Handling Data 5: Cumulative Frequency, Box Plots and Histograms | 数据处理5：累积频率、箱形图与直方图

📚 Handling Data 5: Cumulative Frequency, Box Plots and Histograms | 数据处理5：累积频率、箱形图与直方图

In IGCSE Edexcel Mathematics, Handling Data 5 brings together three essential statistical graphs: cumulative frequency curves, box plots (also called box-and-whisker diagrams) and histograms. These tools allow you to summarise large grouped datasets, find key measures such as the median and interquartile range, and compare distributions with clarity. Whether you are drawing a smooth S-curve, interpreting a box plot's five-number summary, or working out frequency densities for unequal class widths, mastering this chapter will sharpen your data-analysis skills and boost your exam confidence. The following bilingual guide walks you through every key concept, example and common pitfall.

在IGCSE Edexcel数学中，数据处理5汇集了三种重要的统计图：累积频率曲线、箱形图（也称箱须图）和直方图。这些工具能够总结大型分组数据集，确定中位数、四分位距等关键度量，并清晰地比较分布。无论你是在绘制平滑的S形曲线、解读箱形图的五数概括，还是计算不等组距的频率密度，掌握本章都会提升你的数据分析能力并增强应考信心。以下双语指南将带你逐步掌握每个关键概念、示例和常见误区。

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1. Understanding Cumulative Frequency | 理解累积频率

Cumulative frequency is a running total of frequencies as you progress through ordered data. For grouped data, it is the sum of the frequencies for all class intervals up to the upper boundary of the current class. The final cumulative frequency always equals the total number of observations. This running-total approach helps us quickly see how many data points lie below a particular value, which is the foundation for estimating medians and quartiles from a graph.

累积频率是数据按顺序排列时频率的累加总和。对于分组数据，它是截至当前组上界之前所有组区间频率的总和。最终的累积频率始终等于观测值的总数。这种累加方法能让我们快速看出有多少数据点落在某个值以下，这是从图中估算中位数和四分位数的基础。

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2. Constructing a Cumulative Frequency Table | 构造累积频率表

To build a cumulative frequency table, add two extra columns to your grouped frequency table: one for the upper class boundary and one for the cumulative frequency. Always use the upper boundary because a value such as 109.5 includes all measurements up to that limit. Complete the cumulative frequency column by adding each frequency to the sum of all preceding frequencies. The table below summarises the lengths of 40 leaves.

要构建累积频率表，在分组频率表中额外添加两列：一列是上组界，一列是累积频率。务必使用上组界，因为例如109.5这个值包含了该界限以下的所有度量值。将每个频率与前面所有频率之和相加，即可填满累积频率列。下表汇总了40片叶子的长度。

Length (mm)	Frequency	Upper boundary (mm)	Cumulative frequency
10 – 19	4	19.5	4
20 – 29	10	29.5	14
30 – 39	15	39.5	29
40 – 49	8	49.5	37
50 – 59	3	59.5	40

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3. Drawing a Smooth Cumulative Frequency Curve | 绘制平滑的累积频率曲线

Plot the upper boundary on the horizontal axis against the cumulative frequency on the vertical axis. Always begin the curve at the lower boundary of the first class with a cumulative frequency of zero. Then plot each point and join them by hand with a smooth curve – do not use a ruler to draw straight segments. The resulting S-shape rises gently at first, becomes steeper in the middle and flattens towards the end. This shape reflects the concentration of data in the central classes.

将上组界标在横轴上，累积频率标在纵轴上。曲线始终从第一组的下组界、累积频率为零的点开始。然后标出每个点，并用手绘平滑曲线连接——不要用直尺画出直线段。结果得到的S形曲线起初缓慢上升，中间变陡，最后趋于平坦。这种形状反映出数据集中在中间组。

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4. Reading Medians, Quartiles and Percentiles from the Graph | 从图中读取中位数、四分位数和百分位数

quency, and the upper quartile (Q₃) at three-quarters. The interquartile range is IQR = Q₃ – method also works for any percentile; the kth percentile is at (k/100) × total frequency.

要找到中位数（Q₂），在纵轴上定位总频率的一半，画一条水平线与曲线相交，再向下作垂直线到横轴。读取的值就是中位数。下四分位数（Q₁）在总频率的四分之一处，上四分位数（Q₃）在四分之三处。四分位距 IQR = Q₃ – Q₁。这种方法同样适用于任何百分位数；第k百分位数位于 (k/100) × 总频率 处。

Median position = Total frequency ÷ 2

中位数位置 = 总频率 ÷ 2

IQR = Q₃ – Q₁

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5. Introduction to Box Plots | 箱形图简介

A box plot (box-and-whisker diagram) is a compact visual summary of a dataset using five key numbers: minimum, Q₁, median, Q₃ and maximum. A rectangular box is drawn from Q₁ to Q₃ with a vertical line at the median. Whiskers extend from the box to the minimum and maximum, unless outliers are present. Box plots are extremely useful for comparing two or more distributions at a glance, especially in Edexcel IGCSE exam questions that ask you to comment on central tendency and spread.

箱形图（箱须图）是利用五个关键数字对数据集进行紧凑可视化总结：最小值、Q₁、中位数、Q₃和最大值。从Q₁到Q₃绘制一个矩形盒子，并在中位数处画一条垂直线。须线从盒子延伸至最小值和最大值（除非存在异常值）。箱形图非常适合一眼比较两个或多个分布，特别是在Edexcel IGCSE考题中要求你评论集中趋势和离散程度时。

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6. Drawing and Interpreting Box Plots | 绘制与解读箱形图

Start by identifying the five-number summary. If you are given a raw data list, find the median and then the medians of the lower and upper halves to get Q₁ and Q₃ – using the (n+1)/2 rule for discrete data is common. Plot these values on a well-labelled scale. Draw the box, whiskers and median line. From the box plot you can comment on the symmetry or skewness. A median closer to Q₁ signals positive skew; closer to Q₃ indicates negative skew. Also, a long upper whisker relative to the lower whisker suggests a tail on the high end.

首先确定五数概括。若给定原始数据列表，找出中位数，然后找出下半部分和上半部分的中位数分别得到Q₁和Q₃——对离散数据通常使用 (n+1)/2 规则。将这些值标在带标签的刻度上。画出盒子、须线和中

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