📚 Handling Data 6: Cumulative Frequency, Box Plots & Histograms | 数据处理6：累积频数、箱线图和直方图
In IGCSE Mathematics, handling continuous data involves more than just bar charts and pie charts. When we work with grouped frequency distributions, we need advanced tools such as cumulative frequency curves, box plots and histograms. These representations help us to visualise the spread, find medians and quartiles, and compare data sets effectively. In this revision guide, we will break down the key concepts behind cumulative frequency, box-and-whisker diagrams and frequency density histograms – all essential for Edexcel IGCSE higher tier questions.
在IGCSE数学中，处理连续数据不仅仅涉及条形图和饼图。在处理分组频数分布时，我们需要累积频数曲线、箱线图和直方图等高级工具。这些表示法有助于可视化数据的离散程度、找到中位数和四分位数，并有效地比较数据集。在这份复习指南中，我们将分解累积频数、箱线图和频率密度直方图背后的核心概念——这些都是Edexcel IGCSE更高等级题目的关键内容。
1. Understanding Cumulative Frequency | 理解累积频数
Cumulative frequency is the running total of frequencies as you move through the classes in ascending order. It tells you how many observations are less than or equal to a certain value. For example, if a frequency table shows the scores of students, the cumulative frequency for the third class is the sum of the frequencies from the first class up to the third class.
累积频数是按升序遍历各组时频数的累计总和。它告诉你小于或等于某个值的观测数量。例如，如果一个频数表显示学生的分数，第三组的累积频数就是从第一组到第三组的频数之和。
2. Building a Cumulative Frequency Table | 构建累积频数表
To create a cumulative frequency table, start with a frequency distribution that includes class intervals. Add an extra column for the upper class boundary of each interval. Then, in a cumulative frequency column, write the first frequency as the initial cumulative frequency. For each subsequent row, add the frequency of that class to the previous cumulative frequency. The final total should match the total frequency.
要创建累积频数表，首先要有包含组区间的频数分布。为每个区间添加上限值列。然后，在累积频数列中，将第一个频数写为初始累积频数。对于后续每一行，将该组的频数加到前一个累积频数上。最终总和应与总频数一致。
Here is an example table for the heights of seedlings (in cm):
下面是一个幼苗高度（厘米）的示例表格：
| Height (h cm) | Frequency | Upper boundary | Cumulative frequency |
|---|---|---|---|
| 0 < h ≤ 10 | 4 | 10 | 4 |
| 10 < h ≤ 20 | 7 | 20 | 11 |
| 20 < h ≤ 30 | 15 | 30 | 26 |
| 30 < h ≤ 40 | 8 | 40 | 34 |
| 40 < h ≤ 50 | 2 | 50 | 36 |
Notice how the cumulative frequency in the final row equals the total number of seedlings (36).
注意最后一行的累积频数等于幼苗的总数（36）。
3. Plotting the Cumulative Frequency Curve | 绘制累积频数曲线
To draw a cumulative frequency diagram, plot points at the upper boundary of each class against its cumulative frequency. Join the points with a smooth curve – never straight line segments. The curve should start at the lowest boundary (where cumulative frequency is zero, unless data begins at a non-zero value) and rise to the total frequency. In the seedling example, you would begin the curve at (0,0) and plot the next point at (10,4), (20,11), (30,26), (40,34), (50,36).
要绘制累积频数图，根据每组的上限值与其累积频数描点。用平滑曲线将各点连接起来——切勿使用直线段。曲线应从最低边界开始（除非数据起始非零，累积频数为零），并上升到总频数。在幼苗示例中，曲线从(0,0)开始，然后描点(10,4)、(20,11)、(30,26)、(40,34)、(50,36)。
The use of upper class boundaries ensures that the curve accurately reflects the continuous nature of the data. A common mistake is using midpoints instead of boundaries – this would make it impossible to correctly find medians and quartiles from the graph.
使用组上限确保了曲线准确反映数据的连续性。一个常见错误是使用区间中点而不是上限——这会导致无法从图形中正确找到中位数和四分位数。
4. Finding the Median and Quartiles from the Curve | 从曲线中查找中位数和四分位数
Once the cumulative frequency curve is drawn, you can estimate the median (Q₂), lower quartile (Q₁) and upper quartile (Q₃). The median corresponds to the value on the horizontal axis at ½ of the total frequency. The lower quartile is at ¼ of the total frequency, and the upper quartile at ¾. Simply draw a horizontal line from the appropriate fraction of the cumulative frequency to the curve, then drop a vertical line to the x-axis to read the value.
累积频数曲线绘制完成后，你可以估算中位数（Q₂）、下四分位数（Q₁）和上四分位数（Q₃）。中位数对应总频数的一半在横轴上的取值。下四分位数在总频数的四分之一处，上四分位数在四分之三处。只需从累积频数轴的相应比例位置画水平线到曲线，然后向下画垂直线到x轴读取数值。
curve. The interquartile range (IQR) = Q₃ − Q₁, which measures the spread of the middle 50% of data.
对于幼苗数据（总频数=36），中位数在第18个值处，Q₁在第9个值处，Q₃在第27个值处。你会从曲线上读取对应的高度。四分位距（IQR） = Q₃ − Q₁，它衡量数据中间50%的离散程度。
5. Constructing a Box Plot (Box-and-Whisker Diagram) | 构建箱线图（箱须图）
A box plot uses the five-number summary: minimum value, Q₁, median, Q₃ and maximum value. Draw a horizontal scale, then draw a box from Q₁ to Q₃ with a vertical line inside at the median. Extend 'whiskers' from the box to the minimum and maximum values. All lengths are directly proportional to the data values.
箱线图使用五个数据概括：最小值、Q₁、中位数、Q₃和最大值。绘制一条水平标尺，然后从Q₁到Q₃画一个矩形箱，箱内在中位数处画一条垂直线。从箱体向最小值和最大值延伸出“须”。所有长度与数据值成正比。
Box plots give a clear visual summary of the distribution, making it easy to see the centre, spread and skewness. In IGCSE, you may need to draw
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