Drawing Histograms | 绘制直方图

📚 Drawing Histograms | 绘制直方图

Histograms are a fundamental topic in the Edexcel IGCSE Mathematics syllabus. Unlike bar charts, histograms display continuous data using bars whose areas represent frequencies. Mastering how to draw histograms correctly is essential for achieving top marks in both Paper 1 and Paper 2 of the examination.

直方图是 Edexcel IGCSE 数学大纲中的一个基础主题。与条形图不同，直方图通过矩形的面积来表示频数，用于展示连续数据。掌握正确绘制直方图的方法，对于在卷一和卷二考试中取得好成绩至关重要。

1. What Is a Histogram? | 什么是直方图？

A histogram is a graphical representation of a frequency distribution for continuous data. The horizontal axis represents the variable being measured, divided into class intervals, while the vertical axis represents frequency density. The height of each bar depends on the frequency density, and the area of each bar is proportional to the frequency of that class.

直方图是连续数据频数分布的图形表示。横轴表示被测量的变量，按组距分段；纵轴表示频率密度。每个矩形的高度取决于频率密度，而矩形的面积与该组的频数成正比。

In a histogram, bars are drawn adjacent to one another with no gaps. This reflects the continuous nature of the underlying data — there are no breaks between one class interval and the next. The absence of gaps is one of the most distinctive visual features of a histogram.

在直方图中，矩形之间紧密相连，没有空隙。这反映了底层数据的连续性——相邻组距之间没有间断。矩形之间没有间隙是直方图最显著的视觉特征之一。

2. Histogram vs Bar Chart | 直方图与条形图的区别

Many students confuse histograms with bar charts, but these two graphs serve very different purposes. A bar chart is used for discrete or categorical data, where each bar represents a distinct category and the height shows the frequency. There are spaces between bars, and the width of each bar carries no meaning.

许多学生容易将直方图与条形图混淆，但两者用途截然不同。条形图用于离散数据或分类数据，每个条形代表一个独立的类别，高度表示频数。条形之间有空隙，条形宽度没有实际意义。

Bar Chart: Categories are separate; gaps between bars; height = frequency; width has no significance.

条形图：类别之间相互独立；条形间有空隙；高度 = 频数；宽度无特殊含义。
Histogram: Data is continuous; no gaps between bars; area = frequency; width = class interval.

直方图：数据是连续的；条形间无空隙；面积 = 频数；宽度 = 组距。

Another key distinction lies in how the bars are labelled. In a bar chart, each bar has a category label beneath it. In a histogram, the horizontal axis is a continuous number line, and bars are labelled by their class boundaries rather than by category names.

另一个关键区别在于条形下方的标注方式。条形图的每个条形下方有类别标签。而直方图的横轴是连续的数轴，条形按其组界而非类别名称来标注。

3. Continuous Data and Class Boundaries | 连续数据与组界

Histograms are only appropriate for continuous data — data that can take any value within a range. Examples include height, weight, time, temperature, and distance. Continuous data must be grouped into class intervals, such as “10 ≤ x < 20" or "10–20 cm".

直方图仅适用于连续数据——即在一定范围内可取任意数值的数据。例如身高、体重、时间、温度和距离等。连续数据需要按组距分组，例如“10 ≤ x < 20”或“10–20 cm”。

Each class interval has a lower class boundary and an upper class boundary. For example, in the interval 10 ≤ x < 20, the lower boundary is 10 and the upper boundary is 20. The class width is the difference between these boundaries: 20 − 10 = 10. Note that the upper boundary of one interval is the same as the lower boundary of the next interval, which is why bars in a histogram touch.

每个组距都有下组界和上组界。例如，在区间 10 ≤ x < 20 中，下组界为 10，上组界为 20。组距即两个组界之差：20 − 10 = 10。注意，一个区间的上组界与下一个区间的下组界相同，这就是直方图中各矩形彼此相接的原因。

When recording continuous data, class boundaries are determined by the precision of measurement. For instance, a height recorded to the nearest centimetre as “160 cm” actually spans from 159.5 cm to 160.5 cm. When drawing histograms from such data, you must use the true boundaries, not the rounded values.

在记录连续数据时，组界由测量精度决定。例如，身高精确到厘米记录为“160 cm”，实际上涵盖了 159.5 cm 到 160.5 cm 的范围。在根据此类数据绘制直方图时，应使用真实边界，而非四舍五入后的数值。

4. The Frequency Density Formula | 频率密度公式

The central formula for drawing histograms relates frequency, class width, and frequency density. Since the area of each bar must equal the frequency of its class, the height of each bar is calculated using the frequency density.

绘制直方图的核心公式将频数、组距与频率密度联系起来。由于每个矩形的面积必须等于该组的频数，因此每个矩形的高度通过频率密度来计算。

Frequency Density = Frequency ÷ Class Width
频率密度 = 频数 ÷ 组距

Rearranging this formula gives two equivalent forms that may be needed in exam questions:

将该公式变形，可得到两种等价形式，在考试中可能会用到：

Frequency = Frequency Density × Class Width
频数 = 频率密度 × 组距

Class Width = Frequency ÷ Frequency Density
组距 = 频数 ÷ 频率密度

When all class intervals have equal widths, the frequency density is directly proportional to the frequency, and you may draw the histogram with frequency on the vertical axis. However, when class widths are unequal, the vertical axis MUST represent frequency density — this is non-negotiable for a correct solution.

当所有组距相等时，频率密度与频数成正比，此时纵轴可直接用频数绘制直方图。但当组距不相等时，纵轴必须表示频率密度——这是解题时不可妥协的原则。

5. Setting Up the Axes | 坐标轴的设置

Before drawing any bars, you must establish an appropriate scale for both axes. The horizontal axis should cover the full range of the data, extending slightly beyond the lowest lower boundary and the highest upper boundary to ensure the graph looks balanced.

在绘制任何矩形之前，必须为两个坐标轴设定合适比例。横轴应覆盖数据的完整范围，并稍微延伸至最低下组界以下和最高上组界以上，使图形看起来均衡。

For the vertical axis, the scale must accommodate the largest frequency density value. Choose a scale that is easy to read, such as 1 square = 1 unit or 1 square = 2 units. A well-chosen scale prevents the histogram from appearing either excessively tall or squashed.

对于纵轴，比例尺必须能够容纳最大的频率密度值。选择易于读数的比例，例如 1 格 = 1 个单位或 1 格 = 2 个单位。选择恰当的比例可以避免直方图过高或过于扁平。

Always label both axes clearly: the horizontal axis should show the variable name and its units (e.g., “Height (cm)”), and the vertical axis must be labelled “Frequency Density” (without units). The label “Frequency Density” is one of the specific details examiners look for when awarding method marks.

务必清晰标注两个坐标轴：横轴应标明变量名称及其单位（例如“身高 (cm)”），纵轴必须标注“频率密度”（无需单位）。“频率密度”这一标注是考官在评分时特别关注的方法分要点之一。

6. Drawing Bars Step by Step | 分步绘制矩形

Once the axes are prepared, follow this systematic procedure to draw each bar of the histogram:

坐标轴准备好后，按以下系统步骤绘制直方图的每个矩形：

Step 1: Identify the lower and upper boundary for each class interval.

第一步：确定每个组距的下组界和上组界。
Step 2: Calculate the class width for each interval: upper boundary − lower boundary.

第二步：计算每个组距的组距：上组界 − 下组界。
Step 3: Calculate the frequency density for each class: frequency ÷ class width.

第三步：计算每个组的频率密度：频数 ÷ 组距。
Step 4: On the horizontal axis, mark the boundaries of each class interval.

第四步：在横轴上标出每个组距的边界。
Step 5: For each class, draw a vertical line from the lower boundary to the appropriate frequency density height, then draw the top edge horizontally to the upper boundary, and close the rectangle.

第五步：对每个组，从下组界开始画一条垂直线到相应的频率密度高度，然后水平画出顶边至上组界，最后闭合矩形。
Step 6: Ensure adjacent bars share a common side — there must be no gaps between consecutive bars.

第六步：确保相邻矩形共享一条公共边——连续矩形之间不得留有空隙。

It is strongly recommended to use a pencil and a ruler when drawing histograms in the examination. A freehand sketch will not earn full marks, and a ruler is necessary to make the bars neat and the boundaries accurate.

强烈建议在考试中使用铅笔和直尺来绘制直方图。徒手绘制的草图无法获得满分，使用直尺才能确保矩形整洁、边界准确。

7. Worked Example: Student Heights | 例题：学生身高分布

A class of 50 students has their heights recorded. The results are grouped as follows:

一个班级 50 名学生的身高记录如下分组：

Height (cm)	Frequency	Class Width	Frequency Density
140 < h ≤ 145	5	5	5 ÷ 5 = 1
145 < h ≤ 155	12	10	12 ÷ 10 = 1.2
155 < h ≤ 160	15	5	15 ÷ 5 = 3
160 < h ≤ 175	18	15	18 ÷ 15 = 1.2

From this table we find the frequency densities: 1, 1.2, 3, and 1.2 respectively. The horizontal axis spans from 140 to 175 cm. The vertical axis must reach at least 3, which is the largest frequency density.

由此表得出各组的频率密度值：分别为 1、1.2、3 和 1.2。横轴范围从 140 到 175 cm。纵轴至少应达到 3，即最大的频率密度值。

For the first bar, draw from 140 to 145 on the horizontal axis and raise the bar to a height of 1. For the second bar, draw from 145 to 155 and raise to 1.2. Note that the second bar is twice as wide as the first but only slightly taller, so its area (12) correctly represents its larger frequency.

对于第一个矩形，在横轴上从 140 画到 145，并将矩形高度设为 1。第二个矩形从 145 画到 155，高度设为 1.2。注意第二个矩形比第一个宽一倍但仅稍高一些，其面积（12）正确地反映了较大的频数。

For the third bar, draw from 155 to 160 with height 3. This bar is the tallest because it has both a high frequency and a narrow width. The fourth bar spans 160 to 175 at height 1.2, creating a wide, short rectangle whose area represents 18 students.

第三个矩形从 155 画到 160，高度为 3。这个矩形最高，因为该组频数高且组距窄。第四个矩形横跨 160 到 175，高度为 1.2，形成一个宽而矮的矩形，其面积代表 18 名学生。

8. Handling Unequal Class Widths | 处理不等组距

One of the most common histogram questions on the Edexcel IGCSE paper involves data with unequal class widths. When class widths differ, you cannot simply plot frequency on the vertical axis — doing so would create a misleading graph where the area of each bar is not proportional to the frequency.

Edexcel IGCSE 试卷中最常见的直方图题型之一是组距不相等的数据。当组距不同时，不能简单地将频数画在纵轴上——这样做会生成误导性的图形，因为每个矩形的面积不再与频数成正比。

Consider this frequency table where class widths are clearly unequal:

考虑以下频数表，其中组距明显不同：

Time (minutes)	Frequency	Class Width	Frequency Density
0 ≤ t < 5	20	5	20 ÷ 5 = 4
5 ≤ t < 15	30	10	30 ÷ 10 = 3
15 ≤ t < 20	10	5	10 ÷ 5 = 2
20 ≤ t < 40	40	20	40 ÷ 20 = 2

The frequency densities are 4, 3, 2, and 2 respectively. Notice that the second class has a higher frequency than the first, but its frequency density is lower because the data is spread over a wider interval. This is why the vertical axis must show frequency density — only then does the visual area accurately tell the true story of the data.

各组的频率密度分别为 4、3、2 和 2。注意第二组的频数高于第一组，但其频率密度较低，因为数据分布在更宽的区间上。这就是为什么纵轴必须表示频率密度——只有这样才能通过视觉面积准确反映数据的真实情况。

9. Common Mistakes to Avoid | 需要避免的常见错误

Through years of marking IGCSE scripts, examiners consistently identify several recurring errors students make when drawing histograms. Being aware of these pitfalls can easily gain you valuable marks.

经过多年的 IGCSE 阅卷经验，考官们发现学生在绘制直方图时反复出现若干典型错误。了解这些陷阱，可以帮助你轻松获得宝贵的分数。

Mistake 1: Using frequency instead of frequency density when class widths are unequal. This produces bars with incorrect areas and loses multiple marks. Always calculate frequency density for every class interval.

错误一：当组距不等时，用频数代替频率密度。这会导致矩形的面积不正确，并丢失多个分值点。始终计算每个组距的频率密度。
Mistake 2: Leaving gaps between adjacent bars. Histograms must have touching bars because the data is continuous. Spaces are only appropriate in bar charts.

错误二：相邻矩形之间留下空隙。直方图的矩形必须彼此相接，因为数据是连续的。空隙只适用于条形图。
Mistake 3: Misidentifying class boundaries. For example, using 145 instead of 145.5 when the data was measured to the nearest whole number. Always consider the true lower and upper boundaries.

错误三：错误识别组界。例如，当数据精确到整数时，本应使用 145.5 却用了 145。始终考虑真实的下界和上界。
Mistake 4: Forgetting to label the vertical axis as “Frequency Density”. This is a required element for full method marks.

错误四：忘记将纵轴标注为“频率密度”。这是获得完整方法分的必要元素。
Mistake 5: Drawing the histogram from memory without a frequency density table first. Doing the calculation in your head often leads to errors.

错误五：不先列出频率密度表就直接凭记忆绘制直方图。心算往往容易出错。

10. Edexcel IGCSE Exam Tips | Edexcel IGCSE 考试技巧

In the Edexcel IGCSE examination, histogram questions typically appear in Section B of Paper 1 and in Paper 2. They are often worth 4 to 6 marks, covering both drawing and interpretation. Here are targeted strategies to maximise your score.

在 Edexcel IGCSE 考试中，直方图题通常出现在卷一 B 部分和卷二。这类题通常占 4 到 6 分，涵盖绘图和读图两个方面。以下是一些针对性策略，帮助你获取最高分。

First, always show your working for frequency density calculations even if you are confident in your mental arithmetic. Examiners award method marks for the correct calculations even when the graph itself has minor inaccuracies.

首先，即使你对自己的心算能力非常自信，也要展示频率密度的计算步骤。即使图形本身有轻微误差，考官也会根据正确的计算过程给予方法分。

Second, if the question asks you to estimate the number of data values in a particular interval, use the formula: area of the bar = number of values. For example, if a bar has a frequency density of 3 and a width of 10, then the frequency is 3 × 10 = 30. This “area equals frequency” bridge is the single most important concept you can carry into the exam room.

其次，如果题目要求你估计某个区间内的数据个数，请使用以下方法：矩形的面积 = 数据个数。例如，若一个矩形的频率密度为 3、宽度为 10，则该组频数为 3 × 10 = 30。“面积等于频数”这一桥梁概念，是你带进考场最重要的知识点。

Third, pay close attention to whether the question provides a histogram and asks you to complete a frequency table, or provides a table and asks you to draw the histogram. Both directions are examined regularly. For the former, work backwards: multiply each frequency density by its corresponding class width.

第三，注意审题：题目是给直方图让你补全频数表，还是给表格让你绘制直方图？这两种方向都会考到。如果是前者，请反向操作：将每个频率密度乘以其对应的组距即可得到频数。

Finally, practise with past papers specifically from the Edexcel IGCSE Mathematics A (4MA1) specification. The histogram questions follow very predictable patterns, and familiarity with the style will give you a significant confidence boost on exam day.

最后，请使用 Edexcel IGCSE 数学 A（4MA1）考纲的历年真题进行练习。直方图题型的模式非常固定，熟悉出题风格能让你在考试当天自信倍增。

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Drawing Histograms | 绘制直方图