📚 Handling Data 7: Cumulative Frequency and Box Plots | 数据处理 7：累积频率与箱形图
Handling Data 7 focuses on two powerful tools for understanding distributions: cumulative frequency graphs and box plots. These methods let you estimate the median, quartiles, and interquartile range, and they are essential for comparing two or more data sets on the IGCSE Edexcel Mathematics course.
数据处理 7 重点讲解理解数据分布的两个强大工具：累积频率图和箱形图。这些方法可以估算中位数、四分位数和四分位距，在 Edexcel IGCSE 数学考试中，也是比较两组或多组数据集的核心内容。
1. What is Cumulative Frequency? | 什么是累积频率？
Cumulative frequency is a running total of frequencies. For each class interval, the cumulative frequency tells you how many data values are less than or equal to the upper boundary of that interval.
累积频率是频率的累加总和。对于每个组区间，累积频率表示小于或等于该区间上边界的数据个数。
To calculate it, add the frequencies together in order, keeping a running total. Here is an example using ages of 70 people:
计算时按顺序把各组的频率依次相加，并保持一个累加总数。下面是一个使用 70 位年龄数据的例子：
| Age (years) | Frequency | Cumulative frequency |
| 0–10 | 8 | 8 |
| 10–20 | 15 | 23 |
| 20–30 | 22 | 45 |
| 30–40 | 19 | 64 |
| 40–50 | 6 | 70 |
The last cumulative frequency always equals the total number of data values. In this example, the total is 70.
最后一个累积频率一定等于数据总个数。在这个例子中，总数为 70。
2. Drawing a Cumulative Frequency Graph | 绘制累积频率图
To plot a cumulative frequency graph, use the upper class boundary of each interval on the x-axis and the cumulative frequency on the y-axis. For example, for the interval 10–20, plot the point (20, 23).
绘制累积频率图时，x 轴应使用每个区间的上边界，y 轴使用累积频率。例如，区间 10–20 应描点 (20, 23)。
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Choose suitable scales that allow all points to fit comfortably.
选择合适的坐标比例，使所有点都能比较宽松地画在图上。
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Plot the cumulative frequency against the upper boundary of each class.
把每个区间的累积频率对应该区间的上边界描点。
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Join the points with a smooth curve or straight-line segments; never use a bar chart.
用平滑曲线或直线段把相邻点连接起来；不要画成条形图。
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Label both axes clearly and give the graph a title.
清晰标注两个坐标轴，并给图表加上标题。
Sometimes the first point starts at the lower boundary of the first class with cumulative frequency 0, for example (0, 0). This makes the graph start from the origin.
有时为了让图形从原点开始，会从第一个区间的下边界处取累积频率 0，例如点 (0, 0)。
3. Finding the Median from a Cumulative Frequency Graph | 从累积频率图中求中位数
rve, then a vertical line down to the x-axis. The x-coordinate at this point is the median.
设 n 为总频率。中位数是第 n/2 个数据值。在累积频率图上，从 y = n/2 处画一条水平线与曲线相交，再垂直向下到 x 轴，交点的 x 坐标就是中位数。
If you are using a grouped frequency table and no graph, you can estimate the median by linear interpolation:
如果使用分组频数表而不作图，也可以用线性插值来估算中位数：
Median = L + ( (n/2 − F) / f ) × w
where L is the lower class boundary of the median class, F is the cumulative frequency before that class, f is the frequency of the median class, and w is the class width.
其中 L 是中位数所在组的下边界，F 是该组之前的累积频率，f 是该组的频率，w 是组距。
4. Quartiles and the Interquartile Range | 四分位数与四分位距
tive frequency graph, use the same method as for the median but with y = n/4 and y = 3n/4.
下四分位数 Q1 是第 n/4 个数据值，上四分位数 Q3 是第 3n/4 个数据值。在累积频率图中，分别用 y = n/4 和 y = 3n/4 重复求中位数的方法即可。
The interquartile range (IQR) measures the spread of the middle 50% of the data:
四分位距（IQR）用于衡量中间 50% 数据的离散程度：
IQR = Q3 − Q1
The IQR is less affected by extreme values than the full range. This makes it a more reliable measure of spread when outliers are present.
与极差相比，四分位距受极端值影响更小，因此在存在离群值时，它是更可靠的离散程度度量。
5. Drawing Box Plots | 绘制箱形图
A box plot (also called a box-and-whisker plot) is a visual summary of five key values: the minimum value, Q1, median, Q3, and maximum value.
箱形图（也叫箱线图）用五个关键值来概括数据：最小值、Q1、中位数、Q3 和最大值。
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Draw a horizontal axis covering the full range of data.
先画一条能覆盖整个数据范围的横轴。
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Draw a box from Q1 to Q3.
画一个从 Q1 到 Q3 的矩形箱体。
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Draw a vertical line inside the box at the median.
在箱体内部中位数位置画一条竖线。
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Draw whiskers from Q1 to the minimum and from Q3 to the maximum.
从 Q1 向最小值画左须，从 Q3 向最大值画右须。
Box plots are excellent for comparing two or more data sets on the same scale.
在同一坐标轴上，箱形图非常适合互相对比两组或多组数据。
6. Comparing Data Using Box Plots | 使用箱形图比较数据
When comparing two distributions, always comment on both an average and a measure of spread. Use the context of the data in your answer.
比较两组分布时，既要评论平均数（或中心趋势），也要评论离散程度，并且要把结论放到数据的具体背景中。
For example, if one box plot has a higher median, that data set tends to be larger overall. If one box plot has a larger interquartile range, its middle 50% of values are more spread out.
例如，如果一个箱形图的中位数更高，说明该组数据的整体数值更大。如果四分位距更大，则说明其中间 50% 的数据更加分散。
Remember to mention the range as well: a larger range means a wider spread across the whole data set, but it can be affected by outliers.
同时也要提及全距：全距越大，说明整组数据的波动范围越大，但全距容易受到离群值影响。
7. Interpreting Cumulative Frequency Graphs | 解读累积频率图
Cumulative frequency graphs allow you to estimate how many data values lie below or above a given value. For a given value x, read up from x to the curve, then across to the y-axis to find the cumulative number less than or equal to x.
累积频率图可以估算小于或大于某个数值的数据个数。对于给定的数值 x，从 x 向上读曲线，再水平对应到 y 轴，即可得到小于或等于 x 的累积个数。
You can also estimate percentiles. The pth percentile is the value below which p% of the data lie. For example, the 90th percentile is found using y = 0.90n.
还可以估算百分位数。第 p 百分位数表示 100 个数据中有 p 个数据都小于它的值。例如，第 90 百分位数可用 y = 0.90n 来查找。
These interpretations are common in longer IGCSE exam questions, so practise reading graphs carefully and rounding your answers sensibly.
这些解读在 IGCSE 考试的长题中非常常见，所以务必练习精确读图，并注意合理取近似值。
8. Worked Exam-Style Example | 考试风格例题
The table below shows the time, in minutes, that 80 students spent on homework one evening.
下表显示了 80 名学生某晚做家庭作业的时间（分钟）。
| Time (minutes) | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
| Frequency | 6 | 12 | 18 | 24 | 14 | 6 |
| Cumulative frequency | 6 | 18 | 36 | 60 | 74 | 80 |
To find the median, use n = 80, so n/2 = 40. The 40th student lies in the class 30–40. Using interpolation:
求中位数时，n = 80，所以 n/2 = 40。第 40 名学生位于 30–40 这一组。使用插值法：
Median ≈ 30 + ( (40 − 36) / 24 ) × 10 = 30 + 1.67 = 31.7 minutes
Similarly, Q1 is the 20th value, which lies in the 20– ndary 40, so Q3 = 40 minutes. The IQR is therefore approximately 40 − 21.1 = 18.9 minutes.
类似地，Q1 是第 20 个数据，位于 20–30 组：Q1 ≈ 20 + ( (20 − 18) / 18 ) × 10 ≈ 21.1 分钟。Q3 是第 60 个数据，正好在边界 40 处，所以 Q3 = 40 分钟。因此 IQR 约为 40 − 21.1 = 18.9 分钟。
These values can then be used to draw a box plot on a suitable scale.
得到这些数值后，就可以选择合适的比例尺绘制箱形图。
9. Common Misconceptions and Pitfalls | 常见误区与易错点
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Plotting at the midpoint instead of the upper class boundary.
误用组中值，而不使用上边界来描点。
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Forgetting to take the square root in IQR? No, IQR is simply Q3 − Q1; many students confuse it with standard deviation.
把四分位距 IQR 误认为是标准差；实际上 IQR 就是 Q3 − Q1。要熟练掌握中位数、四分位数的定义。
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Using the lower boundary when reading the median from the graph.
在累积频率图上求中位数时，错误使用下边界对应当前的 y 值。
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Joining points with a curve that does not pass through all plotted points.
连接各点时画的曲线没有穿过所描的所有点。
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Drawing a box plot with whiskers to the class limits instead of the actual minimum and maximum.
绘制箱形图时，把须画到组边界，而不是真正的最小值和最大值。
To avoid these mistakes, always label your axes, check the direction of cumulative frequency, and practise with past exam papers.
要避免上述错误，请务必标注坐标轴、检查累积频率的递增方向，并用真题加强练习。
10. Summary and Key Formulas | 总结与关键公式
Cumulative frequency and box plots provide a reliable way to summarise, interpret, and compare data distributions. Remember the following key ideas:
累积频率和箱形图为总结、解读和比较数据分布提供了可靠的方法。请记住以下要点：
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Plot cumulative frequency against upper class boundaries.
用上边界与累积频率描点。
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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。
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IQR = Q3 − Q1.
四分位距 IQR = Q3 − Q1。
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A box plot shows min, Q1, median, Q3, max.
箱形图显示最小值、Q1、中位数、Q3 和最大值。
Use these tools together: cumulative frequency graphs are for reading values accurately, while box plots are for comparing distributions visually.
要把两种工具结合使用：累积频率图用于准确读取数值，箱形图用于直观比较分布。
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