📚 Histograms | 直方图
In the Edexcel IGCSE Mathematics syllabus (4MA1), histograms appear regularly in Paper 1 and Paper 2 under the statistics and data-handling strand. Many students lose marks because they treat a histogram as a bar chart. In a histogram, the area of each bar represents the frequency, not the height. This single idea is the key to drawing, reading, and interpreting histograms correctly.
在爱德思IGCSE数学大纲（4MA1）中，直方图在Paper 1和Paper 2的统计与数据处理部分频繁出现。许多学生因为把直方图当作柱状图而失分。在直方图中，每个条形的面积代表频数，而不是高度。这一个核心概念是正确绘制、读取和解读直方图的关键。
1. What Is a Histogram? | 什么是直方图？
A histogram is a graphical representation of grouped continuous data, such as times, lengths, masses, or ages. Because the data is continuous, the bars are drawn touching one another with no gaps. The horizontal axis shows the class intervals, and the vertical axis shows the frequency density. The visual message of a histogram comes from the area of each bar: wider intervals produce wider bars, and taller bars indicate a greater frequency density.
直方图是分组连续数据（如时间、长度、质量或年龄）的图形表示。由于数据是连续的，条形之间必须紧挨着绘制，不留空隙。横轴表示组区间，纵轴表示频率密度。直方图的视觉信息来自每个条形的面积：更宽的组产生更宽的条形，而更高的条形表示更大的频率密度。
The table below summarises the differences between a histogram and a bar chart:
下表总结了直方图与柱状图之间的区别：
| Feature | Histogram | Bar Chart |
|---|---|---|
| Type of data | Continuous | Discrete or categorical |
| Gaps between bars | None | Gaps usually present |
| What the bars represent | Area = frequency | Height = frequency |
| Vertical axis label | Frequency density | Frequency |
2. The Frequency Density Formula | 频率密度公式
When class intervals are equal, you can use the frequency as the bar height. But when class intervals are unequal, you must adjust for the width. The quantity that measures how concentrated the data is within a class is called the frequency density.
当组距相等时，可以直接用频数作为条形高度。但当组距不相等时，必须考虑宽度的影响。衡量数据在一组内集中程度的量称为频率密度。
Frequency density = Frequency ÷ Class width
Frequency density = 频数 ÷ 组距
The reverse rearrangement is equally important:
其逆运算同样重要：
Frequency = Frequency density × Class width
频数 = 频率密度 × 组距
The class width is the difference between the upper class boundary and the lower class boundary. For example, the class 10 ≤ t < 15 has a width of 15 − 10 = 5. Always check the width carefully: a class written as 20–35 has width 15, not 15–1.
组距等于上组界减下组界。例如，区间 10 ≤ t < 15 的组距为 15 − 10 = 5。务必仔细判断组距：写作 20–35 的区间，其组距是15，而不是14。
3. Why Area Represents Frequency | 为什么面积代表频数
In a histogram, each bar is a rectangle whose width is the class width and whose height is the frequency density. The area of the bar is therefore:
在直方图中，每个条形都是矩形，其宽为组距，高为频率密度。因此条形的面积为：
Area = Class width × Frequency density = Frequency
面积 = 组距 × 频率密度 = 频数
This is the fundamental property of a histogram. Because frequency is represented by area, the total area of all bars equals the total frequency. When you are asked to find a frequency from a histogram, always multiply the vertical reading by the horizontal width. Never simply read the height and call it the frequency.
这就是直方图的基本性质。由于频数由面积表示，所有条形的总面积等于总频数。当题目要求从直方图中求频数时，一定要用纵轴读数乘以横轴宽度。绝不能只读取高度就当作频数。
4. Drawing a Histogram Step by Step | 分步绘制直方图
Follow this reliable sequence when drawing a histogram from grouped data:
按以下可靠顺序根据分组数据绘制直方图：
- Calculate the frequency density for every class: Frequency ÷ Class width.
- Draw the horizontal axis using a continuous scale that covers all class boundaries.
- Label the vertical axis “Frequency density”.
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