Data Representation | 数据表示

📚 Data Representation | 数据表示

In Edexcel A Level Mathematics, data representation is not simply about drawing a graph. It is about selecting the most appropriate diagram to expose key features such as central location, spread, shape, outliers and relationships between variables.

在 Edexcel A Level 数学中，数据表示不仅仅是画图。更重要的是选择最合适的图示来揭示数据集的核心特征，例如集中位置、离散程度、分布形状、离群值以及变量之间的关系。

This topic underpins many statistical exam questions, from constructing histograms with unequal class widths to interpreting cumulative frequency curves and comparing box plots.

这一主题是许多统计考试题目的基础，从绘制组距不等的直方图，到解读累积频率曲线和比较箱线图，都需要扎实掌握数据表示方法。

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1. Types of Data and Their Representations | 数据类型及其表示方法

Before choosing a diagram, you must identify whether the data are categorical, discrete or continuous. Categorical data are usually shown in bar charts or pie charts, discrete data in vertical line charts or stem-and-leaf diagrams, and continuous data in histograms or cumulative frequency graphs.

在选择图示之前，必须先判断数据是分类数据、离散数据还是连续数据。分类数据通常用条形图或饼图表示，离散数据用垂线图或茎叶图表示，连续数据用直方图或累积频率图表示。

For example, the number of pets owned by students is discrete, while the time taken to complete a puzzle is continuous. Choosing a histogram for discrete number of pets would be less effective than a vertical line chart or a stem-and-leaf diagram.

例如，学生养宠物的数量是离散数据，而完成拼图所用的时间是连续数据。对于离散的宠物数量，使用直方图不如使用垂线图或茎叶图有效。

Data type | 数据类型	Example | 示例	Suitable diagram | 合适图示
Categorical | 分类	Favourite colour | 最喜欢的颜色	Bar chart, pie chart | 条形图、饼图
Discrete | 离散	Number of goals | 进球数	Vertical line chart, stem-and-leaf | 垂线图、茎叶图
Continuous | 连续	Height, time, mass | 身高、时间、质量	Histogram, cumulative frequency graph | 直方图、累积频率图

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2. Frequency Tables and Grouped Data | 频数表与分组数据

A frequency table summarises raw data by listing values or groups alongside their frequencies. When continuous data are grouped, we use class intervals such as 0 ≤ x < 10, 10 ≤ x < 20 and so on.

频数表通过列出数值或分组及其对应频数来汇总原始数据。当连续数据被分组时，我们使用组距，例如 0 ≤ x < 10、10 ≤ x < 20 等。

Grouping data loses the exact individual values. For grouped data, the midpoint of each class is often used as an approximate representative value when calculating an estimate of the mean.

分组会丢失确切的单个数值。对于分组数据，在计算均值的估计值时，通常用每个组的中点作为近似代表值。

Class boundaries are important for continuous data. A class written as 10−19 is often treated as 9.5 ≤ x < 19.5 if the data are rounded to the nearest integer. Always check whether boundaries are given in the question.

对于连续数据，组边界非常重要。写作 10−19 的组如果数据四舍五入到整数，通常应视为 9.5 ≤ x < 19.5。解题时必须检查题目给出的边界。

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

A histogram is used for continuous data. The area of each bar is proportional to the frequency, not necessarily the height. When class widths are unequal, the height must be calculated as frequency density.

直方图用于连续数据。每个条形的面积与频数成正比，而不一定是高度。当组距不相等时，高度必须用频率密度来计算。

frequency density = frequency ÷ class width

频率密度 = 频数 ÷ 组距

For example, a class 10 ≤ x < 20 with frequency 12 has class width 10, so its frequency density is 1.2. A class 20 ≤ x < 25 with frequency 9 has class width 5, so its frequency density is 1.8.

例如，组 10 ≤ x < 20 的频数为 12，组距为 10，因此频率密度为 1.2。组 20 ≤ x < 25 的频数为 9，组距为 5，因此频率密度为 1.8。

Many exam mistakes come from using frequency as the bar height for unequal class widths. Always label the vertical axis as ‘frequency density’ and check that the total area is proportional to the total frequency.

许多考试错误源于在组距不相等时直接用频数作为条形高度。务必在纵轴标注 ‘频率密度’，并检查总面积与总频数成正比。

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4. Stem-and-Leaf Diagrams | 茎叶图

A stem-and-leaf diagram preserves the original data values while showing the shape of the distribution. Each data value is split into a stem and a leaf, and leaves are usually ordered from smallest to largest.

茎叶图在展示分布形状的同时保留了原始数据值。每个数据值被分为茎和叶，叶通常按从小到大的顺序排列。

A key must always be given, for example ‘6 | 3 represents 63 cm’ or ‘6 | 3 represents 6.3’. Without a key, the diagram is not valid in Edexcel exam answers.

必须始终给出图例，例如 ‘6 | 3 表示 63 cm’ 或 ‘6 | 3 表示 6.3’。没有图例，图示在 Edexcel 考试答案中是无效的。

Back-to-back stem-and-leaf diagrams are useful for comparing two data sets, such as test scores for boys and girls. They share a common stem and allow quick comparison of shape and spread.

背靠背茎叶图适用于比较两个数据集，例如男生和女生的考试成绩。它们共享同一个茎，并能快速比较分布形状和离散程度。

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5. Box Plots and Outliers | 箱线图与离群值

A box plot displays the minimum, lower quartile Q1, median Q2, upper quartile Q3 and maximum. It is useful for showing spread, skewness and unusual values without displaying every raw data point.

箱线图显示最小值、下四分位数 Q1、中位数 Q2、上四分位数 Q3 和最大值。它适合展示离散程度、偏态和异常值，而无需显示每个原始数据点。

The interquartile range is IQR = Q3 − Q1. An outlier is defined as any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Outliers are plotted as individual crosses on a box plot.

四分位距为 IQR = Q3 − Q1。离群值被定义为低于 Q1 − 1.5 × IQR 或高于 Q3 + 1.5 × IQR 的任何数值。离群值在箱线图上用单独的叉号表示。

outlier if x < Q1 − 1.5 × IQR or x > Q3 + 1.5 × IQR

离群值条件：x < Q1 − 1.5 × IQR 或 x > Q3 + 1.5 × IQR

When outliers are present, the whiskers extend only to the smallest and largest non-outlier values. Box plots are excellent for comparing several data sets because they summarise location and spread visually.

当存在离群值时，须线只延伸到最小和最大的非离群值。箱线图非常适合比较多组数据，因为它们以可视化方式汇总了位置和离散程度。

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6. Cumulative Frequency Diagrams | 累积频率图

A cumulative frequency diagram shows the running total of frequencies plotted against the upper class boundaries. It usually produces a smooth S-shaped curve and is used to estimate medians, quartiles and percentiles.

累积频率图显示频率的累计总和与组上边界之间的关系。它通常产生一条平滑的 S 形曲线，用于估计中位数、四分位数和百分位数。

For grouped continuous data, plot the cumulative frequency at the upper class boundary of each interval. For example, for 0 ≤ x < 10, plot the point at x = 10, using the cumulative frequency up to the end of that class.

对于分组连续数据，应在每个区间的组上边界处绘制累积频率点。例如，对于 0 ≤ x < 10，在 x = 10 处绘制点，并使用截至该组结束时的累积频率。

To estimate the median, draw a horizontal line from half the total frequency on the vertical axis to the curve, then drop a vertical line to the horizontal axis. The same method works for Q1 at one quarter and Q3 at three quarters of the total.

要估计中位数，从纵轴上总频数的一半处画水平线到曲线，再向下画垂直线到横轴。Q1 在总频数的四分之一处、Q3 在总频数的四分之三处，方法相同。

Cumulative frequency diagrams are also useful for comparing two distributions, especially when you need to estimate percentiles such as the 90th percentile.

累积频率图也适合比较两个分布，尤其是在需要估计第 90 百分位等百分位数时。

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7. Scatter Diagrams and Correlation | 散点图与相关性

A scatter diagram represents paired bivariate data by plotting one variable on the horizontal axis and the other on the vertical axis. It is used to investigate whether an association or correlation exists between the two variables.

散点图通过将两个变量中的一个画在横轴、另一个画在纵轴来表示成对的双变量数据。它用于考察两个变量之间是否存在关联或相关性。

Correlation may be positive, negative or zero. Positive correlation means that as one variable increases, the other tends to increase. Negative correlation means that as one variable increases, the other tends to decrease.

相关性可以是正相关、负相关或零相关。正相关意味着一个变量增大时，另一个变量往往也增大。负相关意味着一个变量增大时，另一个变量往往减小。

A line of best fit can be drawn on a scatter diagram to make predictions, but you must be careful with extrapolation beyond the data range. Correlation does not imply causation; a strong correlation can arise from a third underlying factor.

散点图上可以绘制最佳拟合线进行预测，但必须谨慎对超出数据范围的外推。相关性并不意味着因果关系；强相关可能来自第三个潜在因素。

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8. Measures of Location from Diagrams | 从图示中读取位置度量

Different diagrams allow different summary statistics to be read directly. A box plot gives the median and quartiles directly, but a histogram does not show exact individual values because data are grouped.

不同的图示可以直接读取不同的汇总统计量。箱线图直接给出中位数和四分位数，但直方图由于数据分组而不显示确切的单个数值。

From an ordered stem-and-leaf diagram, the median and quartiles can be found by counting leaves. For n data values, the median is at the (n+1)/2-th position when using the usual discrete data convention.

从有序茎叶图中，可以通过数叶数找到中位数和四分位数。对于 n 个数据值，按照通常的离散数据约定，中位数位于第 (n+1)/2 个位置。

For a cumulative frequency graph, interpolation is used to estimate the median from grouped data. The formula is based on the lower class boundary L, the cumulative frequency before the class F, the class frequency f, and the class width w.

对于累积频率图，使用插值法从分组数据中估计中位数。公式基于组下边界 L、该组之前的累积频率 F、组频数 f 和组距 w。

median = L + ((n/2 − F) ÷ f) × w

中位数 = L + ((n/2 − F) ÷ f) × w

Similar interpolation works for any percentile by replacing n/2 with the relevant proportion, such as n/4 for Q1 or 3n/4 for Q3.

类似的插值可用于任何百分位数，只需将 n/2 替换为相应的比例，例如 Q1 用 n/4、Q3 用 3n/4。

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9. Choosing an Appropriate Representation | 选择合适的表示方法

Edexcel exam questions often ask you to suggest a suitable diagram for a given data set and justify your choice. The best choice depends on the type of data and the feature you want to highlight.

Edexcel 考试题目经常要求你为给定数据集建议一个合适的图示并说明理由。最佳选择取决于数据类型以及你想要突出的特征。

Use a box plot to compare medians, quartiles and outliers. Use a histogram to show the shape and spread of continuous data. Use a stem-and-leaf diagram to retain original discrete values. Use a cumulative frequency curve to estimate percentiles.

用箱线图比较中位数、四分位数和离群值。用直方图显示连续数据的形状和离散程度。用茎叶图保留原始离散数值。用累积频率曲线估计百分位数。

For bivariate data, a scatter diagram is the natural first choice. For categorical data, a bar chart or pie chart is appropriate, but pie charts are less useful for comparing categories with similar proportions.

对于双变量数据，散点图是自然的第一选择。对于分类数据，条形图或饼图是合适的，但饼图在比较比例相近的类别时不太有用。

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10. Common Exam Mistakes and Useful Tips | 常见考试错误与实用提示

One of the most common errors is using frequency instead of frequency density when drawing a histogram with unequal class widths. Always calculate height as frequency ÷ class width.

最常见的错误之一是在绘制组距不等的直方图时使用频数代替频率密度。必须始终用频率密度 = 频数 ÷ 组距来计算条形高度。

In cumulative frequency diagrams, plot against upper class boundaries, not midpoints. Also remember to draw smooth curves and use a ruler when reading median or quartiles from the graph.

在累积频率图中，应针对组上边界描点，而不是中点。同时要记住绘制平滑曲线，并在从图中读取中位数或四分位数时使用直尺。

For box plots, do not forget to mark outliers as crosses and to extend whiskers only to non-outlier extremes. Incorrect outlier limits often come from misidentifying Q1 and Q3 or using the range instead of the IQR.

对于箱线图，不要忘记将离群值标记为叉号，并只将须线延伸到非离群极值。离群值界限错误往往源于 Q1 和 Q3 识别错误，或使用了极差而不是四分位距。

Finally, always label axes, include keys for stem-and-leaf diagrams, use correct units, and avoid rounding until the final answer. These small details can make a critical difference in Edexcel statistics questions.

最后，务必标注坐标轴、为茎叶图提供图例、使用正确单位，并避免在最终答案之前提前舍入。这些细节在 Edexcel 统计题中可能产生关键影响。

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