📚 Mastering Frequency Density & Histograms (Topic 157) | 掌握频率密度与直方图（主题157）
Histograms are one of the most commonly tested statistical diagrams in the Edexcel IGCSE Mathematics syllabus. Unlike simple bar charts, histograms use the concept of frequency density to display grouped continuous data accurately, especially when class intervals are unequal. This article, structured around Topic 157, will guide you through every essential aspect of frequency density and histograms – from basic definitions to exam-style application.
直方图是Edexcel IGCSE数学大纲中最常考的统计图之一。与简单的条形图不同，直方图借助频率密度的概念来准确展示分组连续数据，特别适用于组距不等的情况。本文围绕主题157展开，将带你系统掌握频率密度与直方图的每一个关键点——从基本定义到考试型的应用。
1. What Is a Histogram? | 什么是直方图？
A histogram is a type of statistical diagram used to represent the distribution of continuous data. Data are grouped into classes (intervals), and the area of each bar is proportional to the frequency of that class. The bars in a histogram touch each other, reflecting the continuous nature of the data.
直方图是一种用于表示连续数据分布的统计图。数据被分入不同的组（区间），每个条形的面积与该组的频率成正比。直方图中的条形相互紧贴，体现了数据的连续性。
In Edexcel IGCSE, you are expected to construct histograms with equal or unequal class widths and to interpret them to find frequencies, total number of observations, or specific proportions.
在Edexcel IGCSE考试中，要求你能够绘制组距相等或不等的直方图，并能通过读图计算频率、观测总数或特定比例。
2. Bar Chart vs Histogram | 条形图与直方图的区别
Many students confuse histograms with bar charts. A bar chart is used for discrete or categorical data, with spaces between bars, and the height of the bar usually represents frequency. In a histogram, the area of the bar is what matters, and the height corresponds to frequency density, not raw frequency.
许多学生容易混淆直方图和条形图。条形图用于离散数据或分类数据，条形之间有间隙，且条形的高度通常代表频率。而在直方图中，关键是条形的面积，高度对应的是频率密度，而不是原始频率。
This distinction becomes extremely important when class intervals have different widths. Using a bar chart for unequal-width grouped data would give a misleading visual impression.
当组距不同时，这一区别就至关重要。对不等宽分组数据使用条形图会产生误导性的视觉效果。
| Bar chart | Histogram |
| For discrete/categorical data | For continuous grouped data |
| Gaps between bars | No gaps (bars touch) |
| Height = frequency | Area = frequency; height = frequency density |
中文对照：条形图高度代表频率；直方图面积代表频率，高度代表频率密度。
3. Understanding Frequency Density | 理解频率密度
Frequency density is the key concept that makes histograms accurate when class widths vary. It is defined as the frequency divided by the class width.
频率密度是使直方图在组距变化时依然准确的核心概念。它被定义为频率除以组距。
Frequency density = Frequency ÷ Class width
频率密度 = 频率 ÷ 组距
0 has a frequency of 14, the class width is 10, so the frequency density is 14 ÷ 10 = 1.4.
例如，若组 10 ≤ x < 20 的频率为 14，组距为 10，则频率密度为 14 ÷ 10 = 1.4。
Always remember: on a histogram, the vertical axis is labelled 'Frequency density', not 'Frequency'.
始终记住：直方图的纵轴标注为'频率密度'，而不是'频率'。
4. Calculating Class Width Correctly | 正确计算组距
For continuous data, boundaries must be considered carefully. If classes are written as 0–9, 10–19, the real boundaries are 0 ≤ x < 10 and 10 ≤ x < 20, giving widths of 10.
组距是区间的长度。若分组为 20 ≤ t < 30，宽度为 30 − 20 = 10。对于连续数据，必须仔细处理边界。若分组写为 0–9, 10–19，那么实际边界应为 0 ≤ x < 10 和 10 ≤ x < 20，组距均为 10。
A very common mistake is to use the difference between the highest and lowest values in a class without adjusting for continuous boundaries. Always express the class interval using inequality notation to identify the exact width.
一个非常常见的错误是直接使用组内最大值与最小值之差，而没有根据连续边界进行调整。务必用不等式表示组区间，以确定准确宽度。
5. Constructing a Histogram Step by Step | 逐步绘制直方图
Step 1: Group the continuous data into suitable classes. Ensure you know the frequency for each class.
第一步：将连续数据分入适当的组。确保知道每组的频率。
Step 2: Calculate the class width for each interval.
第二步：计算每个区间的组距。
mpute the frequency density for each class using the formula FD = frequency ÷ class width.
第三步：用公式 频率密度 = 频率 ÷ 组距 计算每组的频率密度。
Step 4: Draw axes. The horizontal axis represents the variable (with a continuous scale), and the vertical axis represents frequency density. Label them clearly.
第四步：绘制坐标轴。横轴表示变量（连续刻度），纵轴表示频率密度。清晰地标上标签。
Step 5: For each class, draw a bar whose width spans the class interval and whose height equals the frequency density. Adjacent bars must touch.
第五步：对每个组，绘制一个宽度等于该组组距、高度等于其频率密度的条形。相邻条形必须接触。
bar corresponds correctly to the frequency (area = frequency density × width = frequency).
第六步：检查每个条形的面积是否与频率一致（面积 = 频率密度 × 宽度 = 频率）。
6. Interpreting Histograms: Finding Frequencies | 解读直方图：求频率
Given a histogram, you can recover the original frequency for any class by multiplying the frequency density (the bar's height) by the class width.
给出直方图，你可以通过将频率密度（条形高度）乘以组距来恢复任何组的原始频率。
Frequency = Frequency density × Class width
频率 = 频率密度 × 组距
× 20 = 50. The total number of data values is the sum of all these individual frequencies.
若直方图中区间 30 ≤ m < 50 的条形高度为 2.5，组距为 20，则频率为 2.5 × 20 = 50。数据值的总数就是所有这些单组频率的总和。
This reverse calculation is frequently examined in IGCSE papers. You should also be able to estimate the number of observations in a given sub-interval by using proportional areas.
这种反向计算在IGCSE试卷中常考。你还需要能够通过面积比例来估计某个子区间内的观测值个数。
7. Dealing with Unequal Class Widths | 处理不等组距
When class widths are unequal, drawing a histogram without adjusting heights would make wider classes seem artificially more significant. Frequency density solves this problem: a wider class will have a lower frequency density for the same frequency, keeping the visual impression proportional to actual frequency.
当组距不等时，如果不调整高度直接绘制直方图，较宽的组会显得人为地更重要。频率密度解决了这一问题：对于相同的频率，较宽的组会有较低的频率密度，从而保持视觉效果与实际频率成正比。
In many Edexcel IGCSE questions, a table with unequal intervals is provided, and you must first calculate all frequency densities before drawing. Never skip this step.
在许多Edexcel IGCSE题目中，会提供一个组距不等的表格，你必须先计算出所有的频率密度再作图。千万不要跳过这一步。
A helpful check: the sum of all bar areas should match the total frequency of the dataset.
一个有用的检验方法：所有条形面积之和应等于数据集的总频数。
8. Worked Example: Constructing a Histogram | 例题解析：构造直方图
Example: The table shows the lengths of 60 leaves in mm.
示例：表格显示了60片叶子的长度（mm）。
| Length (x mm) | Frequency |
| 10 ≤ x < 15 | 8 |
| 15 ≤ x < 20 | 14 |
| 20 ≤ x < 30 | 24 |
| 30 ≤ x < 45 | 14 |
, 15. Then compute frequency densities: 8/5 = 1.6, 14/5 = 2.8, 24/10 = 2.4, 14/15 ≈ 0.933.
首先找出组距：5, 5, 10, 15。然后计算频率密度：8/5 = 1.6, 14/5 = 2.8, 24/10 = 2.4, 14/15 ≈ 0.933。
its frequency. The histogram accurately shows the distribution despite the varying widths.
绘制条形，其底边位于给定区间，高度等于这些频率密度。20-30区间的条形面积为 2.4 × 10 = 24，恰好等于其频率。虽然组距不同，该直方图仍准确地展示了分布。
9. Worked Example: Extracting Information from a Histogram | 例题解析：从直方图提取信息
, calculate frequency = frequency density × class width = 6 × (5-2) = 6 × 3 = 18 students.
假设直方图展示了80名学生解谜题所花的时间。对于区间 2 ≤ t < 5 分钟，条形高度为6。要计算花费2到5分钟的学生数，频率 = 频率密度 × 组距 = 6 × (5-2) = 6 × 3 = 18名学生。
idth 2), so add to the frequency of any previous interval. This proportional area method is essential for IGCSE exam success.
如果题目要求计算少于4分钟的学生总数，不能简单地读取高度。你需要结合2-5区间中至4分钟的部分面积，即面积的三分之二，因为4分钟对应该区间宽度的三分之二。估计的频率 = 6 × 2 = 12（宽度为2），加上之前区间的频率。这种面积比例法是IGCSE考试成功的关键。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Labelling the vertical axis 'Frequency' instead of 'Frequency density'. This loses marks instantly. Always write 'Frequency density' if a scale is required.
错误一：将纵轴标为'频率'而不是'频率密度'。这会立即丢分。如果需要标刻度，务必写上'频率密度'。
Mistake 2: Forgetting to use correct class boundaries for continuous data. When a class is given as 20-29, 30-39, the real boundaries are 20 and 30, giving width 10, not 9, because the data are continuous (e.g. 29.5) and are rounded.
错误二：忘记对连续数据使用正确的组边界。当分组为20-29, 30-39时，实际边界是20和30，组距为10，而不是9，因为数据是连续的（如29.5），并且经过了四舍五入。
Mistake 3: Creating gaps between histogram bars. In a histogram, bars must always touch to show the continuity of the data.
错误三：在直方图的条形之间留缝隙。直方图中条形必须始终紧贴，以体现数据的连续性。
Exam tip: Always show your working when calculating frequency densities; many marking schemes award method marks even if the final histogram has a minor scale error. And check that the total area is consistent with total frequency.
考试技巧：计算频率密度时务必展示步骤；即使最终直方图的刻度有轻微错误，许多评分方案都会给方法分。并检查总面积是否与总频数一致。
11. Connecting Histograms to Other Statistical Topics | 直方图与其他统计题型的联系
Histograms are often linked with cumulative frequency curves and frequency polygons in IGCSE questions. You might be asked to complete a cumulative frequency table from a histogram or to compare two distributions using their shapes. Understanding frequency density deepens your grasp of how grouped data behave, which is also helpful when estimating medians and quartiles.
在IGCSE试题中，直方图常与累积频率曲线和频率多边形结合考查。你可能会被要求根据直方图完成累积频率表，或通过图形形状比较两个分布。理解频率密度能加深你对分组数据特性的掌握，这在估算中位数和四分位数时也很有帮助。
Moreover, the idea of area representing frequency is conceptually similar to density in physics or probability density functions, providing a foundation for further study in mathematics and statistics.
此外，面积表示频率的理念在概念上类似于物理学中的密度或概率密度函数，为进一步学习数学和统计学奠定了基础。
12. Summary and Final Practice | 总结与最终练习
ncy by class width before plotting, and when reading a histogram, multiply the bar height by the class width to find the count. Practice with both equal and unequal intervals until these calculations become automatic.
直方图是可视化连续分组数据的强大工具。请记住：高度 = 频率密度，面积 = 频率。绘图前始终用频率除以组距，读图时用条形高度乘以组距得出计数。用等距和不等距的练习反复训练，直到这些计算变成你的本能。
For your revision, try assembling your own data set, group it with mixed widths, and construct the full histogram. Then swap with a partner and check each other's frequency density calculations and bar areas. This hands-on approach mirrors the Edexcel IGCSE requirement to both draw and interpret histograms fluently.
复习时，尝试自行收集一组数据，用混合组距分组，然后绘制完整的直方图。与同学交换，互相检查频率密度计算和条形面积。这种动手实践的方式，正好对应了Edexcel IGCSE对熟练绘制与解读直方图的要求。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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