📚 Cumulative Frequency and Box Plots | 累积频率与箱线图
In IGCSE Statistics, cumulative frequency graphs and box plots are essential tools for summarising large data sets. They help you quickly identify the median, quartiles, and spread of the data without needing every individual value. This guide covers everything you need to know for the Edexcel IGCSE Mathematics exam, from drawing cumulative frequency curves to interpreting and comparing box plots.
在 IGCSE 统计部分，累积频率图与箱线图是总结大型数据集的重要工具。它们能帮助你快速找到中位数、四分位数和数据的分布范围，而不需要每一个单独的数据值。本指南涵盖了 Edexcel IGCSE 数学考试所需的全部内容，包括绘制累积频率曲线以及解读和比较箱线图。
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 a data set arranged in order. For a grouped frequency table, you add each class frequency to the sum of all previous frequencies. This running total tells you how many data points lie at or below the upper boundary of that class.
累积频率是当你按顺序遍历数据时，频数的累计总和。对于分组频数表，你将每个组频数与之前所有组的频数之和相加。这个累计总和可以告诉你，有多少个数据点位于该组上边界或低于该上边界。
2. Setting Up a Cumulative Frequency Table | 制作累积频率表
Start with a grouped frequency table that includes class intervals and their frequencies. Add an extra column for cumulative frequency. The first cumulative frequency is simply the first frequency. For each subsequent class, add its frequency to the previous cumulative frequency. The final cumulative frequency must equal the total number of data points.
从包含组距和频数的分组频数表开始。额外添加一列用于累积频率。第一个累积频率就是第一个频数。对于后续的每一个组，将其频数与之前的累积频率相加。最终的累积频率必须等于数据点的总数。
| Class Interval (kg) | Frequency | Cumulative Frequency |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 12 | 5 + 12 = 17 |
| 20–30 | 20 | 17 + 20 = 37 |
| 30–40 | 18 | 37 + 18 = 55 |
| 40–50 | 5 | 55 + 5 = 60 |
3. Plotting the Cumulative Frequency Graph | 绘制累积频率图
Plot the points using the upper boundary of each class interval on the horizontal axis and the cumulative frequency on the vertical axis. The first point is plotted at the upper boundary of the first class. Join the points with a smooth curve. Always start the curve at the lower boundary of the first class with a cumulative frequency of zero, unless the data starts at zero. The graph should be an S-shaped curve (ogive) that rises from left to right.
绘制时，以每个组距的上边界为横坐标、累积频率为纵坐标描点。第一点标在第一组的上边界处。用平滑曲线连接各点。曲线通常从第一组的下边界开始，此时累积频率为零（除非数据从零开始）。图形应为一条从左向右上升的 S 形曲线（卵形线）。
4. Finding the Median from the Graph | 从图中求中位数
The median is the middle value when the data is ordered. On the cumulative frequency graph, find half of the total frequency on the vertical axis, draw a horizontal line to the curve, and then draw a vertical line down to the horizontal axis. The reading on the horizontal axis is the median. If the total frequency is n, the median is the (n/2)th value.
中位数是数据排序后位于中间的值。在累积频率图上，在纵轴上找到总频数的一半，画一条水平线与曲线相交，再向下画一条垂直线到横轴。横轴上的读数就是中位数。如果总频数为 n，则中位数是第 (n/2) 个值。
5. Finding Quartiles and Interquartile Range | 求四分位数与四分位距
The lower quartile (Q₁) is at one-quarter of the total frequency, and the upper quartile (Q₃) is at three-quarters of the total frequency. Use the same method: from these positions on the vertical axis, draw lines to the curve and then down to the horizontal axis. The interquartile range (IQR) is Q₃ – Q₁. The IQR measures the spread of the middle 50% of the data.
下四分位数 (Q₁) 位于总频数的四分之一处，上四分位数 (Q₃) 位于四分之三处。使用同样的方法：从纵轴上的这些位置画线到曲线，再向下引垂直线到横轴。四分位距 (IQR) 为 Q₃ – Q₁。IQR 衡量中间 50% 数据的离散程度。
6. Estimating Percentiles | 估算百分位数
xis, draw a horizontal line to the curve, then a vertical line down to read the value. This technique is especially useful for comparing data sets, such as finding the score below which 90% of candidates lie.
你还可以使用累积频率曲线估算任何百分位数。对于第 p 百分位数，在纵轴上找到 (p/100) × 总频数，画水平线与曲线相交，再向下画垂直线读数。这一技巧在比较数据集时特别有用，比如找出 90% 考生低于的分数。
7. Constructing a Box Plot (Box-and-Whisker Plot) | 构建箱线图（箱形图）
A box plot displays the five-number summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. Draw a horizontal scale. The box is drawn from Q₁ to Q₃ with a vertical line inside at the median. Whiskers extend from the box to the minimum and maximum values, provided they are not outliers. Outliers can be marked with a separate symbol, but at IGCSE you typically only deal with clean data without outliers.
箱线图展示了五数总结：最小值、下四分位数 (Q₁)、中位数 (Q₂)、上四分位数 (Q₃) 和最大值。画一条水平标尺。箱体从 Q₁ 画到 Q₃，箱内画一条垂直线表示中位数。须线从箱体延伸到最小值和最大值，前提是它们不是异常值。异常值可以用单独的符号标记，但在 IGCSE 阶段通常只处理没有异常值的干净数据。
8. Drawing a Box Plot from a Cumulative Frequency Graph | 从累积频率图绘制箱线图
Once you have read the median, Q₁, and Q₃ from the cumulative frequency curve, you only need the minimum and maximum data values to complete the five-number summary. These can be taken from the raw data or read from the curve at cumulative frequency 0 and n. Then draw the box plot on a suitable number line. Label the five key points clearly.
从累积频率曲线上读出中位数、Q₁ 和 Q₃ 后，你只需要数据的最小值和最大值即可完成五数总结。这些可以从原始数据中获得，或从曲线上累积频率为 0 和 n 处读取。然后在合适的数轴上绘制箱线图，并清晰地标注五个关键点。
9. Interpreting Box Plots | 解读箱线图
The box plot gives a quick visual impression of the data’s centre, spread, and skewness. The length of the box represents the IQR. If one whisker is longer than the other, the data may be skewed in that direction. A longer right whisker suggests a positive skew (more spread above the median), while a longer left whisker suggests a negative skew.
箱线图能快速让人对数据的中心、分布和偏态产生直观印象。箱体的长度代表 IQR。如果某一端的须线比另一端长，数据可能朝那个方向偏斜。右须线较长表示正偏态（中位数以上的分布较广），左须线较长则表示负偏态。
10. Comparing Distributions Using Box Plots | 使用箱线图比较分布
When comparing two or more data sets, draw their box plots on the same scale, one above the other. Compare the medians to see which group has a higher typical value. Compare the IQRs and overall ranges to discuss which group is more spread out or consistent. Always comment on both a measure of central tendency and a measure of spread, and mention skewness if relevant.
比较两个或多个数据集时，将它们的箱线图按相同比例上下并排绘制。比较中位数以确定哪一组的典型值更高。比较 IQR 和总范围，讨论哪一组更分散或更一致。始终既要评论集中趋势的度量，也要评论离散程度的度量，如果相关的话还要提及偏态。
11. Common Exam Pitfalls | 常见考试陷阱
A common mistake is plotting cumulative frequency against the midpoint of the interval instead of the upper boundary. Always use the upper boundary. Another error is forgetting to start the curve at zero cumulative frequency for the first lower boundary. Also, when finding medians or quartiles from grouped data, students sometimes use the class width incorrectly – the graph approach avoids this, but you must read the axis carefully.
一个常见错误是将累积频率相对于组中值描点，而不是上边界。务必使用上边界。另一个错误是忘记在第一个下边界处从累积频率零开始画曲线。此外，在从分组数据求中位数或四分位数时，学生有时会错误地使用组距——图形方法可以避免这一点，但你必须仔细读取轴上的数值。
12. Practice Questions to Test Yourself | 自测练习题
Find a set of grouped data (e.g., test scores or heights). Construct the cumulative frequency table, draw the graph, and determine the median, quartiles, and IQR. Then draw the box plot. Check your answers using the formula-based method if class intervals are small. Practice comparing two box plots from past Edexcel IGCSE papers to strengthen your exam technique.
找一组分组数据（例如考试成绩或身高）。制作累积频率表，绘制图形，确定中位数、四分位数和 IQR。然后画出箱线图。如果组距较小，可以用基于公式的方法检查答案。通过练习往年 Edexcel IGCSE 试卷中比较两个箱线图的题目，来强化你的考试技巧。
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