Statistical Graphs | 统计图表

📚 Statistical Graphs | 统计图表

In IB Statistics, statistical graphs are essential tools for visualising data, uncovering patterns, and supporting statistical inference. From the simplest bar chart to the more advanced cumulative frequency curve, each graph type conveys information in its own unique way. This article explores the key statistical graphs required by the IB syllabus, covering their construction, interpretation, strengths, and common pitfalls.

在 IB 统计中，统计图表是将数据可视化、揭示模式和支持统计推断的重要工具。从最简单的条形图到更复杂的累积频率曲线，每种图形都以独特的方式传达信息。本文深入探讨 IB 教学大纲要求的关键统计图表，涵盖它们的构建、解读、优势以及常见误区。

1. The Role of Statistical Graphs | 统计图表的作用

Statistical graphs translate raw numerical data into a visual format, allowing us to see the shape, centre, spread, and any unusual features at a glance. In IB Mathematics: Applications and Interpretation as well as Analysis and Approaches, students are expected to produce graphs both by hand and using technology such as GDCs. The ability to read and criticise graphs is equally important, as misleading scales or omitted labels can distort the message.

统计图表将原始数值数据转化为可视化形式，使我们能够一眼看清分布的形状、中心、离散程度及任何异常特征。在 IB 数学：应用与解释以及分析与方法中，学生既要学会手工绘制，也要会用图形计算器等工具生成图形。同样重要的还有读图和批判图表的能力，因为误导性的刻度或遗漏的标签会扭曲真实信息。

A well-chosen graph makes comparisons straightforward and highlights trends that might be hidden in a table of numbers. For example, a histogram can immediately show whether a dataset is symmetric, skewed, or bimodal, while a box plot succinctly compares several groups on the same scale.

一幅恰当的图形能让比较变得直截了当，并突显出隐藏在数字表格中的趋势。例如，直方图可以立即显示数据集是对称的、偏斜的还是双峰的，而箱线图可以在同一尺度上简洁地比较多个组。

2. Bar Charts and Pie Charts | 条形图与饼图

Bar charts display categorical data using rectangular bars whose lengths are proportional to the frequencies or relative frequencies of each category. The bars are usually separated by small gaps to emphasise that the categories are distinct, not continuous. Horizontal or vertical orientation is acceptable, and the order of categories can often be rearranged to highlight a pattern.

条形图用矩形条展示分类数据，条形的长度与每个类别的频数或相对频数成正比。条形之间通常留有小间隙，以强调类别是离散的而非连续的。水平或垂直方向均可，类别的排列顺序常常可以重新调整以突出规律。

Pie charts represent categories as sectors of a circle, with the central angle proportional to the frequency of each category. A full circle corresponds to 360° and the angle for a category with frequency f is (f / total) × 360°. While pie charts give a quick impression of part-to-whole relationships, they become hard to read when there are many small slices or when precise comparisons are needed.

饼图将类别表示为圆的扇形，圆心角与每个类别的频数成正比。整个圆对应 360°，频数为 f 的类别对应的角度为 (f / 总数) × 360°。饼图能快速展示部分与整体的关系，但当扇形过多过小或需要精确比较时，饼图就难以阅读了。

In IB data-based questions, you may be asked to decide which chart is more appropriate. Bar charts are generally preferred when comparing actual frequencies, while pie charts work better for showing proportions in a small number of categories.

在 IB 数据分析题中，你可能会被要求判断哪种图表更合适。比较实际频数时通常偏好条形图，而类别较少时展示比例关系则饼图更合适。

3. Dot Plots | 点图

A dot plot is a simple graph that uses a dot for each observation above a horizontal number line. It is particularly useful for small, ungrouped datasets where individual values can be clearly seen. Dot plots reveal clusters, gaps, and potential outliers without any smoothing.

点图是一种简单图形，在水平数轴上方用点标记每个观测值。它特别适用于小型未分组数据集，可以清晰地看到每个值。点图无需任何平滑处理就能揭示聚集区、空隙和潜在的异常值。

To construct a dot plot, draw a scaled axis and stack dots vertically for repeated values. For instance, the scores 5, 6, 6, 7, 8, 8, 8, 9 would create a column of three dots at score 8. The shape is immediately visible, and the mode is simply the tallest stack.

构建点图时，先画出带刻度的轴，然后对重复值垂直堆叠圆点。例如，得分 5, 6, 6, 7, 8, 8, 8, 9 将在分数 8 处形成三个点的一列。分布形状即刻可见，众数就是最高的那堆点。

IB exam papers sometimes ask students to identify the median and range from a dot plot. The median is found by counting inwards from both ends of the ordered dots, and the range is the difference between the largest and smallest values on the plot.

IB 考试有时要求学生从点图中找出中位数和极差。中位数可以通过从有序点的两端向中间计数找到，极差则是图上最大值与最小值的差。

4. Stem-and-Leaf Plots | 茎叶图

A stem-and-leaf plot splits each data value into a stem (all but the final digit) and a leaf (the final digit), preserving the original data while showing its distribution. For example, 43 becomes stem 4, leaf 3. Leaves are ordered, and a key must be provided to explain the place value, e.g. ‘4 | 3 represents 43’ or ‘4 | 3 represents 4.3’ if scaled.

茎叶图将每个数据值拆分为茎（除去最后一位数字的部分）和叶（最后一位数字），既保留了原始数据又展示了分布。例如 43 变成茎 4、叶 3。叶须按序排列，且必须提供图例说明位值，如“4 | 3 表示 43”或缩放后“4 | 3 表示 4.3”。

Back-to-back stem-and-leaf plots allow comparison of two related datasets by sharing a central column of stems. This is common in IB questions where, for instance, exam scores of boys and girls are compared. The stem is written once, with leaves for one group extending to the left and leaves for the other group extending to the right.

背靠背茎叶图通过共享中间的茎列来比较两个相关数据集。这在 IB 题目中很常见，例如比较男生和女生的考试成绩。茎只写一次，一组数据的叶向左延伸，另一组的叶向右延伸。

Advantages of stem-and-leaf plots include retaining the original data and showing the shape of the distribution, but they are most effective for small to moderate datasets with roughly similar stem structures.

茎叶图的优点包括保留原始数据和显示分布形状，但对于茎结构大致相似的小到中等数据集最为有效。

5. Histograms and Frequency Density | 直方图与频率密度

Histograms display the distribution of continuous data by dividing the range into intervals (bins) and drawing adjacent rectangles. The area of each rectangle is proportional to the frequency it represents. When all intervals have the same width, the rectangle heights are proportional to frequencies. However, if the class widths are unequal, the height must represent frequency density instead of raw frequency.

直方图通过将数据范围划分为区间（组）并绘制相邻矩形来展示连续数据的分布。每个矩形的面积与其所代表的频数成正比。当所有组距相等时，矩形的高度与频数成正比。但如果组距不相等，高度必须表示频率密度而非原始频数。

Frequency density = Frequency ÷ Class width

This formula is central to the IB syllabus. Always check whether class widths are equal before labelling the vertical axis. A histogram with equal-width intervals may have the vertical axis labelled ‘Frequency’, while a histogram with unequal intervals should use ‘Frequency density’.

该公式是 IB 教学大纲的核心。标注纵轴前，务必检查组距是否相等。等距区间的直方图纵轴可以标注“频数”，而组距不等的直方图应使用“频率密度”。

When constructing a histogram, the modal class is the interval with the tallest bar (or greatest frequency density). The shape—symmetric, positively skewed, negatively skewed, or bimodal—gives clues about the underlying distribution. The histogram is a precursor to understanding probability density functions in later topics.

构建直方图时，众数区间是条形最高（或频率密度最大）的区间。形状——对称、正偏、负偏或双峰——为底层分布提供线索。直方图是理解后续概率密度函数的基础。

6. Cumulative Frequency Graphs | 累积频率图

A cumulative frequency graph, or ogive, plots the running total of frequencies against the upper class boundary of each interval. The points are joined smoothly, usually by a curve or straight line segments. Cumulative frequency always rises from zero to the total frequency, never decreasing.

累积频率图（或 Ogive）将频数的累积总和相对于每个区间的上组界描点。各点平滑连接，通常用曲线或折线。累积频率总是从零上升到总频数，绝不会下降。

From an ogive, we can estimate key percentiles: the median is found at 50% of the total frequency, the lower quartile Q₁ at 25%, and the upper quartile Q₃ at 75%. Draw a horizontal line from the desired cumulative frequency on the vertical axis to the curve, then drop a vertical line to the horizontal axis to read the value. This visual method underpins many IB questions.

从累积频率图中可以估计关键百分位数：中位数对应总频数的 50%，下四分位数 Q₁ 对应 25%，上四分位数 Q₃ 对应 75%。从纵轴上所需的累积频率处画水平线到曲线，再向下画垂线到横轴读取数值。这种可视化方法是许多 IB 题目的基础。

The interquartile range IQR = Q₃ – Q₁ can be read directly from the graph and gives a measure of spread that is resistant to outliers. Cumulative frequency graphs also allow us to answer questions like ‘How many items are less than a given value?’ or ‘What value is not exceeded by 90% of the data?’

四分位距 IQR = Q₃ − Q₁ 可直接从图中读出，它给出一个不受异常值影响的离散度度量。累积频率图还可以回答诸如“有多少个体小于给定值？”或“90% 的数据不超过什么值？”之类的问题。

7. Box-and-Whisker Plots | 箱线图

A box-and-whisker plot (box plot) is a compact display of the five-number summary: minimum, Q₁, median, Q₃, and maximum. The box spans the interquartile range, with a line inside marking the median. Whiskers extend to the minimum and maximum, unless outliers are present, in which case they stop at the smallest and largest values within the fences.

箱线图（盒须图）是对五数概括——最小值、Q₁、中位数、Q₃ 和最大值——的紧凑展示。盒子覆盖四分位距，内部有一条线标示中位数。须线延伸到最小值和最大值，除非存在异常值，此时须线停在界限内的最小和最大值处。

Outliers are identified using the 1.5 × IQR rule. A data point is considered an outlier if it lies below Q₁ − 1.5 × IQR (lower fence) or above Q₃ + 1.5 × IQR (upper fence). The IB syllabus expects students to identify and plot outliers individually as open or solid circles beyond the whiskers.

异常值通过 1.5 × IQR 规则识别。数据点若落在 Q₁ − 1.5 × IQR（下界限）以下或 Q₃ + 1.5 × IQR（上界限）以上，就被视为异常值。IB 大纲要求学生识别异常值并将其在须线之外单独绘制为空心或实心圆点。

Box plots are especially useful for side-by-side comparisons of several datasets, as they highlight differences in centre, spread, and skewness. When the median is closer to Q₁, the distribution is positively skewed; when closer to Q₃, negatively skewed. Symmetry shows the median roughly in the middle of the box.

箱线图在并排比较多个数据集时特别有用，因为它能突出中心、离散度和偏斜程度的差异。中位数靠近 Q₁ 表示分布正偏；靠近 Q₃ 表示负偏。如果中位数大致在盒子中间，则说明接近对称。

8. Scatter Plots and Correlation | 散点图与相关性

A scatter plot displays paired numerical data (x, y) as points on a Cartesian plane, revealing the relationship between two variables. The pattern, direction, and strength of the association are investigated visually before numerical measures like Pearson’s product-moment correlation coefficient r are calculated.

散点图将成对的数值数据 (x, y) 在笛卡尔平面上显示为点，揭示两个变量之间的关系。在计算皮尔逊积矩相关系数 r 等数值度量之前，先通过视觉方式考察关联的模式、方向和强度。

Positive correlation means y tends to increase as x increases; negative correlation means y tends to decrease as x increases. If no clear pattern exists, the variables may have zero correlation. The IB course also distinguishes between linear and non-linear association; r only measures linear correlation strength.

正相关意味着 y 随 x 增大而增大；负相关意味着 y 随 x 增大而减小。若没有明显模式，变量可能零相关。IB 课程还区分线性和非线性关联；r 只度量线性相关的强度。

Regression lines, particularly the least-squares regression line y = a + bx, are often superimposed on scatter plots to model the relationship. The line of best fit should pass through the mean point (x̄, ȳ). Outliers and influential points can significantly affect the regression equation and must be discussed in IB investigations.

回归直线，尤其是最小二乘回归线 y = a + bx，常被叠加在散点图上以模拟关系。最佳拟合线应通过均值点 (x̄, ȳ)。异常值和强影响点会显著影响回归方程，必须在 IB 探究中加以讨论。

9. Comparative Graphs: Population Pyramids and Radar Charts | 比较图形：人口金字塔与雷达图

While less common in core IB statistics, population pyramids (back-to-back horizontal bar charts) and radar charts appear in some project-based tasks. A population pyramid shows age distributions of males and females simultaneously, with horizontal bars extending left and right. Radar charts overlay multiple quantitative variables on axes radiating from a centre, useful for comparing profiles of individuals or groups.

尽管在 IB 核心统计中不常见，人口金字塔（背靠背水平条形图）和雷达图会出现在某些项目任务中。人口金字塔同时展示男性和女性的年龄分布，用左右延伸的水平条形表示。雷达图从中心辐射出多根轴，叠加多个定量变量，便于比较个体或群体的特征画像。

Although the IB exam rarely asks students to construct these by hand, an awareness of them broadens statistical literacy. In both graphs, careful scaling and clear labelling are essential to avoid misinterpretation.

尽管 IB 考试很少要求手工绘制这些图形，但了解它们能拓宽统计素养。在这两种图形中，仔细标注尺度和清晰标签对于避免误解至关重要。

10. Choosing the Right Graph | 选择正确的图形

Selecting an appropriate graph depends on the data type (categorical, discrete, continuous) and the purpose of the analysis. Categorical data usually call for bar charts or pie charts. Small discrete datasets work well with dot plots or stem-and-leaf plots. Large continuous datasets are best summarised with histograms. To compare distributions, box plots or back-to-back stem plots are excellent. To explore relationships between two continuous variables, a scatter plot is indispensable.

选择合适的图形取决于数据类型（分类、离散、连续）和分析目的。分类数据通常适合条形图或饼图。小型离散数据集用点图或茎叶图表现良好。大型连续数据集最好用直方图概括。比较分布时，箱线图或背靠背茎叶图非常出色。探究两个连续变量之间的关系时，散点图不可或缺。

IB questions often present a scenario and ask which graph would best represent the data, or they may give you a graph and ask what can be concluded. Demonstrating knowledge of each graph’s purpose and limitations is key to earning high marks.

IB 题目常给出一个情境，询问哪种图形最适合表示数据，或者给出图形并要求得出什么结论。展示对每种图形用途和局限性的认识是获得高分的关键。

11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One frequent mistake is confusing histograms with bar charts. Remember, a histogram has no gaps between bars (unless a class has zero frequency) and the horizontal axis is a continuous number line. Labelling the vertical axis ‘Frequency’ for unequal-width intervals is incorrect; use ‘Frequency density’ instead.

一个常见错误是混淆直方图与条形图。请记住，直方图的条形之间没有空隙（除非某组频数为零），且横轴是连续数轴。对于不等距区间，纵轴标为“频数”是不正确的；应使用“频率密度”。

Misreading cumulative frequency graphs is another common pitfall. Always check whether the question asks for the number or percentage of items below a value. When estimating the median, draw your lines carefully: the median is where the ogive reaches half the total frequency, not half the horizontal axis length.

误读累积频率图是另一个常见陷阱。务必检查问题是问低于某值的数量还是百分比。估计中位数时，线条要画准确：中位数是 ogive 达到总频数一半的位置，而不是横轴一半长度的位置。

With box plots, students sometimes forget to identify outliers or misplace the whiskers. The whiskers are drawn to the most extreme points that are not outliers, not automatically to the observed minimum and maximum if outliers exist. Also, always label a box plot’s scale and the five-number summary values.

对于箱线图，学生有时忘记识别异常值或错误放置须线。须线应画到非异常值的最极端点，如果存在异常值，不能机械地画到观测的最小值和最大值。此外，务必标注箱线图的刻度和五数概括值。

When constructing stem-and-leaf plots, forgetting to order the leaves or to provide a key is penalised in IB exams. The key is a non-negotiable requirement, as it defines the place value of every observation.

构建茎叶图时，忘记给叶子排序或提供图例都会在 IB 考试中被扣分。图例是强制要求，因为它定义了每个观察值的位值。

12. Technology and Statistical Graphs | 技术与统计图形

IB students are expected to use a graphical display calculator (GDC) or approved software to produce graphs quickly and accurately. Tools like TI-Nspire, Casio fx-CG50, and Desmos can generate histograms, box plots, scatter plots and regression lines with ease. However, you still need to know how to interpret the output and to sketch graphs manually for certain exam questions.

IB 学生应能使用图形计算器或经批准的软件快速准确地生成图形。TI-Nspire、Casio fx-CG50 和 Desmos 等工具可以轻松生成直方图、箱线图、散点图和回归直线。但某些考试题目仍需要你知道如何解读输出结果并手工绘制草图。

When using technology, always adjust window settings, check the scale, and verify that the graph accurately represents the data. A histogram on a GDC with poorly chosen bin width can hide important features. The statistical reasoning behind the graph is more important than the button-pressing sequence.

使用技术工具时，务必调整窗口设置，检查刻度，并验证图形是否准确反映了数据。在图形计算器上，组距选择不当的直方图可能掩盖重要特征。图形背后的统计推理比按键顺序重要得多。

Mastering both hand-drawn and technology-generated graphs gives you flexibility and deepens your understanding of data visualisation, a skill that extends well beyond the IB exam room.

掌握手绘和技术生成两种图形方式可以让你拥有灵活性，并加深对数据可视化的理解，这项技能远超 IB 考场本身。

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Statistical Graphs | 统计图表