UQ)，首先在累积频率轴上找到相应的位置：
Median position = n/2
中位数位置 = n/2
LQ position = n/4
下四分位数位置 = n/4
UQ position = 3n/4
上四分位数位置 = 3n/4
where n is the total frequency. Draw horizontal lines from these positions to meet the curve, then draw vertical lines down to the horizontal axis. The x‑coordinates give the median, LQ and UQ.
其中 n 为总频数。从这些位置画水平线与曲线相交，再向下画垂线至横轴，其 x 坐标即为中位数、LQ 和 UQ。
5. Interquartile Range (IQR) | 四分位距
The interquartile range is the difference between the upper and lower quartiles:
四分位距是上四分位数与下四分位数之差：
IQR = UQ – LQ
IQR = UQ – LQ
It measures the spread of the middle 50% of the data. A small IQR indicates the central values are tightly clustered; a large IQR suggests greater variability.
它衡量中间 50% 数据的离散程度。IQR 小表示中心值集中；IQR 大则表示变异性较大。
The IQR is also used to identify outliers, but at IGCSE level outliers are rarely tested for box plots — you only need the basic five‑number summary.
IQR 也用于识别异常值，但在 IGCSE 阶段，箱形图很少考查异常值 —— 你只需掌握基本的五数概括。
6. Constructing a Box Plot | 构建箱形图
A box plot (or box‑and‑whisker diagram) visually represents the five‑number summary: minimum, LQ (Q₁), median (Q₂), UQ (Q₃) and maximum. These values can be read from the cumulative frequency curve or given in a question.
箱形图（也称箱线图）用图形表示五数概括：最小值、下四分位数 (Q₁)、中位数 (Q₂)、上四分位数 (Q₃) 和最大值。这些值可从累积频率曲线读取，或由题目给出。
Draw a horizontal scale covering the data range. Draw a box from Q₁ to Q₃ and add a vertical line inside the box at the median. Extend the left whisker from Q₁ down to the minimum, and the right whisker from Q₃ up to the maximum.
画一条覆盖数据范围的横轴。从 Q₁ 到 Q₃ 画一个矩形盒子，并在盒内中位数处画一条竖线。左须线从 Q₁ 画至最小值，右须线从 Q₃ 画至最大值。
The box must be drawn accurately and clearly labelled. It is often helpful to write the key values above or below the axis.
箱体必须准确绘制并清晰标示。在轴上或轴下写出关键数值通常是很有帮助的。
7. Interpreting Box Plots | 解读箱形图
The position of the median inside the box reveals the skewness of the data. If the median is closer to the left side of the box, the data is positively skewed; if it is closer to the right side, the data is negatively skewed. A roughly centred median suggests a symmetric distribution.
中位数在箱体内的位置揭示了数据的偏态。如果中位数靠近箱体左侧，数据呈正偏态；若靠近右侧，则为负偏态；中位数大致居中则表明对称分布。
The length of the box (IQR) and the whiskers show the spread. A short box means the middle half of the data is concentrated, while long whiskers indicate values that are far from the centre.
箱体长度（IQR）和须线长度显示分散程度。短箱体表示中间一半数据集中，长须线则表明存在远离中心的值。
Always refer to the context — for example, a box plot of exam scores tells you about typical performance and range, not just statistical numbers.
务必结合具体情境 —— 例如，考试成绩的箱形图不仅提供统计数字，还能反映典型表现和分数范围。
8. Comparing Distributions Using Box Plots | 用箱形图比较分布
Box plots are excellent for comparing two or more data sets side by side. To make a comparison, comment on the median and the IQR (spread), and also mention any differences in skewness or range.
箱形图非常适合并排比较两个或多个数据集。进行比较时，需对中位数和四分位距（离散程度）加以评论，并指出偏态或全距的差异。
For instance, if the median for boys is higher than that for girls, boys typically scored higher. If the IQR for girls is larger, their scores were more spread out in the middle half.
例如，如果男生的中位数高于女生，则男生通常得分更高。如果女生的 IQR 较大，则女生成绩的中间一半更加分散。
Avoid simply listing numbers — use comparative language such as “higher median”, “smaller interquartile range” and “less consistent”.
避免仅仅罗列数字 —— 应使用比较性语言，如“中位数更高”、“四分位距更小”、“一致性较低”。
9. Exam Tips and Common Pitfalls | 考试技巧与常见错误
A common mistake is plotting cumulative frequency against class midpoints or lower boundaries. Always use upper boundaries and start the curve at the lower boundary of the first class with a cumulative frequency of 0.
常见错误是将累积频率相对于组中点或下界描点。务必使用上界，并从第一组的下界、累积频率为 0 开始绘制曲线。
Another frequent error is joining the points with straight lines instead of a smooth curve. The curve should be drawn freehand with a consistent, sweeping shape.
另一个常见错误是用直线连接各点，而非光滑曲线。曲线应用手绘出连续平滑的形状。
When reading the median and quartiles, some students forget to go from the cumulative frequency axis to the curve and then down to the data axis. Always follow the horizontal‑then‑vertical rule and show construction lines on your graph.
在读取中位数和四分位数时，一些学生忘记应从累积频率轴画至曲线、再向下画至数据轴。请务必遵循“先水平后垂直”的规则，并在图上画出作图线。
For box plots, a classic error is drawing whiskers that extend to Q₁ – e whiskers reach the minimum and maximum values of the data set, unless the question specifies otherwise.
对于箱形图，一个典型错误是将须线画至 Q₁ – 1.5×IQR 和 Q₃ + 1.5×IQR。在 IGCSE 考试中，除非题目另有说明，须线应延伸至数据集的最小值和最大值。
Always label your axes and give your box plot a title if required. Neat, clearly scaled diagrams earn additional communication marks.
始终标注坐标轴，并在需要时给箱形图加上标题。整洁、比例清晰的图表能为你赢得额外的表达分。
10. Summary | 总结
Cumulative frequency curves and box plots allow you to summarise large grouped data sets efficiently. Remember the key steps: build a cumulative frequency table using upper boundaries, draw a smooth curve, read off the five‑number summary, and construct a box plot to visualise the distribution.
累积频率曲线和箱形图能高效地总结大型分组数据集。牢记关键步骤：使用上界构建累积频率表、绘出光滑曲线、读取五数概括，并构建箱形图以可视化分布。
With careful practice and attention to detail, these statistical diagrams become a reliable source of marks in your Edexcel IGCSE Mathematics exam.
通过仔细练习并注意细节，这些统计图将成为你在爱德思 IGCSE 数学考试中可靠的得分点。
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