📚 Representing Data | 数据表示
In statistics, raw data is often difficult to interpret directly. Representing data involves selecting appropriate tables, charts, and summary measures to reveal patterns, trends, and distributions. The International Baccalaureate Mathematics courses require students to choose suitable representations for discrete and continuous data, understand properties of each graph, and use visual displays to calculate statistical measures such as quartiles or to identify outliers. Effective data representation is a key skill in both Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabuses.
在统计学中，原始数据往往难以直接解读。数据表示就是选择合适的表格、图表和汇总度量，以揭示模式、趋势和分布。国际文凭数学课程要求学生能够为离散数据和连续数据选择恰当的表示方法，理解各图表的特性，并利用可视化展示计算四分位数等统计量或识别异常值。有效的数据表示是分析数学和应用数学两门课程中的核心技能。
1. Types of Data | 数据类型
Before choosing a representation, it is essential to classify data. Data can be categorical (qualitative) or numerical (quantitative). Numerical data is further split into discrete data, which takes only isolated values such as counts, and continuous data, which can take any value within an interval. IB problems often ask students to decide whether a given set is discrete or continuous and to choose the most suitable diagram accordingly.
在选择表示形式之前，必须先对数据进行分类。数据可以是类别型（定性）或数值型（定量）。数值型数据又分为离散数据（只能取孤立的数值，如计数）和连续数据（可以在一个区间内取任意值）。IB问题中经常要求学生判断给定数据集是离散的还是连续的，并据此选择最合适的图表。
2. Frequency Tables and Grouped Data | 频率表与分组数据
A frequency table organises raw data by listing each distinct value alongside its frequency. For large or continuous datasets, data are grouped into class intervals. When constructing a grouped frequency table, it is crucial to define class boundaries without gaps, using notation like 0 ≤ x < 10, 10 ≤ x < 20, and so on, so that every possible value belongs to exactly one class. The midpoint of each interval is used to estimate the mean of grouped data.
频率表通过列出每个不同数值及其频数来组织原始数据。对于大型或连续数据集，数据会被分入组距。在构建分组频率表时，必须确保组界之间没有空隙，使用诸如0 ≤ x < 10，10 ≤ x < 20这样的记法，使得每个可能的值恰好属于一个组。组距的中点被用来估计分组数据的均值。
For estimating the mean from a grouped frequency table, the formula estimated mean = Σ(f × xₘ) / Σf is applied, where f is the frequency and xₘ is the class midpoint. This is an approximation because individual data points are assumed to lie at the centre of their interval.
从分组频率表估计均值时，使用公式估计均值 = Σ(f × xₘ) / Σf，其中f是频数，xₘ是组中点。这是一个近似值，因为假定每个数据点都位于其区间的中心。
estimated mean = Σ(f × xₘ) / Σf
3. Bar Charts and Pie Charts | 条形图与饼图
g a circle into sectors whose angles are proportional to the frequencies; the angle for a category with frequency f out of total n is (f / n) × 360°.
条形图用于类别数据或离散数值数据。每个类别有一根矩形条，其高度代表频数或相对频率。条与条之间有间隙，以强调类别是独立的。饼图通过将圆分割为扇形来展示整体中的比例，扇形的角度与频数成正比；对于一个频数为f且总数为n的类别，其角度为 (f / n) × 360°。
Bar charts can also display grouped discrete data, but when the horizontal axis represents numerical ranges, a histogram may be more appropriate. Pie charts become difficult to read when there are many small categories, so they are most effective for fewer than six or seven categories.
条形图也可以显示分组的离散数据，但当横轴表示数值范围时，直方图可能更为合适。当有多个小类别时，饼图变得难以阅读，因此当类别数少于六或七个时饼图最为有效。
4. Histograms and Frequency Density | 直方图与频数密度
A histogram is used for continuous data or grouped data with equal or unequal class widths. Unlike a bar chart, the bars in a histogram touch each other to reflect the continuous nature of the data. The area of each bar is proportional to the frequency. When class widths are unequal, frequency density must be used as the vertical axis. The frequency density for a class is calculated as frequency divided by class width.
直方图用于连续数据或具有相等或不等组距的分组数据。与条形图不同，直方图中的条彼此相连，以反映数据的连续性。每个条的面积与频数成正比。当组距不相等时，必须使用频数密度作为纵轴。一个组的频数密度等于频数除以组距。
frequency density = frequency ÷ class width
A common IB exam task is to complete a histogram where some bars have unequal widths and to use the area to retrieve frequencies. Understanding that frequency = frequency density × class width helps in drawing and interpreting these graphs accurately.
IB考试中常见的一个任务是补全组距不等的直方图，并根据面积反推频数。理解频数 = 频数密度 × 组距有助于准确地绘制和解读这类图形。
5. Cumulative Frequency Graphs | 累积频数图
A cumulative frequency graph, or ogive, plots the running total of frequencies against the upper class boundary of each interval. The points are joined with a smooth curve or straight line segments. The graph allows estimation of the median, quartiles, and percentiles without needing the raw data. The lower quartile Q₁, median Q₂, and upper quartile Q₃ correspond to the values on the horizontal axis at the cumulative frequencies that are one-quarter, one-half, and three-quarters of the total frequency.
累积频数图（或称折线图）将累积频数相对于每个区间的上组界进行描点，并用光滑曲线或线段连接。该图允许在不需要原始数据的情况下估计中位数、四分位数和百分位数。下四分位数Q₁、中位数Q₂和上四分位数Q₃分别对应纵轴累积频数为总频数的四分之一、一半和四分之三时横轴上的数值。
From a cumulative frequency curve, the interquartile range (IQR = Q₃ − Q₁) can be read directly. The IQR measures the spread of the middle 50% the median and the IQR, as well as to find the number of observations below a given value.
从累积频数曲线上可以直接读出四分位距（IQR = Q₃ − Q₁）。IQR衡量了中间50%数据的离散程度，且不受异常值影响。在IB试题中，学生常被要求绘制累积频数曲线，然后利用它估计中位数和IQR，并求出低于某一给定值的观测个数。
6. Box-and-Whisker Plots | 箱线图
d-whisker plot provides a five-number summary of a dataset: the minimum, lower quartile Q₁, median, upper quartile Q₃, and maximum. The box spans from Q₁ to Q&# ximum values that are not outliers. Outliers are typically defined as data points below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR. These are plotted as individual points beyond the whiskers.
箱线图提供了数据集的五数概括：最小值、下四分位数Q₁、中位数、上四分位数Q₃和最大值。箱体从Q₁延伸到Q₃，箱内有一条线标示中位数。触须线延伸到非异常值的最小值和最大值。异常值通常定义为小于Q₁ − 1.5×IQR或大于Q₃ + 1.5×IQR的数据点，它们被画在触须线之外，用孤立点表示。
Box plots are particularly useful for comparing the centre, spread, and skewness of several datasets side by side. A box plot with a longer whisker on the right and the median closer to Q₁ indicates positive skew. The symmetrical case has the median centred and whiskers of similar length.
箱线图特别适用于并列比较多个数据集的中心、离散程度和偏态。右侧触须较长且中位数靠近Q₁的箱线图表明正偏态。对称情况则是中位数居中，且两侧触须长度相似。
7. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram organises data while preserving each individual value. Each observation is split into a stem (usually the leading digits) and a leaf (the last digit). The stems are listed in a vertical column, and the leaves are written in order next to their stem. This display has the advantage of showing the shape of the distribution and allowing quick identification of the mode, median, and range.
茎叶图在组织数据的同时保留了每一个原始值。每个观测值被分为茎（通常是前导数字）和叶（最后一位数字）。茎按竖列排列，叶按顺序写在对应的茎旁边。这种表示法的优点是能够显示分布的形状，并快速识别众数、中位数和极差。
For example, the data 12, 15, 21, 23, 23, 27 could be represented with stem 1 | 2 5 and stem 2 | 1 3 3 7. A key must always be provided, such as "1 | 2 means 12". Back-to-back stem-and-leaf diagrams are an elegant way to compare two related datasets by sharing a central stem.
例如，数据12, 15, 21, 23, 23, 27可以表示为茎1 | 2 5和茎2 | 1 3 3 7。必须提供图例，如“1 | 2 表示12”。背靠背茎叶图通过共享中间的茎，可以优雅地比较两个相关的数据集。
IB questions may ask candidates to find the median directly from an ordered stem-and-leaf diagram. If the number of leaves n is odd, the median is the middle leaf; if n is even, it is the average of the two middle leaves. The diagram also makes it easy to locate Q₁ and Q₃ without a calculator.
IB题目可能要求考生直接从有序茎叶图中找出中位数。如果叶子数量n为奇数，中位数是居中的那个叶子；如果n为偶数，则是中间两个叶子的平均值。该图也便于在不使用计算器时确定Q₁和Q₃。
8. Scatter Plots and Correlation | 散点图与相关性
A scatter plot displays the relationship between two numerical variables by plotting each pair (x, y) as a point on the Cartesian plane. The independent variable is placed on the horizontal axis, and the dependent variable on the vertical axis. The pattern of points can reveal the direction (positive or negative), form (linear or non-linear), and strength of the relationship.
散点图通过在笛卡尔平面上描出每一对(x, y)点来展示两个数值变量之间的关系。自变量放在横轴上，因变量放在纵轴上。点的分布形态可以揭示关系的方向（正或负）、形式（线性或非线性）以及强度。
IB students learn to describe correlation qualitatively: strong, moderate, weak, or zero. In AI, they also calculate Pearson's product-moment correlation coefficient r and interpret its value. It is important to remember that correlation does not imply causation. A scatter plot showing a clear curve may suggest a non-linear model, which can be linearised by transformation.
IB学生学会定性描述相关性：强、中等、弱或零。在AI课程中，还要计算皮尔逊积矩相关系数r并解释其值。重要的是记住相关性并不意味着因果关系。散点图中出现明显的曲线可能暗示非线性模型，这可以通过变量变换进行线性化处理。
9. Measures of Central Tendency | 集中趋势的度量
The three principal measures of central tendency are the mean, median, and mode. The mean x̄ is the arithmetic average, calculated as Σx/n. The median is the middle value when the data are ordered, and the mode is the most frequently occurring value. These measures are often displayed alongside data visualisations. For skewed data, the median is a better measure of centre because the mean is pulled in the direction of the skew.
集中趋势的三个主要度量是均值、中位数和众数。均值x̄是算术平均，计算为Σx/n。中位数是数据排序后居中的值，而众数是出现频率最高的值。这些度量常与数据可视化一并展示。对于偏态数据，中位数是更好的中心度量，因为均值会被拉向偏态的方向。
x̄ = Σx / n
In grouped data, the mode is found in the modal class, the interval with the highest frequency. The choice of which measure to report depends on the data type and the presence of outliers. For symmetric distributions with no extreme values, the mean is usually preferred.
在分组数据中，众数位于具有最高频数的众数组内。选择报告哪种度量取决于数据类型以及是否存在异常值。对于没有极端值的对称分布，通常首选均值。
10. Measures of Spread | 离散程度的度量
ation. The range is simply the maximum minus the minimum. The interquartile range IQR = Q₃ − Q&# is sensitive to outliers. For a sample, it is calculated using the formula with (n − 1) in the denominator.
离散程度通过极差、四分位距、方差和标准差来描述。极差就是最大值减去最小值。四分位距IQR = Q₃ − Q₁描述了中间一半数据的离散程度且具有稳健性。标准差s衡量数据点偏离均值的典型距离，对异常值敏感。对于样本，计算时公式的分母使用(n − 1)。
s = √[ Σ(x − x̄)² / (n − 1) ]
When comparing two datasets, using both the centre and the spread provides a more complete picture. Box plots are excellent for displaying the IQR and outliers visually, while presenting the standard deviation alongside the mean works well in normal-like distributions.
在比较两个数据集时，同时使用中心度和离散度可以提供更全面的图景。箱线图非常适合直观展示IQR和异常值，而在类正态分布中，将标准差与均值一并呈现效果很好。
11. Choosing the Right Representation | 选择恰当的表示方法
Selecting an appropriate diagram depends on the data type and the purpose of the analysis. For comparing parts of a whole, a pie chart is suitable if there are few categories. For showing the frequency distribution of continuous data, a histogram is preferred. To find quartiles and identify outliers, a box plot or cumulative frequency graph works best. Stem-and-leaf diagrams offer a quick way to order data and preserve individual values, while scatter plots reveal relationships between two variables.
选择恰当的图表取决于数据类型和分析目的。若要比较整体中的部分，且类别较少时，饼图是合适的。要展示连续数据的频数分布，直方图更为可取。要找出四分位数并识别异常值，箱线图或累积频数图最好。茎叶图提供了一种快速排序数据并保留原始值的方式，而散点图则揭示了两个变量之间的关系。
In an IB examination, marks are often awarded for justifying the choice of representation. For instance, 'A histogram is chosen because the data are continuous and group widths vary' or 'A box plot is used to compare the median working hours across two groups because it clearly shows skew and outliers'. Avoiding misleading scales, truncated axes, or distorted areas is equally important.
在IB考试中，通常会有分数奖励给对表示方法选择进行论证的答案。例如，“选择直方图是因为数据是连续的且各组组距不同”，或者“使用箱线图来比较两组的工时中位数，因为它清晰地显示了偏态和异常值”。避免误导性的尺度、截断的坐标轴或扭曲的面积同样重要。
12. Common Pitfalls and IB Exam Tips | 常见误区与IB考试提示
Students frequently confuse bar charts with histograms; remember that bar charts have gaps and are for categorical or discrete data, while histograms have touching bars and represent continuous or grouped data. Another common error is neglecting to use frequency density on the vertical axis of a histogram with unequal class widths. When constructing a cumulative frequency graph, always plot points at the upper boundary of each interval, not the midpoint. For box plots, clearly label the scale and indicate any outliers with crosses or dots.
学生常常混淆条形图和直方图；要记住条形图有间隙，用于类别数据或离散数据，而直方图的条形相互接触，表示连续或分组数据。另一个常见错误是在组距不等的直方图中忘记在纵轴使用频数密度。绘制累积频数图时，描点总是在每个区间的上组界，而不在中点。对于箱线图，要清晰地标注尺度，并用叉号或圆点标出异常值。
In any question that asks for an interpretation, link the statistical measure back to the context. Saying 'the median mark was 65' might be accurate, but saying 'on average, half the students scored below 65 marks' shows deeper understanding. When describing a scatter plot, refer to direction, form, strength, and any unusual features. Practising past paper questions where you must select the graph and justify it will build confidence for the IB exam.
在任何要求解释的题目中，要将统计量与上下文联系起来。只说“中位数分数是65”也许没错，但如果说“平均看，一半学生的分数低于65分”则展示出更深的理解。描述散点图时，应提及其方向、形式、强度和任何异常特征。通过练习以前试卷中需要你选择图形并加以论证的题目，将为IB考试建立信心。
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