📚 Drawing Histograms | 绘制直方图
A histogram is a graphical representation of grouped continuous data, widely tested in the Edexcel IGCSE Mathematics exam. Unlike a bar chart, a histogram uses area to represent frequency, making it essential when class intervals have different widths. In this revision guide, you will learn how to draw and interpret histograms accurately, with a focus on calculating frequency density and avoiding common pitfalls.
直方图是分组连续数据的图形表示，在Edexcel IGCSE数学考试中广泛考查。与条形图不同，直方图用面积表示频率，因此在组距不同的情况下尤为重要。在本复习指南中，你将学习如何准确绘制和解读直方图，重点掌握频率密度的计算并避免常见错误。
1. What is a Histogram? | 什么是直方图？
A histogram is a diagram consisting of rectangles whose area is proportional to the frequency of a variable and whose width is equal to the class interval. It is used for continuous data, such as time, length, or mass, grouped into intervals.
直方图是由矩形组成的图表，矩形的面积与变量的频率成正比，矩形的宽度等于组距。它用于连续数据，如时间、长度或质量，这些数据被分入不同的区间。
The key features are: the horizontal axis shows the continuous variable with a continuous scale; the vertical axis shows frequency density, not frequency; and there are no gaps between the bars (unless a frequency is zero and the interval is absent).
主要特征是：水平轴用连续刻度表示连续变量；垂直轴表示频率密度，而非频率；条形之间没有间隙（除非频率为零且该区间缺失）。
In IGCSE, histograms are particularly important when class widths are unequal.
在IGCSE中，当组距不相等时，直方图尤为重要。
2. Histograms vs Bar Charts | 直方图与条形图的区别
Many students confuse histograms with bar charts. A bar chart is used for discrete or categorical data, has gaps between bars, and the height of each bar represents frequency. In a histogram, the data are continuous, bars touch, and the area (not height) represents frequency.
许多学生混淆直方图和条形图。条形图用于离散或分类数据，条形之间有间隙，每个条形的高度表示频率。而在直方图中，数据是连续的，条形相互接触，面积（而非高度）表示频率。
ls the frequency. For equal-width intervals, the height is proportional to frequency, but in unequal-width intervals, using frequency as height would misrepresent the data.
由于直方图的条形宽度可能不同，高度被调整为频率密度，使得面积（频率密度 × 组距）等于频率。对于等宽区间，高度与频率成正比，但在不等宽区间中，若以频率作为高度，将会歪曲数据。
3. Understanding Frequency Density | 理解频率密度
Frequency density is a calculated value that standardizes the height of each bar so that the area of the bar matches the frequency. It is found by dividing the frequency of a class by its class width.
频率密度是一个计算值，它标准化每个条形的高度，使条形的面积与频率相匹配。它由组的频率除以组距求得。
Imagine a class with a large frequency but a very wide interval. If you simply used frequency as the height, the bar would look disproportionately tall, giving a false impression. By using frequency density, the bar height is reduced to reflect the 'concentration' of data.
假设一个组频率很大但区间非常宽。如果直接使用频率作为高度，条形会显得过高，造成错误的印象。通过使用频率密度，条形高度被降低以反映数据的“集中程度”。
4. The Frequency Density Formula | 频率密度公式
The formula is straightforward:
公式很简单：
Frequency density = Frequency ÷ Class width
n the upper and lower boundaries, e.g. for the interval 20 ≤ t < 30, width = 30 − 20 = 10.
用符号表示，如果 f 是频率，w 是组距，那么频率密度 = f / w。使用正确的组距至关重要。对于连续数据，宽度是上界与下界之差，例如区间 20 ≤ t < 30，宽度 = 30 − 20 = 10。
er bound is inclusive, the width is still 20 − 10 = 10. Always check the boundaries given.
对于像 10 ≤ x ≤ 20 这样的区间，即使上界包含在内，宽度仍然是 20 − 10 = 10。始终要检查给出的边界。
5. Worked Example: Calculating Frequency Density | 计算频率密度的工作示例
Let's work through an example. The table shows the time, t minutes, spent on homework by 65 students.
我们通过一个例子来演练。表格显示了65名学生在家庭作业上花费的时间（t 分钟）。
