📚 Cumulative Frequency and Box Plots | 累积频数图与箱线图
Cumulative frequency graphs and box plots are essential statistical tools in the Edexcel IGCSE Mathematics syllabus. They allow you to summarise, interpret, and compare large sets of grouped or ungrouped data by identifying key values such as the median, quartiles, and range. Understanding these diagrams is crucial for scoring high marks on data-handling questions.
累积频数图和箱线图是 Edexcel IGCSE 数学大纲中必不可少的统计工具。它们通过确定中位数、四分位数和极差等关键值，帮助你总结、解释和比较大量分组或未分组数据。理解这些图形对于在数据处理题目中拿到高分至关重要。
1. Introduction to Cumulative Frequency | 累积频数简介
Cumulative frequency is the running total of frequencies as you move through the classes of a frequency table. In other words, the cumulative frequency for a particular class is the sum of all frequencies up to and including that class. This helps you see how many data values lie below a certain upper bound.
累积频数是沿着频数表的组别移动时频数的累加总和。换句话说，某个类别的累积频数就是截至该类别（含）所有频数之和。这能帮你看出有多少数据值落在某个特定上限之下。
2. Constructing a Cumulative Frequency Table | 构建累积频数表
To build a cumulative frequency table, start with your grouped frequency distribution. Add a third column labelled 'Cumulative Frequency'. For the first class, the cumulative frequency equals that class's frequency. For each subsequent class, add the current frequency to the previous cumulative frequency. Always check that the final cumulative frequency equals the total number of data values.
要构建累积频数表，从分组频数分布开始。添加第三列，标记为“累积频数”。第一组的累积频数等于该组的频数。对于之后的每一组，将当前频数加上前一行的累积频数。始终检查最后的累积频数是否等于数据值的总数。
- Example table: as frequency 7, 20 < h ≤ 30 has frequency 10. Cumulative frequencies: 4, 4+7=11, 11+10=21.
- 示例表格：幼苗高度（cm）分组：0 < h ≤ 10 频数4，10 < h ≤ 20 频数7，20 < h ≤ 30 频数10。累积频数：4，4+7=11，11+10=21。
3. Plotting a Cumulative Frequency Curve | 绘制累积频数曲线
per boundary is 20. Join the points with a smooth curve. The graph always starts at the lower boundary of the first class (or zero) with a cumulative frequency of 0, and rises to the total frequency at the last upper boundary.
将每个累积频数对应其组的上限绘制在坐标系中。上限是该组可以包含的最大值。对于 10 < h ≤ 20 组，上限是 20。用平滑曲线连接各点。图形始终从第一组的下限（或零）处开始，累积频数为 0，然后在最后一个上限处上升到总频数。
The horizontal axis represents the variable (e.g. height, mass, time) and the vertical axis represents cumulative frequency. Points should be plotted neatly, and a hand-drawn curve is required in the exam unless instructed otherwise.
横轴表示变量（如高度、质量、时间），纵轴表示累积频数。绘制点应整洁，考试中除非另有说明，否则需要手工绘制平滑曲线。
4. Finding the Median, Quartiles and Percentiles | 求中位数、四分位数和百分位数
From a cumulative frequency curve, you can estimate key statistics. To find the median (Q₂), locate half the total frequency on the vertical axis, draw a horizontal line to the curve, then drop vertically to read the corresponding value. The lower quartile (Q₁) is found at one-quarter of the total frequency, and the upper quartile (Q₃) at three-quarters. Similarly, any percentile can be estimated.
通过累积频数曲线可以估算关键统计量。要找到中位数（Q₂），在纵轴上定位总频数的一半，画水平线交于曲线，再垂直向下读取对应值。下四分位数（Q₁）在总频数的四分之一处找到，上四分位数（Q₃）在四分之三处。类似地，可以估算任何百分位数。
Q₁ = value at n/4, Q₂ = value at n/2, Q₃ = value at 3n/4 (where n is total frequency)
5. Interquartile Range and Semi-Interquartile Range | 四分位距和半四分位距
ween the upper and lower quartiles: IQR = Q₃ − Q₁. It measures the spread of the middle 50% he semi-interquartile range is half of the IQR, sometimes used as a measure of dispersion.
四分位距（IQR）是上四分位数与下四分位数之差：IQR = Q₃ − Q₁。它衡量中间 50% 数据的离散程度，并且不受极端值影响。半四分位距是 IQR 的一半，有时用作离散程度的度量。
In the Edexcel IGCSE exam, you are often asked to calculate the IQR and interpret what it tells you about the consistency of the data.
在 Edexcel IGCSE 考试中，经常要求计算 IQR 并解释它如何反映数据的一致性。
6. Introduction to Box Plots | 箱线图简介
A box plot (or box-and-whisker diagram) displays the five-number summary of a dataset: the minimum value, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum value. It provides a clear visual representation of the spread and skewness of the data.
箱线图（或称盒须图）展示了数据集的五数概括：最小值、下四分位数（Q₁）、中位数（Q₂）、上四分位数（Q₃）和最大值。它清晰地可视化出数据的分布和偏态。
- The box spans from Q₁ to Q₃, with a line inside for the median.
- 盒体从 Q₁ 延伸至 Q₃，内部有一条线表示中位数。
- The whiskers extend from the box to the minimum and maximum values (unless outliers are defined, but at IGCSE they are not).
- 须线从盒体两端延伸至最小值和最大值（IGCSE 阶段不考虑异常值）。
7. Constructing a Box Plot | 绘制箱线图
To draw a box plot, first obtain the five-number summary from the cumulative frequency curve or ordered data. Use a suitable scale on your axis (usually horizontal). Draw vertical lines at Q₁ and Q₃ and join them with horizontal lines to form a rectangle. Mark the median inside this box with a vertical line. Draw whiskers from the ends of the box to the minimum and maximum values, and add small vertical lines at those endpoints.
要绘制箱线图，首先从累积频数曲线或排序数据中获得五数概括。在坐标轴上选择合适的刻度（通常是水平轴）。在 Q₁ 和 Q₃ 处画竖线，并用水平线连接形成矩形。在盒内用竖线标出中位数。从盒体两端画须线到最小值和最大值，并在这些端点处加上小竖线。
In IGCSE, box plots are often drawn on squared paper, and accuracy in marking the scale is important. Always label the axis and give the diagram a title.
在 IGCSE 中，箱线图通常画在方格纸上，准确标记刻度很重要。务必给坐标轴加标签并给图形拟标题。
8. Interpreting Box Plots | 解读箱线图
A box plot can tell you at a glance about the central tendency, spread, and skewness of the data. If the median is not centred inside the box, the data is skewed. For instance, a median closer to the left side of the box suggests positive skew (a long tail to the right). The length of the box shows the IQR; longer boxes indicate greater variability in the middle half. The whiskers show the overall range.
箱线图能让你一眼看出数据的集中趋势、离散程度和偏态。如果中位数不在盒体中央，数据就是偏斜的。例如，中位数更靠近盒体左侧表明正偏态（右尾较长）。盒体长度表示 IQR；盒体越长说明中间半数数据的变异性越大。须线显示整体极差。
You can also compare the lengths of the whiskers; if one whisker is much longer, the data may be stretched in that direction. Always back up observations with numerical values where possible.
你还可以比较须线的长度；如果一条须线明显更长，数据可能在那个方向拉伸。可能的话，始终用数值来支持观察结果。
9. Comparing Distributions Using Box Plots | 用箱线图比较分布
Box plots are particularly useful for comparing two or more datasets on the same scale. You can compare medians to see which set has a higher central value, IQRs to assess consistency, and overall range to see total spread. For example, if dataset A has a smaller IQR than dataset B, then A is more consistent.
箱线图在相同尺度下比较两个或更多数据集时尤为有用。你可以比较中位数来观察哪一组具有更高的中心值，比较 IQR 来评估一致性，比较整体极差来看总分布。例如，如果数据集 A 的 IQR 比 B 小，那么 A 更加稳定一致。
- Always comment on at least one measure of location (median) and one measure of spread (IQR or range).
- 始终至少评论一项位置度量（中位数）和一项离散度量（IQR 或极差）。
- Use comparative language such as 'the median for group A is higher than for group B, suggesting ...'
- 使用比较性语言，例如“A组的中位数高于B组，这表明……”
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
A frequent error is plotting cumulative frequency against class midpoints instead of upper boundaries. Remember: always use the upper endpoint of the interval. Another mistake is not starting the curve at zero cumulative frequency for a value just below the first class. Also, when drawing a box plot, some students confuse Q₃ with the maximum value or forget to label the axis.
一个常见错误是将累积频数与组中值对应绘图，而不是上限。记住：始终使用区间的上端点。另一个错误是没有在第一个组稍前的值处将累积频数从零开始绘制。此外，在绘制箱线图时，有些学生将 Q₃ 与最大值混淆，或者忘记给坐标轴加标签。
To avoid these, double-check your cumulative frequency column, carefully mark your axes, and use the five-number summary exactly. Practice with past papers to build good habits.
为避免这些错误，反复检查累积频数列，仔细标记坐标轴，并精确使用五数概括。通过历年真题练习养成良好习惯。
11. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧
In Edexcel IGCSE Mathematics, questions on cumulative frequency and box plots typically carry 5–8 marks. You will often be asked to complete a table, draw the curve, find the median and IQR, and then construct a box plot. Show all working clearly. When reading values from the curve, draw dashed lines to help the examiner follow your method.
在 Edexcel IGCSE 数学考试中，关于累积频数和箱线图的题目通常占 5–8 分。你通常需要填表、绘制曲线、找出中位数和 IQR，然后绘制箱线图。清晰展示所有步骤。当从曲线读取数值时，画虚线帮助考官理解你的方法。
If you are asked to compare box plots, write at least two specific comments referencing the data. Use numerical values: 'The median for town X is 24 °C, whereas for town Y it is 18 °C, so town X is generally warmer.'
如果要求比较箱线图，至少写出两条引用数据的具体评论。使用数值：“X 城镇的中位数为 24 °C，而 Y 城镇为 18 °C，因此 X 城镇总体上更温暖。”
12. Worked Example | 详细示例
Problem: The frequency table shows the marks of 40 students in a test.
问题：频数表显示了 40 名学生的测验分数。
| Marks ≤ 10 | ≤ 20 | ≤ 30 | ≤ 40 | ≤ 50 |
| Cumulative frequency 2 | 8 | 20 | 35 | 40 |
Draw a cumulative frequency curve, estimate the median and IQR, and construct a box plot.
绘制累积频数曲线，估算中位数和 IQR，并构建箱线图。
Step 1: Plot points (10,2), (20,8), (30,20), (40,35), (50,40) on axes. Join with a smooth curve, starting from (0,0) at 0 marks.
步骤1：在坐标轴上绘制点 (10,2)、(20,8)、(30,20)、(40,35)、(50,40)。用平滑曲线连接，从 (0,0) 开始（0 分处）。
Step 2: tical axis → read across to curve → down to marks axis gives approximately 30. So Q₂ ≈ 30.
步骤2：总频数 n=40。求中位数，在纵轴上找到 40/2=20 → 水平交曲线 → 向下读到分数轴，得出约 30。因此 Q₂ ≈ 30。
Step 3: ately 38. So Q₁ ≈ 23, Q₃ ≈ 38. IQR = 38 − 23 = 15. Minimum ≈ 3 (from graph), maximum = 50.
步骤3：下四分位数在 40/4=10 处 → 曲线给出约 23。上四分位数在 3×40/4=30 处 → 曲线给出约 38。因此 Q₁ ≈ 23，Q₃ ≈ 38。IQR = 38 − 23 = 15。最小值 ≈ 3（从图中读出），最大值 = 50。
Step 4: raw box between 23 and 38 with median line at 30. Whiskers from 23 to 3 and from 38 to 50.
步骤4：在水平刻度上绘制箱线图：标记最小值 ≈ 3、Q₁=23、中位数=30、Q₃=38、最大值=50。在 23 到 38 之间画盒，中位线在 30 处。须线从 23 至 3，从 38 至 50。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图