📚 Common Data Representation Methods in IB Mathematics | IB数学：常见数据表示方法汇总
Data representation is a fundamental skill in IB Mathematics. Whether you are working with statistical analysis, probability, or real-world modelling, choosing the right visual display helps you identify patterns, compare distributions, and communicate findings clearly. This article summarises the most common data representation methods you need to master for your exams.
数据表示是IB数学中的一项基本技能。无论你是在处理统计分析、概率还是现实世界建模，选择合适的可视化图形都能帮助你识别模式、比较分布并清晰地传达研究结果。本文将汇总你在考试中需要掌握的常见数据表示方法。
1. Frequency Table | 频率表
A frequency table organises raw data by listing each value (or class interval) alongside its frequency. It is the foundation for most statistical charts and calculations, such as the mean, median, and mode.
频率表通过列出每个数值（或类别区间）及其频次来整理原始数据。它是大多数统计图表和计算（如均值、中位数和众数）的基础。
For discrete data, list each distinct value and count its occurrences.
对于离散数据，列出每个不同的值并统计其出现次数。
For continuous data, group values into equal-width intervals and record frequencies.
对于连续数据，将数值分组为等宽区间并记录频次。
Always include a total row or column to verify the sum of frequencies equals the sample size.
始终包含合计行或列，以验证频次总和等于样本量。
Scores of 20 students → Score: 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10
Frequency table: Score 2 (1), 3 (2), 4 (3), 5 (4), 6 (3), 7 (2), 8 (2), 9 (2), 10 (1)
2. Bar Chart | 条形图
A bar chart uses rectangular bars of equal width to represent categorical or discrete data. The height (or length) of each bar corresponds to the frequency of that category.
条形图使用等宽的矩形条来表示分类或离散数据。每个条的高度（或长度）对应该类别的频次。
Bars should have gaps between them to show that categories are distinct.
条与条之间应有间隙，以表明类别是离散的。
The bars can be vertical or horizontal, but the axis must be clearly labelled.
条形可以是垂直的或水平的，但坐标轴必须清晰标注。
Use a bar chart when comparing categories or showing frequencies of non-numeric variables.
在比较类别或显示非数值变量的频次时使用条形图。
In IB exams, you may be asked to construct a bar chart from a frequency table or interpret it to find the mode, which is simply the category with the tallest bar.
在IB考试中，你可能会被要求根据频率表绘制条形图，或通过解读条形图来找到众数，即最高条所对应的类别。
3. Histogram | 直方图
A histogram is used for continuous data grouped into intervals. Unlike a bar chart, bars in a histogram touch each other because there is no gap between intervals.
直方图用于按区间分组的连续数据。与条形图不同，直方图中的条彼此相接，因为区间之间没有间隙。
The horizontal axis represents class intervals; the vertical axis represents frequency density (frequency ÷ class width) when intervals have unequal widths.
横轴表示类别区间；当区间宽度不等时，纵轴表示频率密度（频次 ÷ 组距）。
The area of each bar is proportional to the frequency of the interval.
每个条的面积与该区间的频次成正比。
For equal-width intervals, you can simply use frequency on the vertical axis.
对于等宽区间，可以直接在纵轴上使用频次。
Histograms allow you to estimate the mode (the interval with the highest density) and to visualise the overall shape of a continuous distribution.
直方图可以帮助你估计众数（密度最高的区间）并可视化连续分布的整体形状。
4. Pie Chart | 饼图
A pie chart is a circular diagram divided into sectors, each representing a proportion of the whole. The central angle of each sector is proportional to the category's frequency.
饼图是一个被分割成扇区的圆形图，每个扇区代表整体的一部分。每个扇区的圆心角与该类别的频次成正比。
To calculate the angle for a category: angle = (frequency ÷ total) × 360°.
计算类别角度的公式：角度 = (频次 ÷ 总和) × 360°。
Use a pie chart when you want to show relative proportions, not exact numerical comparisons.
当你希望显示相对比例而非精确数值比较时，使用饼图。
Avoid too many categories; too many slices make the chart hard to read.
避免过多类别；太多扇区会使图表难以阅读。
Example: 30 students, 12 prefer Maths → Sector angle = 12 ÷ 30 × 360° = 144°
示例：30名学生，12人偏好数学 → 扇形角度 = 12 ÷ 30 × 360° = 144°
5. Stem-and-Leaf Diagram | 茎叶图
A stem-and-leaf diagram is a quick way to display small sets of numerical data while retaining the original values. Each number is split into a stem (leading digit(s)) and a leaf (final digit).
茎叶图是一种快速显示小规模数值数据并保留原始值的方法。每个数字被拆分为茎（前导数字）和叶（最后一位数字）。
Stems are listed vertically in increasing order; leaves are written horizontally next to their stem.
茎按递增顺序垂直列出；叶水平写在对应的茎旁边。
Include a key to explain the stem and leaf units, e.g., 2 | 3 means 23.
包含图例以解释茎和叶的单位，例如 2 | 3 表示 23。
From a stem-and-leaf diagram, you can easily find the mode, median, and range.
从茎叶图中，你可以轻松找到众数、中位数和极差。
Stem-and-leaf diagrams are excellent for comparing two related distributions using back-to-back or double stems.
茎叶图非常适合通过背靠背或双茎形式比较两个相关分布。
6. Box-and-Whisker Plot | 箱线图
A box-and-whisker plot summarises a distribution using five key numbers: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. It is also called a boxplot.
箱线图使用五个关键数值来概括分布：最小值、下四分位数（Q₁）、中位数（Q₂）、上四分位数（Q₃）和最大值。它也被称为箱形图。
The box spans from Q₁ to Q₃, and the median is marked inside the box.
箱体从Q₁延伸到Q₃，中位数在箱内标记。
lest and largest values within 1.5 × IQR of the quartiles; any points beyond are outliers.
须从箱体延伸到距四分位数1.5 × IQR范围内的最小和最大值；超出该范围的点被视为离群值。
IQR = Q₃ − Q₁; it measures the spread of the middle 50% of the data.
IQR = Q₃ − Q₁；它衡量中间50%数据的离散程度。
Boxplots are ideal for comparing distributions across different groups, especially when checking symmetry, spread, and outliers.
箱线图非常适合比较不同组间的分布，尤其是在检查对称性、离散程度和离群值时。
7. Cumulative Frequency Graph | 累积频率图
A cumulative frequency graph plots the cumulative frequency against the upper boundary of each class interval. It produces an S-shaped (ogive) curve that is used to estimate median, quartiles, and percentiles.
累积频率图将累积频次与每个类区间的上边界绘制成图。它产生一条S形（卵形）曲线，用于估计中位数、四分位数和百分位数。
Plot points at the upper class boundary, then connect them with a smooth curve.
在上类边界处标记点，然后用平滑曲线连接它们。
To find the median, locate n/2 on the cumulative frequency axis and read across to the curve, then down to the data value.
要找到中位数，在累积频率轴上找到 n/2，水平延伸至曲线，再向下读取数据值。
The lower quartile is at n/4 and the upper quartile is at 3n/4.
下四分位数位于 n/4，上四分位数位于 3n/4。
Cumulative frequency diagram: Median ≈ value at n/2, Q₁ ≈ n/4, Q₃ ≈ 3n/4
累积频率图：中位数 ≈ n/2处的值，Q₁ ≈ n/4，Q₃ ≈ 3n/4
8. Frequency Polygon | 频率多边形
A frequency polygon is a line graph that connects the midpoints of the tops of histogram bars. It is a useful alternative to a histogram for comparing multiple distributions overlayed on the same axes.
频率多边形是一种连接直方图条形顶部中点的折线图。它是直方图的替代方案，特别适合在同一坐标轴上叠加比较多个分布。
For each class interval, find the midpoint (class mark) and the frequency.
对于每个类区间，找到中点（组中值）和频次。
Plot points at (midpoint, frequency) and connect them with straight lines.
在（中点，频次）处标记点，并用直线连接。
Start and end the polygon on the horizontal axis to enclose the area like a histogram.
在水平轴处开始和结束多边形，以像直方图一样围出面积。
Frequency polygons are especially helpful when you overlay two datasets because the lines make differences easier to see.
频率多边形特别适合叠加两组数据，因为折线使差异更容易观察。
9. Scatter Plot | 散点图
A scatter plot displays two numerical variables on a Cartesian plane, with one variable on the x-axis and the other on the y-axis. It is used to explore correlation and possible relationships between variables.
散点图在笛卡尔平面上显示两个数值变量，一个变量在x轴，另一个在y轴。它用于探索变量之间的相关性及可能的关系。
Each point represents an observation with coordinates (x, y).
每个点代表一个观测值，坐标为（x, y）。
Positive correlation: y tends to increase as x increases.
正相关：随着x增加，y趋于增加。
Negative correlation: y tends to decrease as x increases.
负相关：随着x增加，y趋于减少。
No correlation: points show no clear linear tendency.
无相关：点没有明显的线性趋势。
In IB Maths AI, you use scatter plots to find the line of best fit (regression line) and the correlation coefficient r.
在IB数学AI中，你使用散点图来找到最佳拟合线（回归线）和相关系数r。
10. Time Series Plot | 时间序列图
A time series plot shows how a variable changes over time. Time is always plotted on the horizontal axis, and the variable is plotted on the vertical axis.
时间序列图显示一个变量如何随时间变化。时间始终绘制在横轴上，变量绘制在纵轴上。
Data points are usually connected with straight lines to highlight trends.
数据点通常用直线连接以突出趋势。
You may need to identify seasonality (regular patterns) and cyclical movement.
你可能需要识别季节性（规律性模式）和周期性波动。
A moving average smooths short-term fluctuations and reveals the long-term trend.
移动平均可以平滑短期波动并揭示长期趋势。
Time series plots are common in applications and interpretation (AI) paper, especially when discussing extrapolation and reliability.
时间序列图在应用与解释（AI）试卷中很常见，尤其是在讨论外推和可靠性时。
11. Choosing the Right Representation | 选择合适的表示方法
To score well in IB Maths, you must be able to justify your choice of data representation. The decision depends on the type of data and the purpose of the visualisation.
要在IB数学中取得好成绩，你必须能够证明你的数据表示选择是合理的。决策取决于数据类型和可视化的目的。
| Data Type | Typical Representation | Key Information |
| Categorical | Bar chart, pie chart | Frequencies, proportions |
| Discrete numeric | Frequency table, bar chart, stem-and-leaf | Individual values, mode |
| Continuous numeric | Histogram, cumulative frequency, boxplot | Distribution, quartiles, spread |
| Two variables | Scatter plot | Correlation, regression |
| Time-based | Time series plot | Trend, seasonality |
Always check the IB question instruction: if it asks for a "statistical diagram", make sure your graph is fully labelled with a title and scales.
始终查看IB题目的指令：如果要求“统计图表”，确保你的图形有完整的标题和刻度标注。
12. Common Mistakes to Avoid | 常见错误避免
Students often lose marks in data representation questions due to minor but preventable errors. Here are the most frequent pitfalls.
学生常常在数据表示题目中因微小但可避免的错误而失分。以下是最常见的陷阱。
Using a bar chart for continuous data instead of a histogram.
对连续数据使用条形图而不是直方图。
Forgetting to use frequency density when class widths are unequal.
当组距不等时忘记使用频率密度。
Drawing a histogram with gaps between bars.
绘制直方图时条之间有间隙。
Misreading the key of a stem-and-leaf diagram.
误读茎叶图的图例。
Plotting cumulative frequency at the lower class boundary instead of the upper boundary.
在累积频率图中将点绘在类区间下边界而不是上边界。
Forgetting to identify outliers in a boxplot.
在箱线图中忘记识别离群值。
By practising each diagram carefully and reviewing these common errors, you will improve both your calculation and interpretation skills.
通过仔细练习每种图形并复习这些常见错误，你将提高计算和解释两方面的技能。
Understanding data representation is not just about drawing graphs — it is about telling a truthful story with data. Master the methods above, practise with past paper questions, and you will be well prepared for the IB Mathematics exams.
理解数据表示不仅仅是绘制图形——它是用数据讲述一个真实的故事。掌握上述方法，练习历年真题，你将充分准备好应对IB数学考试。
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