Interpreting and Drawing Box Plots | 箱线图的解读与绘制

📚 Interpreting and Drawing Box Plots | 箱线图的解读与绘制

A box plot (also called a box-and-whisker diagram) is a standardised graphical summary of a data set based on the five-number summary. In IB Mathematics, you are expected not only to draw box plots accurately but also to interpret them in context, compare distributions, and identify outliers using the 1.5 × IQR rule.

箱线图（又称盒须图）是基于五数概括的一种标准化数据图形摘要。在IB数学中，你不仅需要准确绘制箱线图，还需要结合情境解读它们、比较多个分布，并运用 1.5 × IQR 规则识别离群值。

1. The Five-Number Summary | 五数概括

The five-number summary consists of the minimum value (Min), the first quartile (Q₁), the median (Q₂), the third quartile (Q₃), and the maximum value (Max). These five numbers divide the ordered data set into four parts, each containing approximately 25% of the data.

五数概括由最小值(Min)、第一四分位数(Q₁)、中位数(Q₂)、第三四分位数(Q₃)和最大值(Max)组成。这五个数将有序列数据分成四个部分，每部分约含25%的数据。

To locate Q₁ and Q₃, first arrange the data in ascending order. Find the median; then Q₁ is the median of the lower half of the data, and Q₃ is the median of the upper half. If the total number of data points is odd, the median itself is excluded from both halves when finding the quartiles.

要确定Q₁和Q₃，首先将数据按升序排列。找到中位数后，Q₁是下半部分数据的中位数，Q₃是上半部分数据的中位数。如果数据点总数为奇数，在求四分位数时，中位数本身不包含在任何一半中。

Example: For the data set 4, 7, 10, 12, 15, 18, 22, 26, 31, the median is 15. The lower half is 4, 7, 10, 12, giving Q₁ = 8.5. The upper half is 18, 22, 26, 31, giving Q₃ = 24.

示例：对于数据集4, 7, 10, 12, 15, 18, 22, 26, 31，中位数为15。下半部分为4, 7, 10, 12，得Q₁ = 8.5。上半部分为18, 22, 26, 31，得Q₃ = 24。

2. Drawing a Box Plot Step by Step | 分步绘制箱线图

A box plot consists of a rectangular box spanning from Q₁ to Q₃, with a vertical line drawn at the median. Two whiskers extend from the box: one to the minimum value (or the smallest value that is not an outlier) and one to the maximum value (or the largest value that is not an outlier).

箱线图由一个从Q₁延伸到Q₃的矩形盒构成，盒内中位数处画一条竖线。两条须从盒子向外延伸：一条延伸至最小值（或非离群值的最小值），另一条延伸至最大值（或非离群值的最大值）。

Step 1: Compute the five-number summary. Step 2: Draw a scaled number line that covers the range of the data. Step 3: Draw the box between Q₁ and Q₃. Step 4: Mark the median with a vertical line inside the box. Step 5: Extend whiskers to the smallest and largest data values that are not outliers.

步骤一：计算五数概括。步骤二：画出覆盖数据范围的等距数轴。步骤三：在Q₁与Q₃之间画出矩形盒。步骤四：在盒内中位数处画一条竖线。步骤五：将须延伸至非离群值的最小值和最大值。

Q₁ = 8.5, Median = 15, Q₃ = 24, IQR = 24 − 8.5 = 15.5

Draw the number line from 0 to 35. Place the box from 8.5 to 24, draw the median line at 15, and extend the whiskers from 4 to 8.5 on the left and from 24 to 31 on the right.

画一条从0到35的数轴。将盒子置于8.5到24之间，在15处画中位数线，左侧须从4延伸到8.5，右侧须从24延伸到31。

3. Outliers and the 1.5 × IQR Rule | 离群值与1.5×IQR规则

An outlier is a data point that lies unusually far from the bulk of the data. The IB syllabus requires you to use the 1.5 × IQR rule to determine whether a data point is an outlier.

离群值是偏离数据主体异常远的数据点。IB教学大纲要求你使用1.5×IQR规则来判断某个数据点是否为离群值。

A point is considered an outlier if it lies below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR. These boundaries are called lower and upper fences.

如果某个数据点位于Q₁ − 1.5×IQR之下或Q₃ + 1.5×IQR之上，则该点被视为离群值。这两个边界称为下限栅栏和上限栅栏。

Using the example above, IQR = 15.5, so 1.5 × IQR = 23.25. The lower fence is 8.5 − 23.25 = −14.75 and the upper fence is 24 + 23.25 = 47.25. Since all values lie between these fences, there are no outliers.

以上述示例为例，IQR = 15.5，因此1.5×IQR = 23.25。下限栅栏为8.5 − 23.25 = −14.75，上限栅栏为24 + 23.25 = 47.25。由于所有值都介于这两个栅栏之间，因此不存在离群值。

In a box plot, outliers are usually plotted individually as small dots or asterisks beyond the whiskers. The whiskers then extend only to the most extreme values that are not outliers.

在箱线图中，离群值通常以独立的小圆点或星号绘制在须之外。此时须仅延伸至非离群值中的最极端值。

4. Reading the Shape: Skewness | 解读形状：偏态

Box plots reveal skewness through the relative positions of the median within the box and the relative lengths of the whiskers.

箱线图通过中位数在盒内的相对位置以及两须的相对长度来揭示偏态。

If the median is closer to Q₁ and the right whisker is longer than the left whisker, the distribution is positively skewed (right-skewed). This indicates that the tail extends towards larger values.

如果中位数更靠近Q₁，且右须比左须长，则分布呈正偏态（右偏）。这表明尾部向较大值方向延伸。

If the median is closer to Q₃ and the left whisker is longer than the right whisker, the distribution is negatively skewed (left-skewed). This indicates that the tail extends towards smaller values.

如果中位数更靠近Q₃，且左须比右须长，则分布呈负偏态（左偏）。这表明尾部向较小值方向延伸。

If the median is near the centre of the box and the whiskers are approximately equal in length, the distribution is approximately symmetric.

如果中位数接近盒子的中心，且两须长度大致相等，则分布近似对称。

Note that the box itself shows the interquartile range, which measures the spread of the middle 50% of the data. The whiskers give a rough indication of the spread of the lower and upper 25% of the data.

注意，盒子本身显示的是四分位距，衡量中间50%数据的离散程度。须则粗略反映数据下方和上方各25%的分散范围。

5. Comparing Two or More Distributions | 比较两个或多个分布

Box plots are especially powerful for comparing distributions side by side. When comparing, you should always discuss centre, spread, and shape separately and use numerical values to support your comments.

箱线图特别适合并排比较多个分布。比较时，应始终分别讨论中心、离散程度和形状，并用具体数值支撑你的判断。

Centre: Compare the medians (the lines inside the boxes). A higher median indicates a generally larger central tendency. Spread: Compare the interquartile ranges (the box lengths) and the overall ranges (whisker tip to whisker tip). A larger IQR means more variability in the middle 50% of the data. Shape: Compare the position of the median within each box and the relative whisker lengths to assess skewness.

中心：比较中位数（盒内的线）。中位数越高表明中心趋势越大。离散程度：比较四分位距（盒子的长度）和整体极差（须端到须端）。IQR越大意味着中间50%数据的变异性越大。形状：比较中位数在每个盒中的位置以及须的相对长度，用以评估偏态。

Example: Class A has median 18 and IQR 6; Class B has median 22 and IQR 10. We can say Class B has a higher centre and a greater variability in the middle half of the data.

示例：A班中位数为18，IQR为6；B班中位数为22，IQR为10。我们可以说B班的中心更高，且中间一半数据的变异性更大。

6. Worked Example with an Outlier | 含离群值的完整例题

Consider the data set: 3, 5, 7, 8, 9, 11, 12, 14, 16, 18, 45. There are 11 data points. The median is the 6th value, which is 11.

考虑数据集：3, 5, 7, 8, 9, 11, 12, 14, 16, 18, 45。共有11个数据点。中位数是第6个值，即11。

The lower half is 3, 5, 7, 8, 9, giving Q₁ = 7. The upper half is 12, 14, 16, 18, 45, giving Q₃ = 16. Thus IQR = 16 − 7 = 9.

下半部分为3, 5, 7, 8, 9，得Q₁ = 7。上半部分为12, 14, 16, 18, 45，得Q₃ = 16。因此IQR = 16 − 7 = 9。

Lower fence = 7 − 1.5 × 9 = 7 − 13.5 = −6.5. Upper fence = 16 + 1.5 × 9 = 16 + 13.5 = 29.5. The value 45 exceeds the upper fence, so 45 is an outlier.

下限栅栏 = 7 − 1.5×9 = 7 − 13.5 = −6.5。上限栅栏 = 16 + 1.5×9 = 16 + 13.5 = 29.5。值45超过上限栅栏，因此45是离群值。

When drawing this box plot, the right whisker extends to 18 (the largest value below the upper fence), and the point 45 is plotted individually beyond the whisker.

绘制该箱线图时，右须延伸至18（上限栅栏以下的最大值），而点45则以独立点形式绘制在须之外。

7. Common Pitfalls in IB Exams | IB考试中的常见错误

One frequent error is excluding the median from both halves when the data set has an even number of observations. In fact, for an even number of data points, the data is simply split into two equal halves, and the median of each half gives Q₁ and Q₃.

一个常见错误是，当数据量为偶数时，错误地将中位数从两半中排除。实际上，对于偶数个数据点，数据直接平分为两半，每一半的中位数即为Q₁和Q₃。

Another mistake is drawing whiskers to the fences rather than to the most extreme actual data values. The fences are not part of the data; they are only used to identify outliers. Whiskers must end at actual observed values.

另一个错误是将须画到栅栏处而非最极端的实际数据值。栅栏不属于数据，它们仅用于识别离群值。须必须以实际观测值结束。

A third common issue is failing to label the scale or misplacing the median when the data set contains repeated values. Always check that the median line corresponds exactly to the numerical value on the number line.

第三个常见问题是未标注刻度，或在数据含有重复值时放置错中位数位置。务必检查中位数线与数轴上的数值是否精确对应。

Finally, when comparing box plots, students often describe only one of the three aspects (centre, spread, shape). IB mark schemes typically award marks for discussing all three using quantitative evidence.

最后，在比较箱线图时，学生往往只描述三个方面（中心、离散程度、形状）中的一个。IB评分标准通常要求用量化证据讨论全部三个方面才能得分。

8. Exam Tips and Command Terms | 考试技巧与指令词

When the question says “draw” or “construct” a box plot, you must produce a neat, clearly scaled diagram. Use a ruler and label the five key values. When the question says “interpret” or “comment on”, you must relate the statistics to the context of the question.

当题目要求“draw”或“construct”箱线图时，你必须画出整洁、刻度清晰的图。使用直尺并标注五个关键数值。当题目要求“interpret”或“comment on”时，你必须将统计量联系到题目情境中。

For interpretation questions, always mention the median with its units, the IQR with its units, and the presence or absence of outliers. Use the actual numerical values from your calculation rather than vague descriptions such as “bigger” or “smaller”.

对于解读类问题，务必提及中位数及其单位、IQR及其单位，以及是否存在离群值。使用计算得到的实际数值，而非“更大”“更小”之类模糊描述。

In paper 2, you may be asked to compare box plots for two different groups. Structure your answer as: centre comparison, then spread comparison, then shape comparison, then a concluding sentence that answers the original question.

在卷二（Paper 2）中，你可能会被要求比较两组不同数据的箱线图。组织答案的结构为：先比较中心，再比较离散程度，然后比较形状，最后用一句话回答原题。

With practice and attention to these details, box plots become a straightforward and reliable source of marks in the IB statistics section.

通过练习并关注这些细节，箱线图将成为IB统计部分中稳妥的得分点。

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Interpreting and Drawing Box Plots | 箱线图的解读与绘制