📚 230: Histograms and Frequency Density | 230：直方图与频率密度
Histograms are one of the most powerful tools for displaying grouped continuous data in IGCSE Mathematics. Unlike bar charts, a histogram uses area to represent frequency, making it essential to understand frequency density when class widths vary. This article covers every key concept you need to master for Edexcel IGCSE, from drawing accurate histograms to interpreting and estimating statistics from them.
直方图是 IGCSE 数学中展示分组连续数据最有力的工具之一。与条形图不同，直方图用面积来表示频数，因此当组距不相等时，理解频率密度至关重要。本文涵盖你为 Edexcel IGCSE 需要掌握的每一个关键概念，从绘制精确的直方图到解读并从中估计统计量。
A histogram displays frequency data for continuous variables. The horizontal axis shows the class boundaries, and the vertical axis represents frequency density, not the raw frequency. The area of each bar is proportional to the frequency of that class. This means taller or wider bars indicate higher frequency, but you should never judge by height alone unless all class widths are equal.
直方图展示连续变量的频数数据。横轴显示组界，纵轴表示频率密度，而不是原始频数。每个柱形的面积与该组的频数成正比。这意味着较高或较宽的柱形表示较高的频数，但除非所有组距都相等，否则绝不能仅凭高度来判断。
1. What is a Histogram? | 什么是直方图？
A histogram is a graphical representation of the distribution of numerical data where the data is grouped into intervals. It looks similar to a bar chart, but the bars touch each other to indicate the continuous nature of the data. The key feature is that the area of each bar is equal to the frequency of the observations in that interval.
直方图是将数值数据分组后用图形表示其分布的方式。它看起来与条形图相似，但柱形彼此紧贴，以表明数据的连续性。关键特征是每个柱形的面积等于该区间内观测值的频数。
For Edexcel IGCSE, you will often deal with unequal class widths. In such cases, using frequency on the vertical axis would be misleading because a wider interval would automatically have a larger count, even if the density of data is lower. That is why we use frequency density.
在 Edexcel IGCSE 考试中，你经常会遇到不等组距的情况。此时，如果在纵轴上使用频数会产生误导，因为较宽的区间即使数据密度较低，其计数也会自动变大。这就是我们使用频率密度的原因。
2. Frequency Density Formula | 频率密度公式
The relationship connecting frequency, frequency density, and class width is fundamental. You must memorise it and be able to rearrange it with confidence.
连接频数、频率密度和组距的关系是基础中的基础。你必须记住它，并能自信地进行公式变形。
Frequency density = frequency ÷ class width
频率密度 = 频数 ÷ 组距
This formula can be rearranged into two other useful forms: Frequency = frequency density × class width and Class width = frequency ÷ frequency density. In examinations, you will often be given a table with some gaps, and you need to apply the correct version to complete it.
该公式可以变形为另外两种有用的形式：频数 = 频率密度 × 组距 以及 组距 = 频数 ÷ 频率密度。在考试中，你常常会拿到一张有缺项的表格，需要运用正确的形式来补全它。
Always check that class width is calculated as upper boundary − lower boundary, not the difference between rounded endpoints. For instance, if a class is recorded as 10 ≤ x < 20, the class width is 10, but if boundaries are given as 10.5 and 20.5, it remains 10.
一定要检查组距是按照 上组界 − 下组界 计算的，而不是四舍五入端点之差。例如，如果一个组记录为 10 ≤ x < 20，组距是 10；但如果组界给定为 10.5 和 20.5，组距依然是 10。
3. Histogram vs Bar Chart | 直方图与条形图的区别
Many students confuse histograms with bar charts, but their differences are crucial for IGCSE marks. A bar chart is used for categorical or discrete data; the bars are separated by gaps, and the height represents frequency or count. A histogram is used for continuous data; the bars touch, and the area represents frequency.
许多学生混淆直方图与条形图，但它们之间的差异对 IGCSE 得分至关重要。条形图用于类别数据或离散数据；柱形之间有间隙，高度代表频数或计数。直方图用于连续数据；柱形紧贴，面积代表频数。
In a bar chart, equal-width bars are drawn, and the vertical axis is directly labelled 'Frequency'. In a histogram with unequal class widths, the vertical axis must be labelled 'Frequency density'; otherwise the graph is invalid. Never use a frequency scale on a histogram when widths differ.
在条形图中，绘制的是等宽柱形，纵轴直接标记为“频数”。在组距不等的直方图中，纵轴必须标记为“频率密度”；否则图形无效。当组距不同时，切勿在直方图上使用频数刻度。
4. Drawing an Accurate Histogram | 绘制精确的直方图
To draw a correct histogram, begin by identifying consistent and complete class boundaries. If the data is given as 0–, 10–, 20–30, rewrite it as 0 ≤ x < 10, 10 ≤ x < 20, 20 ≤ x ≤ 30, paying special attention to open-ended boundaries if present.
要绘制正确的直方图，首先确认一致且完整的组界。如果数据以 0–、10–、20–30 的形式给出，重写为 0 ≤ x < 10、10 ≤ x < 20、20 ≤ x ≤ 30，特别注意可能存在的开放端组界。
Calculate each class width and the corresponding frequency density using the formula. Then draw and label axes: the horizontal axis should be continuous and labelled with the variable, and the vertical axis labelled 'Frequency density'. Plot each bar with the correct width (from boundary to boundary) and height equal to frequency density. The bars should touch, and no gaps should appear unless a frequency density is zero.
使用公式计算每个组距和相应的频率密度。然后绘制并标记坐标轴：横轴应是连续的并标记变量，纵轴标记“频率密度”。以正确的宽度（从一个组界到下一个组界）和等于频率密度的高度绘制每个柱形。柱形应紧贴，除非频率密度为零，否则不应出现间隙。
Double-check your work by computing the area of each bar: width × height should give the original frequency. This area check is a powerful way to avoid errors.
通过计算每个柱形的面积来复查你的绘图：宽度 × 高度应该等于原始频数。这种面积检查是避免错误的强大方法。
5. Handling Unequal Class Widths | 处理不等组距
The existence of unequal class widths is what makes histograms meaningful in IGCSE. When class widths differ, a simple frequency bar chart would distort the visual impression. For example, a class of 10 ≤ x < 30 has width 20, while another 30 ≤ x < 40 has width 10. If they both had frequency 15, drawing height 15 would make the wider class look disproportionately small; frequency density solves this.
不等组距的存在正是直方图在 IGCSE 中有意义的原因。当组距不同时，简单的频数条形图会扭曲视觉印象。例如，一个组 10 ≤ x < 30 宽度为 20，而另一组 30 ≤ x < 40 宽度为 10。如果两者频数都是 15，绘制高度 15 会让较宽的组看起来不成比例地小；频率密度解决了这一问题。
.5. The bar with width 20 is drawn at height 0.75, giving area 15; the narrow bar at height 1.5 also gives area 15. The visual weight now reflects actual frequency.
在这里，第一组的频率密度为 15 ÷ 20 = 0.75，第二组的频率密度为 15 ÷ 10 = 1.5。宽度为 20 的柱形以高度 0.75 绘制，面积为 15；窄柱形以高度 1.5 绘制，面积也是 15。视觉权重现在反映了实际频数。
6. Reading Frequency from a Histogram | 从直方图读取频数
Once a histogram is drawn, you can recover the frequency of any class by multiplying the frequency density by the class width. If the vertical scale is missing or you need to deduce frequencies from relative scales, identify a bar where both frequency and dimensions are known, then use proportions.
一旦画出直方图，你可以通过将频率密度乘以组距来恢复任何组的频数。如果纵轴刻度缺失，或你需要从相对尺度推断频数，先确定一个已知频数和尺寸的柱形，然后利用比例。
For example, if one complete bar has width 5, height 4, and frequency 20, then any other bar's frequency is (its frequency density × its width) × (20 / (4 × 5)), but essentially you can set up the relation: frequency ∝ area. Store the constant and apply to other bars.
例如，如果一个完整柱形的宽度为 5、高度为 4、频数为 20，那么任何其他柱形的频数是（其频率密度 × 其宽度）× (20 / (4 × 5))，但本质上你可以建立关系：频数 ∝ 面积。存储该常数并应用于其他柱形。
This technique regularly appears in IGCSE questions where only one bar is fully labelled and using the area-frequency connection is necessary.
这种技巧经常出现在 IGCSE 考题中，此时只有一个柱形被完全标记，必须利用面积–频数关系。
7. Estimating the Median and Quartiles | 估计中位数和四分位数
A histogram allows you to estimate the median, lower quartile (Q1), and upper quartile (Q3) by using cumulative frequency concepts applied to area. Although a cumulative frequency curve is more common, a histogram can give estimates directly, especially when a frequency table is provided alongside.
直方图通过将累积频数概念应用于面积，可以让你估计中位数、下四分位数（Q₁）和上四分位数（Q₃）。尽管累积频数曲线更常见，但直方图也能直接给出估计值，尤其当同时提供了频数表时。
/2 − F) / f) × w, where L is lower boundary, F is cumulative frequency before the class, f is frequency of median class, and w is class width.
要估计中位数，先计算总频数 N。中位数位置为 N/2。然后，从左侧开始累计频数（或面积），直至达到包含中位数的组。在该组内使用线性插值：中位数 = L + ((N/2 − F) / f) × w，其中 L 为下组界，F 为该组之前的累积频数，f 为中位数组的频数，w 为组距。
This interpolation technique is a powerful skill for IGCSE, applicable both in histograms and grouped frequency tables. Practice using it with a clear step-by-step approach to secure method marks.
这种插值技术是 IGCSE 中一项强大的技能，适用于直方图和分组频数表。通过清晰的分步方法练习使用，以确保获得方法分。
8. Adding a Frequency Polygon | 添加频率多边形
You can enhance a histogram by drawing a frequency polygon. To construct it, plot points at the midpoint of the top of each histogram bar (midpoint of class interval, frequency density) and connect them with straight lines. If you have a separate frequency density distribution, you may also plot the frequency polygon alone.
你可以通过绘制频率多边形来增强直方图。要构建它，在每个直方图柱形顶部的中点（组距中点，频率密度）处描点，并用直线连接它们。如果你有一个单独的频率密度分布，也可以只绘制频率多边形。
If you need to compare two distributions on the same axes, frequency density polygons allow a clean visual contrast without overlapping bars. Also, a frequency polygon can be extended to the boundaries by adding endpoints with frequency density zero, ensuring the area underneath approximates the total frequency.
如果你需要在同一坐标轴上比较两个分布，频率密度多边形无需重叠柱形即可实现清晰的视觉对比。此外，可以通过添加频率密度为零的端点将频率多边形扩展至组界，确保下方的面积近似于总频数。
9. Common Mistakes and Misconceptions | 常见错误与误解
One frequent mistake is labelling the vertical axis as 'Frequency' instead of 'Frequency density' when class widths vary. Another is using midpoints as bar widths rather than boundaries. Always use the true class boundaries, even if they are half-way between given limits, and draw bars accurately.
一个常见错误是当组距变化时，将纵轴标记为“频数”而不是“频率密度”。另一个错误是将中点用作柱形宽度，而不是组界。始终使用真实的组界，即使它们处于给定界限之间的一半位置，并精确绘制柱形。
Avoid the assumption that the tallest bar implies the highest frequency. Always compute area. Additionally, when interpolating, ensure you differentiate between cumulative frequency up to the class and frequency within the median class; mixing them up loses marks.
避免假设最高的柱形意味着最高的频数。始终计算面积。此外，在进行插值时，确保你能区分截至该组的累积频数与中位数组内的频数；将它们混淆会丢失分数。
10. Worked Example from Past Paper Style | 来自真题风格的例题
Consider a table with incomplete data: Class 0 ≤ x < 10 has frequency 25; class 10 ≤ x < 30 has frequency density 3.0; class 30 ≤ x < 50 has frequency 36; class 50 ≤ x < 70 has frequency density 1.2. First, you need to complete the table.
考虑一个数据不完整的表格：组 0 ≤ x < 10 的频数为 25；组 10 ≤ x < 30 的频率密度为 3.0；组 30 ≤ x < 50 的频数为 36；组 50 ≤ x < 70 的频率密度为 1.2。首先，你需要补全表格。
, 1.8, and 1.2 respectively. To estimate the median, cumulative frequencies: 25, 85, 121, 145. Median at 72.5th value lies in 10 ≤ x < 30: lower boundary 10, F=25, f=60, w=20, median = 10 + ((72.5−25)/60)×20 = 10 + (47.5/60)×20 ≈ 10 + 15.83 = 25.83.
对于 10 ≤ x < 30，宽度=20，频数=3.0 × 20 = 60。对于 30 ≤ x < 50，宽度=20，频率密度=36 ÷ 20 = 1.8。对于 50 ≤ x < 70，宽度=20，频数=1.2 × 20 = 24。总频数=25+60+36+24=145。直方图将以高度 25/10=2.5、3.0、1.8 和 1.2 显示柱形。要估计中位数，累积频数：25、85、121、145。中位数位于第 72.5 个值，落在 10 ≤ x < 30 组：下组界 10，F=25，f=60，w=20，中位数=10 + ((72.5−25)/60)×20 = 10 + (47.5/60)×20 ≈ 10 + 15.83 = 25.83。
This step-by-step layout secures method marks even if a small arithmetic slip occurs. Always present your working clearly.
这种分步呈现即使出现小的算术错误也能确保方法分。始终清晰展示你的计算过程。
11. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧
Read the question carefully: check whether you are given frequency, frequency density, or area. Often tables are incomplete, and you must fill them before drawing. Any working with class width must use boundary differences, not rounded endpoint subtraction.
仔细审题：检查题目给出的是频数、频率密度还是面积。通常表格是不完整的，你必须在绘图前补全。任何有关组距的计算都必须使用组界之差，而不是四舍五入端点的相减。
Always show your calculation of frequency density or missing frequencies in the working space; examiners allocate marks for these intermediate steps. For the actual drawing, use a sharp pencil, label axes, and mark the scale uniformly. An incorrect scale without a label loses accuracy marks.
始终在作答区域展示频率密度或缺失频数的计算；考官为这些中间步骤分配分数。对于实际绘图，使用削尖的铅笔，标记坐标轴，并均匀标记刻度。没有标签的错误刻度会失去准确性分数。
If estimating median or quartiles, write the formula clearly and substitute values correctly. Use units where appropriate and round final answers to a sensible degree of accuracy, typically three significant figures unless otherwise instructed.
在估计中位数或四分位数时，清晰地写出公式并正确代入数值。在适当的地方加上单位，并将最终答案四舍五入到合理的精度，通常为三位有效数字，除非另有要求。
12. Summary and Key Takeaways | 总结与关键要点
ass width. Unequal class widths make frequency density essential. Recovering frequency means computing area. And don' on allow you to extract median, quartiles, and even estimate the mean using the midpoints.
直方图是基于面积展示连续数据频数的图形。核心关系是频率密度 = 频数 ÷ 组距。不等组距使得频率密度不可或缺。恢复频数即计算面积。同时不要忘记插值技能可以让你提取中位数、四分位数，甚至利用组中值估计平均数。
with past papers, focus on the area concept, and always self-check with area = frequency.
掌握这些技能，Edexcel IGCSE 试卷中的直方图题目将成为可靠的得分来源。通过往年真题练习，聚焦面积概念，并始终用面积=频数进行自我检查。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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