Histograms | 直方图

📚 Histograms | 直方图

A histogram is a graphical display of grouped continuous data. Its key feature is that the area of each bar is proportional to the frequency of the class, not the height. This differs from a bar chart, where height represents frequency and bars are usually separated. In Edexcel A-Level Mathematics, histograms are used when class widths are unequal, because frequency density must be used to keep the display accurate.

直方图是分组连续数据的图形显示。其关键特征是每个条形的面积与组的频数成比例，而不是高度。这与条形图不同，条形图中高度表示频数且条形通常分开。在爱德思 A-Level 数学中，直方图用于组距不相等的情况，因为必须使用频率密度来保持显示准确。

1. What is a Histogram? | 什么是直方图

A histogram is a graphical representation of grouped continuous data. Unlike a bar chart, which uses equal-width bars and where height represents frequency, a histogram uses area to represent frequency. This makes histograms essential when class intervals have unequal widths. In Edexcel A-Level Mathematics, histograms appear in the Statistics part of the course, especially in data presentation and interpretation.

直方图是分组连续数据的图形表示。与条形图不同，条形图使用等宽条形且高度表示频数，而直方图用面积表示频数。这使得直方图在组距不相等时至关重要。在爱德思 A-Level 数学中，直方图出现在统计部分，尤其是数据展示与解释中。

2. Frequency Density Formula | 频率密度公式

The vertical axis of a histogram always represents frequency density. Frequency density is not the same as frequency. It is calculated by dividing the frequency of a class by its class width. If the class widths are all equal, frequency density is proportional to frequency, but you should still label the axis as frequency density.

直方图的纵轴始终表示频率密度。频率密度与频数不同。它的计算方法是用组的频数除以组距。如果所有组距相等，频率密度与频数成比例，但仍应将纵轴标注为频率密度。

frequency density = frequency ÷ class width

The class width is the difference between the upper and lower class boundaries. For example, the interval 10 ≤ x < 20 has width 10. If the frequency is 15, the frequency density is 15 ÷ 10 = 1.5.

组距是上组界与下组界之差。例如，区间 10 ≤ x < 20 的宽度为 10。如果频数为 15，则频率密度为 15 ÷ 10 = 1.5。

3. Equal and Unequal Class Widths | 等距与不等距组距

When all classes have the same width, drawing a histogram with frequency density on the vertical axis gives bars of equal width, so the area and height both represent frequency in a simple proportional way. However, when class widths are unequal, plotting frequency directly would make wider intervals appear more important than they are. The height must be adjusted by dividing by width, so that area remains proportional to frequency.

当所有组具有相同宽度时，以频率密度为纵轴绘制直方图会得到等宽条形，因此面积和高度都以简单的比例方式表示频数。然而，当组距不相等时，直接绘制频数会使较宽的区间看起来比实际更重要。必须通过除以宽度来调整高度，使面积保持与频数成比例。

4. Calculating Frequency Density | 计算频率密度

The table below shows how to calculate frequency density for grouped data. Notice that the frequency density can be the same even when frequencies differ, because the class widths are different.

下表展示了如何计算分组数据的频率密度。请注意，即使频数不同，频率密度也可能相同，因为组距不同。

Using these values, the histogram bars have widths 10, 10, 15, 15 and 30, and heights 1.2, 2.0, 1.6, 1.07 and 0.27 respectively.

使用这些数值，直方图条形的宽度分别为 10、10、15、15 和 30，高度分别为 1.2、2.0、1.6、1.07 和 0.27。

5. Drawing a Histogram | 绘制直方图

To draw a histogram, first calculate the class widths and frequency densities. Then draw axes: the horizontal axis is the continuous variable, and the vertical axis is frequency density. For each class, draw a rectangle whose width is the class width and whose height is the frequency density. The area of each rectangle equals class width × frequency density = frequency. Bars must touch because the data are continuous.

绘制直方图时，首先计算组距和频率密度。然后画坐标轴：横轴是连续变量，纵轴是频率密度。对于每个组，画一个矩形，其宽度为组距，高度为频率密度。每个矩形的面积等于组距 × 频率密度 = 频数。由于数据是连续的，条形应彼此相接。

Start by choosing a suitable scale for the horizontal variable and the frequency density. Draw the first bar from the lower boundary to the upper boundary of the first class at the corresponding frequency density height. Continue for all classes. Bars must touch because the data are continuous; if a class boundary is 10, the next bar starts at 10. Label both axes and give the histogram a title.

首先为横轴变量和频率密度选择合适的刻度。从第一个组的下组界到上组界，以相应的频率密度高度画出第一个条形

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