📚 IGCSE Mathematics: Histograms and Cumulative Frequency | IGCSE 数学：直方图与累积频数
In IGCSE Mathematics, data handling questions often require you to draw and interpret histograms and cumulative frequency diagrams. This Graphs 3 revision guide covers frequency density, grouped frequency tables, cumulative frequency curves, quartiles and box plots. You will also see how to compare distributions and avoid common exam errors.
在 IGCSE 数学中，数据处理题经常要求你绘制并解读直方图和累积频数图。本 Graphs 3 复习指南涵盖频数密度、分组频数表、累积频数曲线、四分位数和箱线图。你还将学会如何比较分布并避免常见考试错误。
1. Frequency Density and Class Width | 频数密度与组距
For a histogram with unequal class intervals, you must not plot the raw frequency on the vertical axis. Instead, first calculate the frequency density for each class using the formula:
对于组距不相等的直方图，你不能在纵轴上直接标出原始频数。相反，首先要计算每一组的频数密度，公式为：
frequency density = frequency ÷ class width
频数密度 = 频数 ÷ 组距
Class width is the difference between the upper and lower boundary of a class interval. For example, 10 ≤ x < 20 has a class width of 10, while 20 ≤ x < 50 has a class width of 30.
组距是组区间上界与下界之差。例如，10 ≤ x < 20 的组距为 10，而 20 ≤ x < 50 的组距为 30。
2. Drawing a Histogram | 绘制直方图
When drawing a histogram, the horizontal axis shows the class boundaries and the vertical axis shows frequency density. Each bar must cover the full class interval, with no gaps between bars. The height of each bar is exactly the frequency density you calculated.
绘制直方图时，横轴表示组界，纵轴表示频数密度。每个条形必须覆盖完整的组区间，条形之间不能有空隙。每个条形的高度恰好就是你计算出的频数密度。
A key idea is that the area of each bar represents its frequency. If the class width is doubled, the height must be halved to keep the area the same. This is why we divide frequency by class width.
一个关键概念是每个条形的面积代表该组的频数。如果组距加倍，高度必须减半才能保持面积不变。这就是我们要用频数除以组距的原因。
3. Interpreting Histogram Areas | 解读直方图面积
To find the number of data values in a class from a histogram, multiply the frequency density by the class width. This recovers the original frequency:
要从直方图中求某一组的数据个数，请用频数密度乘以组距。这样可以还原原始频数：
frequency = frequency density × class width
频数 = 频数密度 × 组距
You may also be asked to estimate the number of values between two points that do not match class boundaries. In that case, find the area of the missing strip by splitting the bar and using its frequency density.
你还可能被要求估计两个不正好落在组界上的点之间的数据个数。此时，可以通过拆分条形并使用其频数密度来求出缺失部分的面积。
4. Frequency Polygons | 频数多边形
A frequency polygon can be drawn by joining the midpoints of the tops of histogram bars with straight lines. For grouped data, the midpoint of each class is the average of the lower and upper boundaries.
频数多边形可以通过用直线连接直方图条形顶部的组中点来绘制。对于分组数据，每组的组中点是下界和上界的平均值。
When comparing two distributions, frequency polygons are often more useful than histograms because two polygons can be drawn on the same axes without the bars overlapping.
当比较两个分布时，频数多边形通常比直方图更有用，因为两个多边形可以绘制在同一坐标轴上，而条形不会重叠。
5. Cumulative Frequency Tables | 累积频数表
A cumulative frequency table is built by adding each frequency to the total of all previous frequencies. The last cumulative frequency must equal the total frequency of the data set.
累积频数表是通过将每个频数与之前所有频数之和相加而建立的。最后一个累积频数必须等于数据集的频数总和。
For grouped data, list the upper class boundary in a separate column and write the running total next to it. Do not use the midpoint or lower boundary for cumulative frequency work.
对于分组数据，请在单独一列中列出上组界，并在一旁写出累计总和。累积频数计算不要使用组中点或下组界。
6. Cumulative Frequency Curve | 累积频数曲线
Plot the cumulative frequency against the upper class boundary for each group. Since cumulative frequency only increases, the curve should never slope downwards.
以每组的上组界为横坐标，累积频数为纵坐标描点。由于累积频数只会增加，曲线绝不应向下倾斜。
Join the points with a smooth curve, not with straight line segments. A common exam instruction is to use a ruler only for the axes and to draw the curve freehand.
用平滑曲线连接各点，不要用直线段连接。考试中常见的要求是只使用直尺画坐标轴，曲线则徒手绘制。
7. Median, Quartiles and Percentiles | 中位数、四分位数与百分位数
To estimate the median from a cumulative frequency curve, first calculate half the total frequency. Draw a horizontal line from this value on the vertical axis to the curve, then draw a vertical line down to the horizontal axis.
要从累积频数曲线估计中位数，请先计算总频数的一半。从纵轴上的这个值画水平线到曲线，再画竖直线向下到横轴。
The lower quartile is at one quarter of the total frequency, the upper quartile is at three quarters of the total frequency, and any percentile can be found by using the appropriate fraction of the total frequency.
下四分位数位于总频数的四分之一处，上四分位数位于总频数的四分之三处，任何百分位数都可以用总频数的相应分数来求得。
8. Box Plots | 箱线图
A box plot displays five key values: the minimum, lower quartile, median, upper quartile and maximum. These are often written as a five-number summary.
箱线图显示五个关键值：最小值、下四分位数、中位数、上四分位数和最大值。这些通常写作五数概括。
Draw a box between the lower and upper quartiles, and mark the median as a line inside the box. The whiskers extend from the box to the minimum and maximum values unless there are outliers.
在下四分位数和上四分位数之间画一个矩形箱，并在箱内用一条线标记中位数。除非有异常值，否则胡须线从箱体延伸到最小值和最大值。
9. Comparing Distributions | 比较分布
When comparing two box plots or cumulative frequency curves, always comment on both a measure of average and a measure of spread. Usually the median and interquartile range are required.
当比较两个箱线图或累积频数曲线时，一定要同时评价集中趋势的度量和离散程度的度量。通常需要使用中位数和四分位距。
Write comparisons in context, for example “the median score for class A is higher than for class B, so class A performed better on average”. Then compare the interquartile range to say which distribution is more consistent.
结合具体情境进行比较，例如“A 班的中位成绩高于 B 班，因此 A 班平均表现更好”。然后比较四分位距，说明哪个分布的波动更小。
10. Worked Example | 例题解析
A grouped frequency table gives the following data: 0 ≤ x < 5 with frequency 8, 5 ≤ x < 15 with frequency 20, 15 ≤ x < 30 with frequency 18, and 30 ≤ x < 50 with frequency 24. Find the frequency densities and state the total frequency.
一个分组频数表给出以下数据：0 ≤ x < 5 频数为 8，5 ≤ x < 15 频数为 20，15 ≤ x < 30 频数为 18，30 ≤ x < 50 频数为 24。求频数密度并说明总频数。
8 ÷ 5 = 1.6, 20 ÷ 10 = 2.0, 18 ÷ 15 = 1.2, 24 ÷ 20 = 1.2
8 ÷ 5 = 1.6，20 ÷ 10 = 2.0，18 ÷ 15 = 1.2，24 ÷ 20 = 1.2
The cumulative frequencies at the upper boundaries 5, 15, 30 and 50 are 8, 28, 46 and 70. The median lies at half of 70, which is 35, inside the class 15 ≤ x < 30.
在上组界 5、15、30 和 50 处的累积频数分别为 8、28、46 和 70。中位数位于 70 的一半，即 35，落在 15 ≤ x < 30 这一组内。
11. Common Errors | 常见错误
One common error is plotting frequency instead of frequency density on a histogram with unequal class widths. This makes tall bars appear more frequent than they really are or hides the true distribution shape.
一个常见错误是在组距不相等的直方图上标出频数而不是频数密度。这会使得高的条形看起来比实际更频繁，或者掩盖真实的分布形状。
Another error is using the middle of each class for cumulative frequency points instead of the upper class boundary. Cumulative frequency must always be plotted at the upper boundary because the total is only complete at the end of the interval.
另一个错误是在累积频数图中使用每组的中点代替上组界。累积频数必须始终标在上组界处，因为总数只有在区间结束时才完整。
12. Exam Tips | 考试技巧
Always check that your cumulative frequency curve starts at or near zero and ends at the total frequency. Before drawing a histogram, double-check that every frequency density has been calculated and labelled with the correct unit.
始终检查你的累积频数曲线是否从零附近开始，并结束于总频数。绘制直方图之前，再次确认每个频数密度都已计算，并且标注了正确的单位。
When reading a value from a curve, draw clear horizontal and vertical construction lines. Even if your final answer is slightly different from the mark scheme, correct working lines can earn method marks.
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