📚 A-Level Mathematics: Box Plot Construction and Interpretation Techniques | A-Level 数学：箱线图绘制与解读技巧
Box plots, also called box-and-whisker diagrams, are a compact visual representation of a numerical data set. In A-Level Mathematics and Statistics, you will be expected to construct them from raw data or from a five-number summary, and to use them to compare two or more distributions quickly and clearly. This article provides a step-by-step guide to drawing box plots, a clear explanation of outlier rules, and practical interpretation skills that are consistently tested in examination questions.
箱线图（又称箱须图）是一种以紧凑方式展示数值型数据分布的统计图形。在 A-Level 数学与统计学中，要求你能够根据原始数据或五数概括绘制箱线图，并借助它们快速、清晰地比较两组或多组数据的分布。本文将提供绘制箱线图的分步指南、离群值规则的清晰讲解，以及在考试中经常考查的实用解读技巧。
1. What Is a Box Plot? | 什么是箱线图？
A box plot displays the distribution of a data set using five key values: the minimum, the first quartile (Q₁), the median (Q₂), the third quartile (Q₃), and the maximum. These five values are often called the five-number summary. The “box” shows the interquartile range (IQR), which contains the middle 50% of the data, while the “whiskers” extend to the rest of the data, except for any values that are classified as outliers.
箱线图利用五个关键值来展示数据集的分布：最小值、第一四分位数（Q₁）、中位数（Q₂）、第三四分位数（Q₃）和最大值。这五个数值通常被称为“五数概括”。“箱体”表示四分位距（IQR），它包含了中间 50% 的数据；“须线”则延伸至其余数据，但排除被判定为离群值的数值。
The main advantage of a box plot is that it allows you to see the centre, spread, and skewness of a distribution at a glance. Unlike a histogram, it does not require a large number of observations to be meaningful, and it is especially helpful when comparing different data sets on the same scale.
箱线图的主要优点是能够让你一眼看出数据分布的中心、离散程度和偏态。与直方图不同，箱线图不需要大量的观测值也能体现分布特征，而且在同一坐标轴上比较不同数据集时尤其有效。
2. The Five-Number Summary | 五数概括
Before you can draw a box plot, you must calculate the five-number summary. The median (Q₂) divides the ordered data set into two halves. The first quartile (Q₁) is the median of the lower half of the data, and the third quartile (Q₃) is the median of the upper half. When the data set has an odd number of values, the median itself is excluded from both halves before finding Q₁ and Q₃. When the data set has an even number of values, the median is the mean of the two middle values and the data set is split evenly.
在绘制箱线图之前，必须先计算五数概括。中位数（Q₂）将有序数据集分为两半。第一四分位数（Q₁）是数据下半部分的中位数，第三四分位数（Q₃）是数据上半部分的中位数。当数据个数为奇数时，在计算 Q₁ 和 Q₃ 之前，需要先将中位数本身从上下两半中排除；当数据个数为偶数时，中位数是中间两个数值的平均数，数据被平均分成两半。
The interquartile range is defined as:
IQR = Q₃ − Q₁
The IQR measures the spread of the middle 50% of the data. It is preferred over the full range in many contexts because it is not affected by extreme values or outliers.
四分位距的定义为：
IQR = Q₃ − Q₁
四分位距衡量中间 50% 数据的离散程度。在许多情况下，它比全距更受青睐，因为它不受极端值或离群值的影响。
3. Outliers and the 1.5 × IQR Rule | 离群值与 1.5 × IQR 规则
An outlier is an observation that falls well outside the overall pattern of the data. The standard rule used at A-Level is the 1.5 × IQR rule. According to this rule, a value is considered to be an outlier if it lies below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
离群值是指明显偏离数据整体模式的观测值。A-Level 考试中使用的标准规则是 1.5 × IQR 规则。根据该规则，如果一个数值低于 Q₁ − 1.5 × IQR 或高于 Q₃ + 1.5 × IQR，则将其判定为离群值。
Lower outlier limit = Q₁ − 1.5 × IQR
Upper outlier limit = Q₃ + 1.5 × IQR
When drawing a box plot, if there are no outliers, the whiskers extend to the minimum and maximum values. If there are outliers, the whiskers extend only to the smallest and largest values that are not outliers; these values are sometimes called the adjacent values, and the outliers are marked individually with a cross (×) or a dot.
绘制箱线图时，如果不存在离群值，则须线延伸至最小值和最大值。如果存在离群值，则须线只延伸至最小的非离群值和最大的非离群值；这些值有时称为“相邻值”，而离群值则用叉号（×）或圆点单独标记出来。
It is important to remember that the “1.5 × IQR” is a convention, not a mathematical law. Different contexts may use different multipliers, but for A-Level examinations you should always apply 1.5 × IQR unless the question states otherwise.
需要记住的是，“1.5 × IQR”是一种约定俗成的规则，而不是数学定律。在不同的应用场景中可能会使用不同的倍数，但在 A-Level 考试中，除非题目另有说明，否则应始终使用 1.5 × IQR。
4. How to Draw a Box Plot: Step-by-Step | 绘制箱线图：分步方法
Follow these steps to draw an accurate box plot from raw data. The same procedure is used whether the data comes from a stem-and-leaf diagram, a table, or a list of values.
按照以下步骤就可以根据原始数据绘制出准确的箱线图。无论数据来自茎叶图、统计表还是一列数值，都可以使用相同的流程。
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Step 1: Order the data from smallest to largest.
步骤一：将数据按从小到大的顺序排列。
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Step 2: Find the median Q₂. If there are n values, the median is the ((n + 1) ÷ 2)-th value when n is odd, or the mean of the two middle values when n is even.
步骤二：求中位数 Q₂。若共有 n 个数值，当 n 为奇数时，中位数是第 ((n + 1) ÷ 2) 个数值；当 n 为偶数时，中位数是中间两个数值的平均数。
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Step 3: Find Q₁ and Q₃ as the medians of the lower and upper halves of the data, ignoring the median itself if the data set has an odd number of values.
步骤三：分别求下半部分和上半部分数据的中位数，得到 Q₁ 和 Q₃。如果数据个数为奇数，需要忽略中位数本身。
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Step 4: Calculate the IQR = Q₃ − Q₁, then find the outlier limits using the 1.5 × IQR rule.
步骤四：计算 IQR = Q₃ − Q₁，然后利用 1.5 × IQR 规则求离群值界限。
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Step 5: Draw a horizontal scale that includes all data values and outlier limits. Draw a box from Q₁ to Q₃, and draw a vertical line inside the box at the median Q₂.
步骤五：画一条包含所有数据值和离群值界限的水平数轴。绘制从 Q₁ 到 Q₃ 的箱体，并在箱体内中位数 Q₂ 的位置画一条竖线。
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Step 6: Draw whiskers from the box to the smallest and largest values that are not outliers. Mark any outliers with a cross (×) beyond the whiskers.
步骤六：从箱体两端画须线，延伸至最小的非离群值和最大的非离群值。若有离群值，在须线之外用叉号（×）标记出来。
5. Worked Example: Drawing a Box Plot | 例题详解：绘制箱线图
Consider the following data set, which shows the ages of 20 visitors to a museum. Notice that the last value, 110, is designed to be an outlier for this demonstration.
考虑以下数据集，它展示了 20 名博物馆参观者的年龄。请注意，最后一个数值 110 是为了演示离群值而特意设置的。
2, 4, 10, 13, 16, 17, 21, 22, 25, 31, 32, 35, 40, 42, 50, 52, 55, 61, 68, 110
Step 1 – Find the median. There are 20 values, so the median is the mean of the 10th and 11th values: (31 + 32) ÷ 2 = 31.5.
步骤一——求中位数。共有 20 个数值，因此中位数是第 10 个和第 11 个数值的平均数：(31 + 32) ÷ 2 = 31.5。
Step 2 – Find Q₁ and Q₃. The lower half is 2, 4, 10, 13, 16, 17, 21, 22, 25, 31. The median of these 10 values is the mean of the 5th and 6th values: (16 + 17) ÷ 2 = 16.5. The upper half is 32, 35, 40, 42, 50, 52, 55, 61, 68, 110. Its median is (50 + 52) ÷ 2 = 51.
步骤二——求 Q₁ 和 Q₃。下半部分为 2, 4, 10, 13, 16, 17, 21, 22, 25, 31。这 10 个数值的中位数是第 5 个和第 6 个数值的平均数：(16 + 17) ÷ 2 = 16.5。上半部分为 32, 35, 40, 42, 50, 52, 55, 61, 68, 110，其中位数为 (50 + 52) ÷ 2 = 51。
Step 3 – Calculate IQR and outlier limits. IQR = 51 − 16.5 = 34.5. The upper limit is 51 + 1.5 × 34.5 = 51 + 51.75 = 102.75. The lower limit is 16.5 − 51.75 = −35.25. Since 110 is greater than 102.75, it is an outlier. There are no lower outliers because no data value is below −35.25.
步骤三——计算 IQR 和离群值界限。IQR = 51 − 16.5 = 34.5。上限为 51 + 1.5 × 34.5 = 51 + 51.75 = 102.75。下限为 16.5 − 51.75 = −35.25。由于 110 大于 102.75，因此它是离群值。因为没有任何数据低于 −35.25，所以不存在低端离群值。
Step 4 – Draw the plot. Draw a scale from 0 to 115. Draw a box from 16.5 to 51, with a vertical line at 31.5. Extend the left whisker to 2 (the minimum) and the right whisker to 68 (the largest non-outlier). Finally, plot a cross at 110 to show the outlier.
步骤四——绘制图形。画一条从 0 到 115 的数轴。绘制从 16.5 到 51 的箱体，并在 31.5 处画竖线。左须线延伸至最小值 2，右须线延伸至最大的非离群值 68。最后在 110 处画一个叉号表示离群值。
6. Interpreting a Box Plot: Shape and Skewness | 解读箱线图：形状与偏态
Once a box plot is drawn, you should be able to describe the distribution of the data. The position of the median inside the box and the relative lengths of the two whiskers tell you about skewness.
箱线图绘制完成后，你应该能够描述数据的分布特征。中位数在箱体内的位置以及左右两条须线的相对长度，可以反映数据的偏态。
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Symmetric distribution: The median is approximately in the centre of the box, and the two whiskers are roughly equal in length.
对称分布：中位数大致位于箱体中央，且两条须线长度大致相等。
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Positive skew (right skew): The median is closer to Q₁ than to Q₃, and the right whisker is noticeably longer than the left whisker. The right tail is stretched out.
正偏（右偏）：中位数更靠近 Q₁ 而非 Q₃，右须线明显长于左须线。右侧尾部被拉长。
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Negative skew (left skew): The median is closer to Q₃ than to Q₁, and the left whisker is noticeably longer than the right whisker. The left tail is stretched out.
负偏（左偏）：中位数更靠近 Q₃ 而非 Q₁，左须线明显长于右须线。左侧尾部被拉长。
You should also comment on the centre and the spread. The median is the measure of centre that is reported directly from a box plot. The interquartile range (the length of the box) is the recommended measure of spread because it is resistant to outliers.
此外，你还需要说明数据的中心趋势和离散程度。中位数是箱线图直接呈现的中心指标；四分位距（箱体的长度）是推荐的离散程度指标，因为它不受离群值影响。
7. Comparing Two or More Distributions | 比较两组或多组分布
One of the most common examination tasks is to compare two box plots placed on the same scale. The key is to make explicit comparisons rather than simply describing each plot separately.
考试中最常见的任务之一是在同一数轴上比较两个箱线图。关键在于进行明确的比较，而不是仅仅分别描述每个图形。
Always compare: median, IQR, range, whisker lengths, skewness, and outliers.
始终需要比较：中位数、IQR、全距、须线长度、偏态和离群值。
Use comparative phrases in your answer. For example: “The median of group A is 10 kg heavier than the median of group B.” “The IQR of group B is smaller, so the middle 50% of the data is more consistent.” “Group A has a positive skew, while group B is approximately symmetric.” “There is an outlier in group A, but none in group B.”
在回答中请使用比较性语言。例如：“A 组的中位数比 B 组的中位数重 10 kg。”“B 组的 IQR 更小，因此中间 50% 的数据更集中。”“A 组呈正偏态，而 B 组大致对称。”“A 组存在一个离群值，而 B 组没有。”
When commenting on outliers, always state whether the outliers are high or low. For example: “The single outlier in group A is an exceptionally large value, which pulls the mean upward but does not affect the median.” This type of comment shows the examiner that you understand the effect of outliers on statistical measures.
在评论离群值时，务必说明离群值是高端还是低端的。例如：“A 组中的单个离群值是一个异常大的数值，它会拉高均值，但不影响中位数。”这种评论能向考官表明你理解离群值对统计量的影响。
8. Common Pitfalls and Exam Traps | 常见错误与考试陷阱
Many students lose marks on box plot questions due to small but avoidable errors. Below are the most common pitfalls, with advice on how to avoid them.
许多学生在箱线图题目中因为一些微小但可以避免的错误而失分。以下是最常见的错误类型以及避免它们的方法。
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Forgetting to check for outliers: If you extend the whiskers to the maximum even when there is an outlier, you lose accuracy. Always compute the 1.5 × IQR limits first.
忘记检查离群值：即使存在离群值，你仍然将须线延伸到最大值，这会丢失准确性。务必先计算 1.5 × IQR 界限。
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Using the wrong quartile method: When the data set has an odd number of values, the median must be excluded before finding Q₁ and Q₃. Some students accidentally include it in both halves, which gives a different (incorrect) quartile.
使用错误的四分位数计算方法：当数据个数为奇数时，计算 Q₁ 和 Q₃ 前必须排除中位数。有些学生会错误地将中位数同时包含在上下两半中，从而导致四分位数计算错误。
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Drawing a box plot without a labelled scale: An unmarked scale makes your plot impossible to read. Always include a clear, evenly spaced numerical scale.
绘制箱线图时未标注数轴刻度：没有刻度的图形无法被准确读取。始终要画出清晰、等距的数值刻度。
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Plotting the mean instead of the median: The line inside the box is the median, not the mean. These two values are equal only for symmetric distributions.
将均值而不是中位数标在箱体内：箱体中的竖线是中位数，而不是均值。只有在对称分布中这两个值才相等。
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Using the range instead of the IQR: For box plot questions, the IQR is the standard measure of spread. Use the length of the box, not the distance from one whisker end to the
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