📚 Cumulative Frequency and Box Plots | 累积频率与箱线图
Cumulative frequency and box plots are essential tools in descriptive statistics that help us summarise and interpret large data sets. By constructing cumulative frequency curves, we can estimate key values such as the median and quartiles, which then allow us to draw informative box plots. These graphical techniques are particularly important for comparing distributions and identifying the spread of data. In Edexcel IGCSE Mathematics, you must be able to construct, read, and interpret cumulative frequency diagrams and box plots accurately.
累积频率和箱线图是描述性统计中的基本工具，有助于我们总结和解读大型数据集。通过绘制累积频率曲线，我们可以估算中位数和四分位数等关键数值，进而绘制信息丰富的箱线图。这些图形技术在比较分布和识别数据离散程度方面尤其重要。在 Edexcel IGCSE 数学考试中，你必须能够准确地构建、阅读和解释累积频率图和箱线图。
1. Understanding Cumulative Frequency | 理解累积频率
Cumulative frequency is the running total of frequencies up to a given value in a data set. It shows the number of observations that fall below a particular boundary. You start with the lowest class, add its frequency, then continue adding each next frequency to the previous sum. The final cumulative frequency equals the total number of data points.
累积频率是数据集中到某个给定值为止的频率累计总和。它表示落在特定边界以下的观察值数量。你从最小的组开始，加上该组的频率，然后继续将下一个频率加到此前的总和上。最后的累积频率等于数据点的总数。
2. Preparing a Cumulative Frequency Table | 准备累积频率表
To build a cumulative frequency table, begin with a grouped frequency distribution. List the class intervals and their frequencies. Add a third column for cumulative frequency. For the first row, the cumulative frequency is simply the frequency of that class. For each subsequent row, add the class frequency to the previous cumulative frequency. Ensure the upper class boundaries are used for plotting later.
要建立累积频率表，首先需要一个分组频率分布。列出组区间及其频率。增加第三列用于累积频率。第一行中，累积频率就是该组的频率。对于之后的每一行，把该组的频率加到前一个累积频率上。确保稍后绘图时使用组的上限。
Example: The table below shows the masses of 40 apples:
例如：下表显示了 40 个苹果的质量：
| Mass (g) | Frequency | Cumulative Frequency |
|---|---|---|
| 100 ≤ m < 120 | 4 | 4 |
| 120 ≤ m < 140 | 10 | 14 |
| 140 ≤ m < 160 | 13 | 27 |
| 160 ≤ m < 180 | 9 | 36 |
| 180 ≤ m < 200 | 4 | 40 |
Notice that cumulative frequency always ends at the total frequency, here 40.
请注意，累积频率始终以总频率结束，此处为 40。
3. Plotting a Cumulative Frequency Curve | 绘制累积频率曲线
Plot points using the upper boundary of each class against its cumulative frequency. For the first class 100–120, plot at (120, 4); for the second class at (140, 14); and so on. Also plot a point at the lower boundary of the first class (100, 0) to anchor the curve. Join the points with a smooth curve, not straight lines. This is called an ogive.
使用每个组的上限为横坐标，对应的累积频率为纵坐标描点。对于第一组 100–120，在 (120, 4) 描点；第二组在 (140, 14) 描点，依此类推。还要在第一组的下限 (100, 0) 处描点，以固定曲线。用光滑的曲线连接各点，而不是直线。这被称为累积频率曲线。
4. Estimating the Median from the Curve | 从曲线上估算中位数
The median is the value below which 50% of the data lies. To find it, locate half the total frequency on the cumulative frequency axis. If the total frequency is n, find n/2 on the vertical axis, draw a horizontal line to the curve, then a vertical line down to the horizontal axis. This point is the estimate of the median.
中位数是 50% 数据所在位置以下的值。要找到它，先在累积频率轴上定位总频率的一半。如果总频率是 n，在纵轴上找到 n/2，画水平线交于曲线，然后向下画垂直线到横轴。这个点就是中位数的估计值。
5. Quartiles and Interquartile Range | 四分位数与四分位距
The lower quartile (Q₁) is at n/4 on the cumulative frequency axis; the upper quartile (Q₃) is at 3n/4. Read these values from the curve using the same method as the median. The interquartile range (IQR) is Q₃ – Q₁, which represents the spread of the middle 50% of the data and is a robust measure of dispersion.
下四分位数 (Q₁) 对应累积频率轴上的 n/4；上四分位数 (Q₃) 对应 3n/4。使用与中位数相同的方法从曲线上读取这些数值。四分位距 (IQR) 为 Q₃ – Q₁，代表中间 50% 数据的离散程度，是一种稳健的离散度量。
h n = 40, median is at 20th value, Q₁ at 10th, Q₃ at 30th. Read the masses from the graph.
例如：对于苹果数据 n = 40，中位数在 20 处，Q₁ 在 10 处，Q₃ 在 30 处。从图中读取质量。
6. Drawing a Box Plot | 绘制箱线图
A box plot (or box-and-whisker plot) displays the five-number summary: minimum value, Q₁, median, Q₃, and maximum value. Draw a scale on a horizontal line. Above it, draw a box from Q₁ to Q₃ with a vertical line at the median. Extend 'whiskers' from the box to the minimum and maximum values. Box plots are particularly useful for comparing distributions side by side.
箱线图（或称盒须图）展示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。在水平线上标出刻度。在其上方，从 Q₁ 到 Q₃ 画一个盒子，并在中位数处画一条竖线。从盒子两端向最小值和最大值延伸出“触须”。箱线图特别适用于并列比较多个分布。
7. Interpreting Box Plots | 解读箱线图
The box represents the middle 50% of the data, so a small box indicates lower variability. The position of the median within the box shows skewness: if it is closer to Q₁, the distribution is positively skewed; if closer to Q₃, negatively skewed. Whisker length can hint at outliers or spread in the tails.
盒子的范围代表中间 50% 的数据，因此小盒子表示变异性较低。中位数在盒子中的位置显示偏斜程度：如果它更靠近 Q₁，分布为正偏态；如果更靠近 Q₃，则为负偏态。触须的长度可以提示异常值或尾部的离散程度。
8. Comparing Distributions Using Box Plots | 使用箱线图比较分布
When comparing two or more data sets, draw box plots on the same scale. Comment on the median (central tendency), the IQR (spread), and overall range. For example, 'Class A has a higher median score than Class B', or 'The IQR for males is larger, showing greater variability in heights'. Always back up statements with numerical values.
比较两个或多个数据集时，要在相同刻度上绘制箱线图。对中位数（集中趋势）、IQR（离散程度）和全距进行评论。例如，“A 班的中位数成绩比 B 班高”，或“男性的 IQR 更大，表明身高的变异性更大”。始终用数值支撑陈述。
9. Identifying Outliers | 识别异常值
t that lies well outside the overall pattern. A common rule is that any value less than Q₁ – s considered an outlier. In a box plot, outliers are sometimes plotted as individual dots. However, in basic IGCSE questions, outliers are usually identified from context or given information.
异常值是指远远偏离总体模式的数据点。一个常见的规则是：任何小于 Q₁ – 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的值都可能被视为异常值。在箱线图中，异常值有时会画成独立的点。但在基础的 IGCSE 题目中，异常值通常根据情境或给定信息来识别。
10. Practical Steps and Common Mistakes | 实际步骤与常见错误
When tackling exam questions, always check: have you plotted cumulative frequency against upper boundary, not mid-point? Did you include the point (lower boundary of first class, 0)? Did you join points with a smooth curve? For box plots, ensure the whiskers are not drawn inside the box and the scale is linear. A frequent mistake is confusing Q₁ with Q₃ or reading values from the wrong axis.
在处理考试问题时，务必检查：累积频率是否对应组的上限而非组中值？是否包括了点（第一组的下限，0）？是否用光滑曲线连接？对于箱线图，确保触须没有画在盒子内部，并且刻度是线性的。一个常见错误是混淆 Q₁ 与 Q₃，或者从错误的坐标轴读取数值。
11. Worked Example: Height Data | 实例分析：身高数据
The heights of 50 students were measured to the nearest cm and grouped as follows:
50 名学生的身高被测量到最接近的厘米，并分组如下：
| Height (cm) | Frequency |
|---|---|
| 140 ≤ h < 150 | 5 |
| 150 ≤ h < 160 | 18 |
| 160 ≤ h < 170 | 20 |
| 170 ≤ h < 180 | 5 |
| 180 ≤ h < 190 | 2 |
0). Median at 25th value ≈ 161 cm; Q₁ at 12.5th ≈ 154 cm; Q₃ at 37.5th ≈ 167 cm. IQR = 167 – 154 = 13 cm. Box plot: box from 154 to 167, median line at 161, whiskers to 140 and 190.
建立累积频率表，绘制曲线，估计中位数、Q₁、Q₃ 和 IQR，然后画出箱线图。解：累积频率为 5, 23, 43, 48, 50。描点：(150,5), (160,23), (170,43), (180,48), (190,50)，以及 (140,0)。中位数在 25 处 ≈ 161 cm；Q₁ 在 12.5 处 ≈ 154 cm；Q₃ 在 37.5 处 ≈ 167 cm。IQR = 167 – 154 = 13 cm。箱线图：盒子从 154 到 167，中位数线在 161，触须延伸到 140 和 190。
12. Using Cumulative Frequency and Box Plots in Context | 累积频率与箱线图的实际应用
These tools are not just exam constructs; they are widely used in quality control, economics, and social sciences. For instance, a manufacturer might use cumulative frequency to monitor product weights and box plots to compare batch consistency. Understanding how to extract and present key statistics from grouped data is a vital analytical skill developed in IGCSE Mathematics.
这些工具并非仅为考试而设；它们广泛应用于质量控制、经济学和社会科学中。例如，制造商可能用累积频率监测产品重量，并用箱线图比较批次的稳定性。从分组数据中提取并呈现关键统计量的能力，是 IGCSE 数学培养的重要分析技能。
Published by TutorHao | IGCSE Edexcel Mathematics Revision Series | aleveler.com
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