Box Plots | 箱线图

📚 Box Plots | 箱线图

A box plot (or box-and-whisker plot) is a graphical representation of a dataset’s five-number summary, used to display the distribution, central tendency, and variability of data. It is a fundamental tool in Edexcel A-Level Mathematics for comparing datasets and identifying outliers.

箱线图（或称盒须图）是数据五数概括的图形展示，用于呈现数据的分布、集中趋势和变异性。在爱德思 A-Level 数学中，它是比较数据集和识别异常值的基本工具。

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1. Introduction to Box Plots | 箱线图简介

Box plots provide a visual summary of the minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. They are especially useful for highlighting the spread and skewness of data. Unlike histograms, box plots do not show the frequency of values but instead focus on the quartiles and extremes.

箱线图提供了最小值、下四分位数（Q₁）、中位数（Q₂）、上四分位数（Q₃）和最大值的可视化摘要。它们特别有助于突出数据的分散程度和偏度。与直方图不同，箱线图不显示数值的频率，而是侧重于四分位数和极值。

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2. The Five-Number Summary | 五数概括法

The five-number summary consists of: Minimum, Q₁, Median, Q₃, Maximum. To find them, the data must first be sorted in ascending order. The median splits the data into two halves; Q₁ is the median of the lower half, and Q₃ is the median of the upper half. If the dataset has an odd number of values, the median is normally excluded when finding the quartiles in Edexcel’s approach. This summary fully describes the spread of a dataset and forms the foundation of any box plot.

五数概括法包括：最小值、Q₁、中位数、Q₃、最大值。要找到它们，数据必须先按升序排列。中位数将数据分成两半；Q₁ 是下半部分的中位数，Q₃ 是上半部分的中位数。若数据集有奇数个值，爱德思通常将中位数排除在外再求四分位数。这个概括完整描述了数据集的分布范围，是箱线图的基础。

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3. Minimum and Maximum | 最小值与最大值

The minimum is the smallest data value, and the maximum is the largest. These values define the range of the data. In a basic box plot without outliers, the whiskers extend exactly to the minimum and maximum. The range = maximum – minimum is a simple measure of dispersion, but it can be heavily influenced by extreme values.

最小值是最小的数据值，最大值是最大的数据值。这些值定义了数据的全距。在无异常值的基本箱线图中，触须恰好延伸到最小值和最大值。全距 = 最大值 – 最小值是分散程度的一个简单度量，但它极易受极端值影响。

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4. Quartiles and the Median | 四分位数与中位数

The median (Q₂) divides the data into two equal parts. The lower quartile (Q₁) is the value below which 25% of the data lie; the upper quartile (Q₃) is the value below which 75% lie. For a dataset of size n, positions are often found using: Q₁ position = (n+1)/4, Q₂ = (n+1)/2, Q₃ = 3(n+1)/4. When positions are not integers, linear interpolation is required, especially for grouped data. These three quartiles create the ‘box’ in a box plot.

中位数（Q₂）将数据分成相等的两部分。下四分位数（Q₁）是 25% 数据低于它的值；上四分位数（Q₃）是 75% 数据低于它的值。对于大小为 n 的数据集，位置通常通过 Q₁ 位置 = (n+1)/4、Q₂ = (n+1)/2、Q₃ = 3(n+1)/4 来计算。若位置非整数，则需线性插值，特别是对于分组数据。这三个四分位数构成了箱线图中的“箱子”。

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5. Interquartile Range (IQR) | 四分位距（IQR）

The interquartile range is defined as IQR = Q₃ − Q₁. It measures the spread of the middle 50% of the data and is resistant to outliers, unlike the range. IQR is essential for constructing box plots and for identifying outliers using the 1.5 × IQR rule. A larger IQR indicates greater variability in the central portion of the data.

四分位距定义为 IQR = Q₃ − Q₁。它测量了中间 50% 数据的分散程度，与全距不同，它不受异常值影响。IQR 对于绘制箱线图和用 1.5 × IQR 规则识别异常值至关重要。较大的 IQR 表示数据中部变异性更大。

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6. Constructing a Box Plot | 绘制箱线图

To draw a box plot: (1) Draw a horizontal or vertical number line with a suitable scale. (2) Mark the five-number summary values: Min, Q₁, Median, Q₃, Max. (3) Draw a rectangular box from Q₁ to Q₃ with a vertical line at the median inside the box. (4) Draw whiskers (horizontal or vertical lines) from the box to the minimum and maximum. (5) Label the axes and give the plot a title. Always use a ruler and plot points accurately; a clear scale is crucial for exam marks.

绘制箱线图的步骤：（1）画出带合适刻度的数轴，可水平或垂直。（2）标出五数概括值：最小值、Q₁、中位数、Q₃、最大值。（3）从 Q₁ 到 Q₃ 画一个矩形箱，箱内中位数处画一条垂直线。（4）从箱子两侧向最小值和最大值画出触须线。（5）标注坐标轴并给出标题。务必使用直尺并精确描点；清晰的刻度对于得分至关重要。

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7. Identifying Outliers | 识别异常值

An outlier is a data value that lies significantly outside the overall pattern. The standard criterion is: any value less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR is considered an outlier. These boundaries are called the lower fence and upper fence. Outliers should be investigated but not simply removed unless there is a valid reason.

异常值是明显偏离整体模式的数据值。标准准则是：任何小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的值被视为异常值。这些边界称为下围栏和上围栏。异常值应被调查，但除非有正当理由，否则不应随意删除。

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8. Box Plots with Outliers | 带异常值的箱线图

When outliers are present, the whiskers do not extend to the original minimum or maximum. Instead, the lower whisker ends at the smallest data value that is ≥ Q₁ − 1.5×IQR, and the upper whisker ends at the largest data value ≤ Q₃ + 1.5×IQR. Outliers are plotted as individual points (often crosses or dots) beyond the whiskers. Always clearly label which points are outliers and show your calculations for the fences.

当存在异常值时，触须不延伸至原始的最小值或最大值。相反，下触须结束于 ≥ Q₁ − 1.5×IQR 的最小数据值处，上触须结束于 ≤ Q₃ + 1.5×IQR 的最大数据值处。异常值用单独的点（常为叉号或圆点）标在触须之外。始终清楚标出哪些点是异常值，并展示围栏的计算过程。

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9. Comparing Distributions | 比较分布

Box plots are excellent for comparing two or more datasets side by side. You can comment on the median (central tendency), the IQR and range (spread), and skewness. For example, a higher median indicates a higher typical value; a larger IQR suggests greater variability. Comparisons should be made in context, using exact numerical values and units. Edexcel exam questions often ask you to compare and interpret, so always make at least two distinct points.

箱线图非常适合并列比较两个或多个数据集。你可以讨论中位数（集中趋势）、IQR 和全距（分散程度）以及偏度。例如，较高的中位数表示典型值较高；较大的 IQR 表示变异性更大。比较时应结合背景，使用精确数值和单位进行说明。爱德思考题常要求比较与解释，因此至少要提出两点不同之处。

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10. Skewness and Symmetry | 偏度与对称性

The shape of a box plot reveals skewness. If the median is closer to Q₁ and the upper whisker is longer, the distribution is positively skewed (right-skewed). If the median is closer to Q₃ and the lower whisker is longer, it is negatively skewed (left-skewed). In a symmetric distribution, the median lies roughly in the centre of the box and the whiskers are of similar length. You can also compare the distances: Q₃ − Median versus Median − Q₁ to confirm skewness.

箱线图的形状可以揭示偏度。如果中位数靠近 Q₁ 且上触须较长，分布呈正偏态（右偏）。如果中位数靠近 Q₃ 且下触须较长，分布呈负偏态（左偏）。在对称分布中，中位数大约在箱子中心，且触须长度相近。你也可以比较 Q₃ − 中位数 与 中位数 − Q₁ 的距离来确认偏度。

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11. Common Mistakes | 常见错误

Common mistakes include: confusing the order of quartiles (e.g. using Q₃ in place of Q₁), miscalculating Q₁ and Q₃ with even/odd datasets, forgetting to sort the data before finding quartiles, misreading the scale when plotting, not adjusting whiskers for outliers, and drawing the median outside the box. Also, some students label the median as the mean, but a box plot displays the median, not the mean. In grouped data, failing to use interpolation correctly is another frequent error.

常见错误包括：混淆四分位数的顺序（例如用 Q₃ 代替 Q₁）、在偶数和奇数数据集时错误计算 Q₁ 和 Q₃、求四分位数前忘记排序数据、绘图时误读刻度、未因异常值调整触须，以及把中位数画在箱子外面。此外，有些学生把中位数标为平均数，但箱线图显示的是中位数而非均值。对于分组数据，未能正确使用插值也是常见问题。

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12. Exam Tips and Practice | 考试技巧与练习

In Edexcel A-Level Mathematics, box plot questions often appear in Statistics sections. You may be asked to draw a box plot from raw or grouped data, identify outliers using the 1.5×IQR rule, or compare two box plots. Always show your calculations for quartiles and fences clearly. Use a ruler, label key values on the box plot, and ensure the scale is linear and consistent. For grouped data, practice interpolation for quartiles using lower class boundaries and class widths. Review past paper questions to become familiar with mark schemes. Time management is key; a box plot question is usually an opportunity to score efficiently if you are well prepared.

在爱德思 A-Level 数学中，箱线图题目常出现在统计部分。你可能需要根据原始数据或分组数据绘制箱线图，用 1.5×IQR 规则识别异常值，或比较两个箱线图。务必清晰展示四分位数和围栏的计算过程。使用直尺，在箱线图上标出关键数值，并确保刻度线性且一致。对于分组数据，要练习使用下限和组距进行四分位数插值。复习历年真题以熟悉评分方案。时间管理很关键；若准备充分，箱线图问题通常是高效得分的良机。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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