📚 IB Mathematics: Data Presentation Techniques | IB数学：数据展示技巧
Data presentation is a core skill in IB Mathematics. It is not enough to calculate statistics; you must display data in a clear, accurate and meaningful way. This article covers the main techniques you need for both standard level and higher level, including frequency tables, histograms, box-and-whisker plots, cumulative frequency graphs and scatter plots.
数据展示是IB数学的核心技能。仅仅计算统计量还不够，你必须能够以清晰、准确且有意义的方式呈现数据。本文涵盖标准级和高等级所需的主要技巧，包括频数表、直方图、箱线图、累积频数图和散点图。
1. Types of Data | 数据类型
The type of data determines which display technique is appropriate. In IB questions, you will often be asked to justify your choice of graph.
数据类型决定哪种展示技巧是恰当的。在IB题目中，你常常需要说明选择某一种图形的理由。
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Categorical data consist of names, labels or categories, such as favourite subject, country or colour.
分类数据由名称、标签或类别组成，例如最喜欢学科、国家或颜色。
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Numerical discrete data can only take certain values, usually whole numbers, such as the number of students in a class.
数值离散数据只能取某些特定值，通常是整数，例如一个班级的学生人数。
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Numerical continuous data can take any value within a range, such as height, mass or time.
数值连续数据可以在某个范围内取任意值，例如身高、质量或时间。
Remember that “temperature” can be continuous, while “number of marks” is usually discrete. The same data set may sometimes be grouped to make it easier to display, but grouping always loses some detail.
请记住，”温度”可以是连续的，而”分数”通常是离散的。同一组数据有时可以进行分组以方便展示，但分组总会丢失一些细节。
2. Frequency Tables and Grouped Data | 频数表与分组数据
A frequency table records how often each value, or group of values, occurs. For ungrouped data, each distinct value is listed with its frequency.
频数表记录每个值或每组值出现的次数。对于未分组数据，每个不同的值都会列出其频数。
| Score | Frequency |
|---|---|
| 2 | 3 |
| 3 | 5 |
| 4 | 7 |
| 5 | 2 |
For grouped data, the class interval must not overlap. Always identify the class boundaries before drawing a histogram.
对于分组数据，组区间不能重叠。在绘制直方图之前，必须先确定组边界。
Class width = Upper class boundary − Lower class boundary
Midpoint = (Lower class limit + Upper class limit) ÷ 2
For example, the interval 10 ≤ x < 20 has lower boundary 10, upper boundary 20, class width 10 and midpoint 15.
例如，区间 10 ≤ x < 20 的下边界是10，上边界是20，组宽为10，组中点为15。
3. Bar Charts and Pie Charts | 条形图与饼图
Bar charts are used for categorical or discrete data. The height of each bar is proportional to the frequency, and bars should have equal width.
条形图用于分类数据或离散数据。每个条形的高度与频数成比例，条形宽度应当相等。
Pie charts are used to show parts of a whole. Each sector angle is calculated from the frequency.
饼图用于显示整体中各部分的比例。每个扇形的角度由频数计算得出。
Sector angle = (Frequency ÷ Total frequency) × 360°
Suppose 30 students choose Maths, Physics, Chemistry and Biology with frequencies 12, 8, 5 and 5. The sector angles are:
假设30名学生选择的科目为数学、物理、化学和生物，频数分别为12、8、5、5，则扇形角度为：
| Subject | Frequency | Angle |
|---|---|---|
| Maths | 12 | 144° |
| Physics | 8 | 96° |
| Chemistry | 5 | 60° |
| Biology | 5 | 60° |
A bar chart compares categories, while a pie chart emphasizes proportions. Do not use a pie chart for many categories, because small sectors become difficult to read.
条形图用于比较类别，而饼图强调比例。不要用饼图展示过多类别，因为小扇形会变得难以辨认。
4. Histograms | 直方图
A histogram displays continuous data. Unlike a bar chart, a histogram has no gaps between bars because the horizontal scale is continuous.
直方图展示连续数据。与条形图不同，直方图的条形之间没有空隙，因为横轴是连续的。
In a histogram, the area of each bar must be proportional to the frequency. When class widths are unequal, use frequency density.
在直方图中，每个条形的面积必须与频数成比例。当组宽不相等时，应使用频数密度。
Frequency density = Frequency ÷ Class width
The vertical axis of a histogram is labelled frequency density, not frequency.
直方图的纵轴标记为频数密度，而不是频数。
| Class interval | Frequency | Width | Frequency density |
|---|---|---|---|
| 0 ≤ x < 10 | 5 | 10 | 0.5 |
| 10 ≤ x < 15 | 12 | 5 | 2.4 |
| 15 ≤ x < 20 | 18 | 5 | 3.6 |
| 20 ≤ x < 30 | 10 | 10 | 1.0 |
In the example above, the third class has the greatest frequency density, so it produces the tallest bar even though total frequencies in other classes may be similar.
在上面的例子中，第三组的频数密度最大，因此它的条形最高，即使其他组的总频数可能相近。
5. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram is a quick way to display a small set of numerical data while preserving every original value.
茎叶图是一种快速展示小型数值数据集的方法，同时保留每一个原始数据值。
For example, consider the data: 12, 15, 22, 27, 31, 34, 38, 42, 45, 49.
例如，考虑数据：12, 15, 22, 27, 31, 34, 38, 42, 45, 49。
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