Cambridge Checkpoint Maths p131: Stem-and-Leaf Diagrams | 剑桥数学 p131：茎叶图

📚 Cambridge Checkpoint Maths p131: Stem-and-Leaf Diagrams | 剑桥数学 p131：茎叶图

In the Cambridge KS3 mathematics curriculum, page 131 introduces stem-and-leaf diagrams – a powerful yet simple way to organize and display data while preserving every original value. This article unpacks everything you need to know about constructing, interpreting and comparing stem-and-leaf diagrams, complete with practice examples and exam tips.

在剑桥 KS3 数学课程中，第 131 页引入了茎叶图——一种既强大又简单的数据整理与展示方式，能够保留每一个原始数值。本文将详细讲解茎叶图的构建、解读和比较，并提供练习示例与考试技巧。

1. What is a Stem-and-Leaf Diagram? | 什么是茎叶图？

A stem-and-leaf diagram (or stemplot) is a method of organising numerical data in order of place value. The ‘stem’ represents the leading digit(s) and the ‘leaf’ shows the trailing digit. It works like a hybrid between a table and a histogram, giving a quick visual snapshot of data distribution while keeping exact values.

茎叶图（又称茎叶图或枝干图）是一种按数位整理数值数据的方法。“茎”代表前导数字，“叶”表示末尾数字。它兼具表格和直方图的特点，能快速呈现数据分布概况，同时保留精确数值。

2. Key Components: Stem and Leaf | 关键组成：茎和叶

Every number is split into two parts: the stem, usually all digits except the last one, and the leaf, the final digit. For example, the number 47 has stem 4 and leaf 7. The stem is written once per row in a vertical column, and the leaves are listed horizontally next to their stem.

每个数被分成两部分：茎通常包含除最后一位以外的所有数字，叶则是最后一位数字。例如，数字 47 中，茎是 4，叶是 7。茎在左侧垂直排列，每行写一个茎值，叶则水平列在对应的茎旁边。

A key must always be provided to show how to read the diagram, e.g. ‘4 | 7 means 47’ or ‘4 | 7 = 4.7’ if scaling.

必须始终提供图例来标明读取方式，例如“4 | 7 表示 47”，或在使用缩放时标明“4 | 7 = 4.7”。

3. How to Construct a Stem-and-Leaf Diagram | 如何构建茎叶图

Building a stemplot follows clear steps:

构建茎叶图的步骤清晰明了：

• Identify the stems: Look at the range of data and decide the stems. For two-digit numbers, tens digit becomes the stem.
  确定茎：观察数据范围，选取茎值。对于两位数，十位数字即为茎。
• Sort the leaves: Write each leaf corresponding to its stem in order from smallest to largest.
  对叶排序：将每个叶按对应茎从小到大依次书写。
• Add a key: Provide a clear key explaining the representation.
  添加图例：提供清晰的图例解释表达方式。
• Label the diagram: Include a title and indicate the units.
  标注图表：加入标题并标明单位。

4. Ordered vs. Unordered Diagrams | 有序图与无序图

An unordered stem-and-leaf diagram lists leaves as they appear in the raw data. While quick to create, it makes finding the median more difficult. An ordered diagram sorts leaves in ascending order, which is essential for statistical calculations and is always preferred in exams.

无序茎叶图按照数据原始顺序列出叶子。虽然创建快捷，但不利于寻找中位数。有序图将叶子按升序排列，这对统计计算至关重要，考试中永远优先使用有序图。

Always rearrange unordered leaves into ordered form before answering questions about median, quartiles or range.

在回答有关中位数、四分位数或极差的问题之前，务必将无序叶子重新排列成有序形式。

5. Reading a Stem-and-Leaf Diagram: Finding the Mode | 读取茎叶图：找出众数

The mode is the data value that appears most often. In a stem-and-leaf diagram, simply scan the leaves and identify which leaf digit occurs most frequently for a given stem, then combine stem and leaf. If multiple modes exist, list them all.

众数是出现频率最高的数据值。在茎叶图中，只需浏览叶子，找出某茎下出现次数最多的叶数字，再将茎与叶组合即可。如果存在多个众数，全部列出。

Example: for the row 2 | 3, 5, 3, 8, 3, the leaf 3 appears three times, so mode = 23 (if stem is 2).
示例：在行 2 | 3, 5, 3, 8, 3 中，叶 3 出现了三次，因此众数 = 23（假设茎为 2）。

6. Finding the Median and Quartiles | 找出中位数和四分位数

The median splits the ordered dataset into two equal halves. Count the total number of values ‘n’ from the leaves. If n is odd, the median is the middle value. If n is even, the median is the mean of the two central values. Quartiles divide the data into quarters: Q1 is the median of the lower half, Q3 is the median of the upper half.

中位数将有序数据分成两等份。先数清叶子总数“n”。若 n 为奇数，中位数即为中间值；若 n 为偶数，中位数是位于中间的两个值的平均数。四分位数将数据分为四等份：Q1 为下半部分的中位数，Q3 为上半部分的中位数。

Position of median = (n + 1)/2

中位数的位置 = (n + 1)/2

To locate Q1, take the lower half of the data (excluding the median if n is odd) and find its median. Similarly, Q3 is the median of the upper half.

要定位 Q1，取数据下半部分（若 n 为奇数则不包括中位数），并找出其中位数。同理，Q3 为上半部分的中位数。

7. Calculating the Range and Interquartile Range | 计算极差和四分位距

The range gives the spread of the data: Range = highest value – lowest value. The interquartile range (IQR) measures the spread of the middle 50%: IQR = Q3 – Q1. The IQR is less affected by outliers and provides a more robust measure of dispersion.

极差反映数据的变化幅度：极差 = 最大值 – 最小值。四分位距 (IQR) 衡量中间 50% 数据的离散程度：IQR = Q3 – Q1。IQR 受异常值影响较小，是一种更稳健的离散程度度量。

If the data set has an even number of values, follow the same process carefully when finding quartiles – some exam boards use different conventions, so always state your method or refer to the method taught in Cambridge Checkpoint.

当数据值为偶数时，寻找四分位数时务必仔细遵循同一流程——部分考试局采用不同惯例，因此务必说明所用的方法或依据剑桥 Checkpoint 教授的方法。

8. Back-to-Back Stem-and-Leaf Diagrams | 背靠背茎叶图

A back-to-back stemplot compares two related datasets using a common stem. Leaves for one dataset extend to the left, and leaves for the other to the right. This visual layout makes it easy to compare distributions, identify differences in spread and central tendency.

背靠背茎叶图利用共同的茎比较两个相关数据集。一组数据的叶子向左延伸，另一组向右延伸。这种可视化布局便于比较分布、识别离散程度和集中趋势的差异。

When constructing, create a central column of stems, then list leaves for the left-hand data in reverse order (smallest nearest the stem). Always provide a key for each side.

构建时，先建立居中的茎列，然后将左侧数据的叶子按逆序排列（最小值靠近茎）。务必为两侧分别提供图例。

9. Advantages and Disadvantages | 优点和缺点

Advantages: preserves original data, easy to find median and mode, shows shape of distribution, useful for small to medium datasets, no loss of information unlike a histogram.

优点：保留原始数据，便于找出中位数和众数，显示分布形态，适用于小到中等规模数据集，与直方图不同，不会丢失信息。

Disadvantages: becomes cluttered with very large datasets or numbers with many digits, not suitable for categorical data, and can be less familiar to some audiences.

缺点：数据集较大或数字位数众多时会变得杂乱，不适合分类数据，有时读者对其不够熟悉。

10. Common Mistakes to Avoid | 常见错误避免

• Forgetting the key: A stem-and-leaf diagram without a key is meaningless in an exam.
  忘记图例：考试中没有图例的茎叶图毫无意义。
• Misaligning stems: Always align stems vertically for clarity.
  茎排列不齐：务必垂直对齐茎以确保清晰。
• Leaves out of order: Always put leaves in ascending order before finding medians.
  叶子未排序：寻找中位数前务必把叶子排为升序。
• Skipping a stem with no leaves: Include the stem even if it has no leaves, to preserve the scale, unless instructed otherwise.
  遗漏无叶茎值：即使某茎没有叶子也要保留该茎，以维持尺度，除非另有说明。
• Incorrect median calculation: Double-check n and (n+1)/2 when locating the middle value.
  中位数计算错误：定位中间值时要重新核对 n 和 (n+1)/2。

11. Practice Example with Step-by-Step Solution | 实践示例与分步解答

The heights (in cm) of 15 students: 152, 148, 155, 150, 149, 157, 154, 153, 151, 150, 156, 152, 160, 147, 158. Construct an ordered stem-and-leaf diagram and find the median and interquartile range.

15 名学生的身高（单位：厘米）：152, 148, 155, 150, 149, 157, 154, 153, 151, 150, 156, 152, 160, 147, 158。构建有序茎叶图，并找出中位数和四分位距。

Step 1: determine stems. Data are three-digit numbers. Use the first two digits as stem (14, 15, 16) and the last digit as leaf.
第 1 步：确定茎。数据为三位数。取前两位作为茎（14, 15, 16），最后一位为叶。

Step 2: sort leaves. Group by stem:
第 2 步：对叶排序。按茎分组：

Stem (茎)	Leaf (叶)
14	7, 8, 9
15	0, 0, 1, 2, 2, 3, 4, 5, 6, 7, 8
16	0

Key: 14 | 7 means 147 cm
图例：14 | 7 表示 147 cm

Step 3: find median. Total n = 15 (odd). Position = (15+1)/2 = 8th value. Counting leaves: 14|7(1),8(2),9(3); 15|0(4),0(5),1(6),2(7),2(8) → the 8th value is 152. So median = 152 cm.
第 3 步：找中位数。总数 n = 15（奇数）。位置 = (15+1)/2 = 第 8 个值。数叶子：14|7(1),8(2),9(3); 15|0(4),0(5),1(6),2(7),2(8) → 第 8 个值为 152。因此中位数 = 152 cm。

Step 4: quartiles. Lower half (7 values): 147,148,149,150,150,151,152. Median of lower half = 150 → Q1 = 150 cm. Upper half (7 values): 153,154,155,156,157,158,160. Median = 156 → Q3 = 156 cm.
第 4 步：四分位数。下半部（7 个值）：147, 148, 149, 150, 150, 151, 152。中位数 = 150 → Q1 = 150 cm。上半部（7 个值）：153, 154, 155, 156, 157, 158, 160。中位数 = 156 → Q3 = 156 cm。

Step 5: IQR. IQR = 156 – 150 = 6 cm.
第 5 步：四分位距。IQR = 156 – 150 = 6 cm。

12. Summary | 总结

Stem-and-leaf diagrams are a fundamental data representation tool in the Cambridge KS3 syllabus. They retain raw data, allow quick estimation of central tendency and spread, and form the basis for box-and-whisker plots. Mastering ordered stemplots, back-to-back comparisons and accurate median/quartile calculations will boost your confidence in handling statistics.

茎叶图是剑桥 KS3 大纲中一种基础的数据表示工具。它保留原始数据，能快速估算集中趋势和离散程度，并构成后续箱形图的基础。掌握有序茎叶图、背靠背比较以及准确的中位数/四分位数计算方法，将极大提升你处理统计问题的信心。

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