📚 IGCSE Mathematics: How to Draw a Histogram — Methods and Steps | IGCSE数学：直方图的绘制方法与步骤
A histogram is a statistical diagram used to display continuous data. It is made up of bars drawn on a continuous horizontal scale. The key idea is that the area of each bar, not just its height, represents the frequency of the data in that interval.
直方图（Histogram）是展示连续数据的统计图形。它由绘制在连续横轴上的多个矩形条组成。其核心要点是：每个矩形条的“面积”代表对应区间的频数，而不仅仅是矩形的高度。
1. What is a Histogram? | 什么是直方图？
In IGCSE Mathematics, a histogram is used when data are continuous, such as height, weight, time, distance, or temperature. The data are first separated into class intervals, and each interval is represented by one bar on the graph.
在 IGCSE 数学中，直方图用于处理连续数据，例如身高、体重、时间、距离或温度。我们先将数据划分为若干组区间，每组在图上用一个矩形条表示。
The horizontal axis shows the class boundaries, and the vertical axis shows the frequency density. Because the bars touch each other, the graph communicates that the data flow continuously from one interval to the next.
横轴表示各组区间的边界，纵轴表示频率密度。由于矩形条彼此紧挨，直方图可以清楚地表达数据从一个区间到下一个区间是连续变化的。
2. Histogram vs Bar Chart | 直方图与条形图的区别
Many students confuse histograms with bar charts, but they are very different. A bar chart is used for discrete or categorical data, and the bars are separated by gaps. The horizontal order can often be changed, and the height of each bar directly shows the frequency.
很多同学会把直方图与条形图（Bar Chart）混淆，但二者完全不同。条形图用于离散数据或分类数据，矩形条之间通常有空隙；横轴上的顺序往往可以调整，条形的高度直接表示频数。
A histogram, however, is used for continuous data that has been grouped into intervals. The bars must touch because there is no gap between class intervals. The horizontal order is fixed because the class intervals follow the natural number line.
而直方图用于已经分组的连续数据。矩形条之间必须没有空隙，因为相邻的组区间之间没有间断；横轴顺序也不能随意调换，因为组区间必须按数值的自然顺序排列。
3. The Key Formula: Frequency Density | 核心公式：频率密度
Before drawing a histogram, you need to know the frequency density for every class interval. This is the most important formula in this topic.
在绘制直方图之前，你需要求出每个组区间的频率密度。这是直方图部分最重要的公式。
Frequency Density = Frequency ÷ Class Width
The class width is the difference between the upper boundary and the lower boundary of a class interval. Frequency density is not the same as frequency; it is the frequency per unit of class width.
组距等于区间上边界减去下边界。频率密度并不等于频数，它表示的是“每单位组距内的频数”。
If class widths are equal, frequency density is proportional to frequency, so the shape of the histogram looks similar to a bar chart. If class widths are unequal, you must always use frequency density on the vertical axis.
如果各组组距相等，频率密度与频数成正比例，因此直方图的外形与条形图相似；如果组距不等，则纵轴必须使用频率密度。
4. Step 1: Identify Class Intervals and Boundaries | 第一步：确定组区间与边界
The first step is to read the class intervals carefully. In a histogram, each bar must be drawn from the lower boundary of the class interval to the upper boundary. These boundaries must be continuous.
第一步是认真阅读组区间。在直方图中，每个矩形条必须从组区间的下边界画到上边界，并且这些边界必须是连续的。
Sometimes intervals are given as inequalities, such as 0 ≤ x < 10, 10 ≤ x < 20. In that case, the boundaries are clearly 0 and 10 for the first interval, and 10 and 20 for the second interval.
有时区间以不等式形式给出，例如 0 ≤ x < 10、10 ≤ x < 20。此时第一个区间的边界就是 0 和 10，第二个区间就是 10 和 20。
If intervals are written as “0-10” and “10-20”, they usually mean the same thing: 0 up to 10, then 10 up to 20. The boundary 10 belongs to both the end of one interval and the start of the next. This is why the bars touch.
如果区间写成“0-10”“10-20”，通常表示 0 到 10、10 到 20。边界 10 既是前一个区间的终点，也是后一个区间的起点。这正是直方图矩形条紧挨的原因。
5. Step 2: Calculate Class Widths | 第二步：计算组距
Once the boundaries are clear, calculate the width of each class interval. The width is simply the upper boundary minus the lower boundary.
边界确定之后，计算每个组区间的组距。组距就是“上边界减下边界”。
For example, if an interval is 5 < t ≤ 10, the lower boundary is 5 and the upper boundary is 10, so the width is 5. If an interval is 10 < t ≤ 20, the width is 10.
例如，区间 5 < t ≤ 10 的下边界是 5，上边界是 10，所以组距是 5；区间 10 < t ≤ 20 的组距是 10。
Be careful with intervals that look similar but have different widths. For 0-5, 5-10, and 10-20, the widths are 5, 5, and 10. For 0-10, 10-20, and 20-35, the widths are 10, 10, and 15.
注意那些看似相似但组距不同的区间。例如 0-5、5-10、10-20 的组距分别是 5、5、10；而 0-10、10-20、20-35 的组距分别是 10、10、15。
6. Step 3: Calculate Frequency Densities | 第三步：计算频率密度
Now divide each frequency by its class width. This gives the height of the bar on the histogram.
接下来，用每组频数除以组距，得到该组矩形条的高度。
For example, if a class has frequency 30 and width 10, the frequency density is 30 ÷ 10 = 3. This means the bar will be 3 units high and 10 units wide.
例如，某组频数为 30，组距为 10，那么频率密度是 30 ÷ 10 = 3。这意味着该矩形条的高度是 3 个单位，宽度是 10 个单位。
Always show your working. If a class has frequency 12 and width 8, write 12 ÷ 8 = 1.5. This method mark can be very helpful in an exam even if you later make a drawing mistake.
一定要写出运算过程。例如某组频数为 12，组距为 8，就应写出 12 ÷ 8 = 1.5。即使之后画图出现失误，这些过程也可以在考试中帮你获得方法分。
7. Step 4: Choose a Suitable Scale | 第四步：选取合适的比例尺
After calculating all the frequency densities, choose a scale for the vertical axis. The scale must be large enough so that all values fit on the graph paper, but not so large that small frequency densities are impossible to plot accurately.
算出所有频率密度后，为纵轴选择合适的比例尺。比例尺要足够大，使所有数值都能画在坐标纸上，但也不能太大，否则较小的频率密度很难精确作图。
For the horizontal axis, mark the continuous scale using the actual class boundaries. Do not use gaps between the bars.
横轴应按实际组边界绘制连续刻度，矩形条之间不能留有空隙。
For example, if the greatest frequency density is 3.6, you could let 1 large square on the vertical axis represent 0.5 units. If the greatest boundary is 40, make sure your horizontal scale reaches at least 40.
例如，若最大频率密度是 3.6，可以让纵轴上一个大格代表 0.5 个单位；若最大边界是 40，则横轴至少要延伸到 40。
8. Step 5: Draw and Label the Axes | 第五步：绘制并标注坐标轴
Draw the two axes clearly with a ruler. The horizontal axis represents the continuous variable, and the vertical axis represents frequency density.
用直尺清晰地画出两条坐标轴。横轴表示连续变量，纵轴表示频率密度。
Label the horizontal axis with the name of the variable and its units, for example “Time (seconds)” or “Length (cm)”. Label the vertical axis as “Frequency density”. Add a title that describes the data.
横轴需要标注变量名称和单位，例如“时间（秒）”或“长度（厘米）”。纵轴标注“Frequency density（频率密度）”。同时为整个直方图加上标题。
Do not forget the axis labels. In the IGCSE examination, missing labels are a common reason for losing marks, even when the bars are drawn correctly.
不要忘记轴标签。在 IGCSE 考试中，即使矩形条画得正确，漏标坐标轴也常常导致失分。
9. Step 6: Draw the Bars Correctly | 第六步：正确绘制矩形条
For each class interval, draw a rectangle starting at the lower boundary and ending at the upper boundary. The width of the rectangle is the class width, and the height is the frequency density.
在横轴上，为每个组区间绘制一个矩形：起点是该组下边界，终点是该组上边界。矩形的宽度是组距，高度是频率密度。
Because the area of a rectangle is width × height, the area of each bar is equal to the class width × frequency density, which is the original frequency. This is why we say “area equals frequency” in a histogram.
因为矩形面积等于宽乘高，所以每个矩形的面积等于组距乘频率密度，也就是原来的频数。这就是直方图中“面积等于频数”的原理。
If all class widths are equal, you will see that the heights are simply a scaled version of the frequencies. If class widths are unequal, you must trust the frequency density calculation, even if a bar with a larger frequency looks shorter than another bar.
若各组的组距都相等，则矩形条的高度其实就是频数按比例缩放后的结果。若组距不等，则一定要相信频率密度的计算结果，可能会出现“频数较大但条形较矮”的情况。
10. Worked Example: Unequal Class Intervals | 实例：不等组距的直方图
Here is a typical IGCSE-style question. The table shows the distances travelled by students to school. Draw a histogram for the data.
下面是一道典型的 IGCSE 风格题目。表格显示了学生们上学的距离，请根据数据绘制直方图。
| Distance (d km) | Frequency |
| 0 < d ≤ 2 | 6 |
| 2 < d ≤ 5 | 12 |
| 5 < d ≤ 10 | 15 |
| 10 < d ≤ 20 | 8 |
Step 1: Find the class widths. The widths are 2, 3, 5, and 10.
第一步：求组距。各组的组距分别是：2、3、5、10。
Step 2: Calculate the frequency densities.
第二步：计算频率密度。
6 ÷ 2 = 3
12 ÷ 3 = 4
15 ÷ 5 = 3
8 ÷ 10 = 0.8
Step 3: Draw bars with widths 2, 3, 5, and 10 and heights 3, 4, 3, and 0.8. The total area should equal the total frequency: 6 + 12 + 15 + 8 = 41.
第三步：绘制矩形条，宽度分别为 2、3、5、10，高度分别为 3、4、3、0.8。所有矩形面积之和应等于总频数：6 + 12 + 15 + 8 = 41。
Notice that the interval 5 < d ≤ 10 has a higher frequency than 2 < d ≤ 5, but both bars have the same height. The interval 5 < d ≤ 10 appears larger only because it is wider.
你会发现，区间 5 < d ≤ 10 的频数比 2 < d ≤ 5 更大，但两个矩形条的高度相同。区间 5 < d ≤ 10 看起来更大，仅仅是因为它的宽度更大。
11. Common Mistakes to Avoid | 常见错误提醒
Mistake 1: Drawing gaps between bars. In a histogram, the data are continuous, so bars must touch.
错误一：矩形条之间留有空隙。直方图表示连续数据，矩形条之间应紧挨。
Mistake 2: Using frequency instead of frequency density on the vertical axis. This is wrong when the class widths are unequal.
错误二：纵轴直接使用频数而不是频率密度。当组距不等时，这样画是错误的。
Mistake 3: Using the midpoint as the width of a class. The class width is the difference between the upper and lower boundaries, not the midpoint.
错误三：把区间的中点当成组距。组距是上边界与下边界之差，而不是中点值。
Mistake 4: Misreading intervals such as 0-10 and 10-20 as having a gap at 10. The boundary 10 is shared, so bars must touch at 10.
错误四：误认为 0-10 与 10-20 在 10 处有间隔。10 是公共边界，因此矩形条应在 10 处相接。
Mistake 5: Forgetting to check that the total area equals the total frequency. This check can help you catch arithmetic errors.
错误五：忘记检查面积总和是否等于总频数。这一步检查可以帮助你发现计算错误。
12. Summary and Exam Tips | 总结与考试技巧
The most important rule in a histogram is that area equals frequency. Always use frequency density on the vertical axis and continuous boundaries on the horizontal axis.
直方图最重要的法则是“面积等于频数”。纵轴必须使用频率密度，横轴必须使用连续边界。
In an exam, if you are asked to draw a histogram, first create a frequency density table. Then choose a clear scale, draw and label the axes, and finally draw the bars so that they touch.
在考试中，如果题目要求绘制直方图，应先制作频率密度表；然后选择合适的比例尺，画好并标注坐标轴；最后绘制彼此紧挨的矩形条。
Remember that the modal class is the class with the highest frequency density, not necessarily the highest frequency. If you are asked to estimate a frequency from a histogram, multiply the class width by the frequency density from the graph.
请记住：众数区间是频率密度最大的区间，而不一定是频数最大的区间。如果题目要求从直方图中估算某组频数，就用“组距 × 频率密度”。
Finally, always show your working. Drawing a neat, labelled histogram with a completed frequency density table will help you earn full marks in the IGCSE Mathematics exam.
最后，务必写出计算过程。画出一张整洁、标注完整的直方图，并附上频率密度表，这将帮助你在 IGCSE 数学考试中拿到满分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导