📚 Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram is a smart way to organise a set of numbers so that you can see the shape of the data without losing any of the original values. It splits each number into a 'stem' (all but the last digit) and a 'leaf' (the final digit), then lists them in a simple table. This tool is especially useful in KS3 mathematics when you need to find the mode, median and range quickly and clearly.
茎叶图是一种巧妙整理一组数据的方式，既能让你看清数据的分布形状，又不会丢失任何一个原始数值。它把每个数拆分成“茎”（除了最后一位以外的所有数位）和“叶”（最后一位数字），然后在一个简单的表格中列出来。在 KS3 数学中，当你需要快速清晰地找出众数、中位数和极差时，茎叶图特别有用。
1. What is a Stem-and-Leaf Diagram? | 什么是茎叶图？
A stem-and-leaf diagram, sometimes called a stem-and-leaf plot, is a method of showing quantitative data that keeps every piece of information visible. In a data set like 34, 38, 41, 45, the tens digits (3, 3, 4, 4) become the stems, and the ones digits (4, 8, 1, 5) become the leaves. A vertical line separates the stems from the leaves, and each stem appears only once, with its leaves listed beside it.
茎叶图（有时也叫茎叶图）是一种展示定量数据的方法，它让每一条信息都清晰可见。像 34, 38, 41, 45 这样的数据集中，十位数（3, 3, 4, 4）成为茎，个位数（4, 8, 1, 5）成为叶。一条竖线把茎和叶分开，每个茎只出现一次，它的所有叶子列在旁边。
This method is different from a bar chart or a histogram because you can see exactly what each original number was. It is like a sorted list of the data that also acts as a picture of the distribution.
这种方法不同于条形图或直方图，因为你可以精确地看到每一个原始数据是什么。它就像一份排好序的数据列表，同时又起到数据分布图的作用。
2. How to Construct a Stem-and-Leaf Diagram | 如何绘制茎叶图
Step 1: Decide on the stem unit. For whole numbers, the stem is usually the tens digit (or hundreds, if the numbers are large). The leaf is the ones digit. For example, for the number 74, the stem is 7 and the leaf is 4. Write the stems in a vertical column, from smallest to largest, even if some stems have no leaves initially.
第一步：确定茎的单位。对于整数，茎通常是十位数（如果数字较大，也可以是百位数）。叶是个位数。例如，对于数字 74，茎是 7，叶是 4。把茎按从小到大排成一列，即使某些茎一开始没有叶子也要列出。
Step 2: Go through each data value and write its leaf next to the appropriate stem. Do not worry about keeping the leaves in order at this stage – just list them as you read the data. If a number appears more than once, write the leaf as many times as it occurs.
第二步：遍历每一个数据值，把它的叶写在对应茎的旁边。这个阶段不用操心叶子顺序，只要按你读取数据的顺序列出即可。如果一个数出现多次，就按照次数重复写叶。
Step 3: After all leaves are recorded, rewrite the diagram with the leaves in each row arranged from smallest to largest. This is called an ordered stem-and-leaf diagram. Always add a key to explain what one stem-and-leaf combination means, e.g. '4|5 means 45'. A key is essential so that anyone can read your diagram correctly.
第三步：所有叶子记录完毕后，将每一行的叶子从小到大重新排列，这称为有序茎叶图。务必添加一个图例，说明一个茎叶组合代表什么，例如 '4|5 表示 45'。图例至关重要，这样任何人都能正确阅读你的图。
Step 4: Give your diagram a title, such as 'Test scores in Class 8A'. The title, stems, leaves and key together make the diagram complete and easy to interpret.
第四步：给你的图加一个标题，如 '8A 班测验成绩'。标题、茎、叶和图例使图完整且易于解读。
3. Reading Data from a Stem-and-Leaf Diagram | 从茎叶图中读取数据
To read the original values, combine each stem with each of its leaves in turn. If a stem is 5 and its leaves are 2, 5, 8, then the numbers are 52, 55 and 58. The key confirms the place value: a key such as '5|2 means 52' tells you that the stem represents tens and leaves represent ones.
要读取原始值，依次把每个茎和它的叶组合起来。如果茎是 5，叶是 2, 5, 8，那么这些数字就是 52, 55 和 58。图例确认位值：比如 '5|2 表示 52' 告诉你茎代表十位、叶代表个位。
When a stem has no leaves, it simply means there are no data values with that stem. Do not make the mistake of thinking it means zero – it just indicates an empty group.
当某个茎没有叶子，仅仅意味着没有这个茎对应的数据值。不要误以为它代表零——它只表示这一组是空的。
4. Finding the Mode | 寻找众数
The mode is the value that appears most often. In an ordered stem-and-leaf diagram, the mode is very easy to spot: just look for a leaf that repeats more than any other. Since the leaves are listed in order, repeated digits stand out. If you have a row with leaves '3, 3, 3, 7, 9', the value corresponding to leaf 3 appears three times, so that value is likely the mode – but always check other rows in case another value appears even more frequently.
众数是出现次数最多的值。在有序茎叶图中，众数极易识别：只要看看哪个叶重复次数最多就可以了。由于叶子排好序，重复的数字一目了然。如果某行叶子是 '3, 3, 3, 7, 9'，叶 3 对应的值出现了三次，那么这个值很可能就是众数——但一定要检查其他行，以防另一个值出现更频繁。
Example: A stem-and-leaf diagram for data 45, 52, 58, 63, 63, 63, 67 has two 3s for stem 6, but leaf 3 appears three times under stem 6, so the mode is 63.
示例：数据 45, 52, 58, 63, 63, 63, 67 的茎叶图中，茎 6 的叶 3 出现了三次，所以众数是 63。
5. Finding the Median | 寻找中位数
The median is the middle value when all data are arranged in order. With a stem-and-leaf diagram, you can count how many leaves there are in total. If the total number of leaves n is odd, the median is the value at position (n + 1) ÷ 2. If n is even, the median is the average of the two middle values: positions n ÷ 2 and (n ÷ 2) + 1.
中位数是所有数据按顺序排列后中间的那个值。借助茎叶图，你可以数出一共有多少片叶子。如果叶子总数 n 是奇数，中位数就是位置在 (n + 1) ÷ 2 的值。如果 n 是偶数，中位数就是中间两个位置 n ÷ 2 和 (n ÷ 2) + 1 的值的平均数。
To find the position efficiently, count the leaves row by row until you reach the middle position. Then combine the stem and leaf to read the value. For an even count, take the two values either side of the 'centre line' and calculate their mean.
为了高效找到位置，可以一行一行地数叶子，直到到达中间位置。然后把茎和叶组合读出数值。对于偶数个数据，取“中线”两侧的两个值，计算它们的平均数。
Worked example: Stems 4|5 ; 5|2 8 ; 6|1 3 3 7 ; 7|2 8 ; 8| 3). The 7th leaf is the 4th leaf in row 6: leaf 7 (value 67). Median = (63 + 67) ÷ 2 = 65.
示例：茎 4|5 ；5|2 8 ；6|1 3 3 7 ；7|2 8 ；8|1 5 9。n = 12（偶数）。中间位置是第 6 和第 7 个值。数叶子：第 4 行 1 片，第 5 行 2 片（共 3 片），第 6 行 4 片（共 7 片）。第 6 片叶是第 6 行的第 3 片叶：叶 3（值 63）。第 7 片叶是第 6 行的第 4 片叶：叶 7（值 67）。中位数 = (63 + 67) ÷ 2 = 65。
6. Finding the Range | 寻找极差
leaf. The largest value comes from the highest stem and its highest leaf. Range = maximum – minimum.
极差是衡量数据分散程度的指标，它等于最大值减去最小值。在茎叶图中，最小值很容易找到：它是最低的茎及其最低叶子的组合。最大值来自最高的茎及其最高叶子。极差 = 最大值 – 最小值。
af 5, giving 45. The highest stem is 8 with leaves 1,5,9, so the maximum is 89. Range = 89 – = 44. Always state the range as a single number, and include units if the data have them.
用前面的例子，最低茎是 4，叶是 5，得到 45。最高茎是 8，叶有 1,5,9，所以最大值是 89。极差 = 89 – 45 = 44。务必将极差表示为一个单一数值，如果数据有单位也要注明。
7. Ordered vs Unordered Stems | 有序与无序茎叶图
When you first sketch a stem-and-leaf diagram, the leaves are often in the order you recorded them, which might be jumbled. This unordered diagram still shows the data, but it is harder to find the median and mode. You should always redraw the diagram so that the leaves in each row run from smallest to largest. This ordered version is what exam questions almost always require or give you to interpret.
当你首次画出茎叶图时，叶子往往按照你记录的顺序排列，可能是混乱的。这种无序图仍然展示了数据，但较难找到中位数和众数。你应该始终将图重画，使每一行的叶子从小到大排列。这种有序版本是考试题目几乎总要求你绘制的，或者直接提供给你进行解读。
Be careful: some students mix up the need to order the stems as well. Stems must be in ascending order from top to bottom, but they are never put out of order – you always write them consecutively anyway. So the main ordering work is arranging the leaves within each stem row.
请注意：有些学生会把茎的排序要求也弄混。茎必须从上到下按升序排列，但它们从来不会被弄乱——你本来就会连续书写它们。所以主要的有序工作是整理每个茎行内的叶子。
8. Stem-and-Leaf Diagrams with Decimals | 包含小数的茎叶图
If your data contain decimals, such as 2.3, 2.7, 3.1, you can still use a stem-and-leaf diagram by choosing the stem as the whole number part and the leaf as the first decimal digit. A key like '2|3 means 2.3' makes the place value clear. You cannot have a decimal point inside the stem or leaf; the key is the only place where the decimal is explained.
如果你的数据包含小数，比如 2.3, 2.7, 3.1，你仍然可以使用茎叶图，方法是选择整数部分作为茎，第一位小数作为叶。图例如 '2|3 表示 2.3' 让位值变得清楚。你不能在茎或叶中写入小数点；图例是解释小数的唯一地方。
For numbers like 1.45 or 0.8, you might decide the stem is units and the leaf is tenths, then ignore or round further digits. The choice of stem unit must be stated clearly in the key. Consistent place value is the most important rule.
对于 1.45 或 0.8 这样的数字，你可以决定茎是个位、叶是十分位，然后忽略或四舍五入后面的数位。选择的茎单位必须在图例中清楚说明。一致的位值是最重要的规则。
9. Back-to-Back Stem-and-Leaf Diagrams | 背靠背茎叶图
A back-to-back stem-and-leaf diagram is used to compare two related sets of data side by side. A single central column of stems is shared, with leaves for one data set extending to the left and leaves for the other set extending to the right. Usually the left-side leaves are still written in order from the stem outwards (smallest closest to the stem), and the same applies to the right side.
背靠背茎叶图用于并排比较两组相关的数据。中间共用一列茎，一组数据的叶子向左延伸，另一组的叶子向右延伸。通常左侧的叶子从茎向外从小到大书写（最小的离茎最近），右侧同理。
This arrangement makes it straightforward to see differences in spread and central tendency. For example, you might compare test scores of boys and girls using the same stem scale. Both keys must be provided: one for the left side (e.g. '5|4 means 54 for boys') and one for the right side (e.g. '5|3 means 53 for girls'), or one combined key if the stems have the same place value.
这种排列让你一眼就能看出分布和集中趋势的差异。例如，你可以用同一个茎刻度比较男生和女生的测验成绩。必须提供两个图例：左边一个（例如 '5|4 表示男生 54'），右边一个（例如 '5|3 表示女生 53'），或者如果茎的位值相同，也可以用一个合并图例。
10. Comparing Two Distributions using Back-to-Back Stems | 用背靠背茎叶图比较两个分布
When you have a back-to-back diagram, you can quickly compare medians: find the median of the left group and the median of the right group. A higher median suggests the values in that group tend to be larger. Likewise, compare ranges to see which group is more spread out. An approximate comparison of modes can also reveal which value is most typical in each group.
当你有一幅背靠背图时，可以快速比较中位数：找出左边组的中位数和右边组的中位数。中位数较高表明该组数值往往偏大。同样，比较极差可以看出哪一组的数据更分散。众数的大致比较也能揭示每组中最典型的数值。
For example, if the boys' data have a median of 65 and a range of 40, while the girls' data have a median of 70 and a range of 30, you can say that, on average, girls scored higher and their scores were more consistent. Always support comparisons with values from the diagram.
举例来说，如果男生数据的中位数是 65、极差是 40，女生数据的中位数是 70、极差是 30，你就可以说，平均而言女生得分更高，且她们的分数更稳定。进行比较时一定要用图中的数值加以支撑。
11. Common Mistakes to Avoid | 常见错误
Mistake 1: Forgetting the key. A stem-and-leaf diagram without a key is meaningless because the reader cannot be sure whether '2|6' means 26, 2.6 or even 260. Always write a clear key.
错误一：忘记图例。没有图例的茎叶图毫无意义，因为读者无法确定 '2|6' 代表 26, 2.6 还是 260。务必写一个清楚明确的图例。
Mistake 2: Skipping empty stems. If there are no data for a stem like 5, you still need to write the stem 5 in the vertical list; otherwise the gaps are lost and the distribution looks wrong. Just leave the leaf space next to 5 blank.
错误二：漏掉空茎。如果某个茎（比如 5）没有数据，你仍然需要在竖列中写上茎 5；否则会丢失这个空隙，分布图看起来就不对了。只需在 5 旁边留白即可。
Mistake 3: Mixing unordered leaves in the final diagram. An unordered diagram is incomplete for presenting final work or exam answers. Always order the leaves after recording them.
错误三：在最终图中留下无序的叶子。无序图在提交最终作业或考试答案时是不完整的。记录叶子之后一定要排序。
Mistake 4: Misaligning leaves in a back-to-back diagram. The leaves closest to the stem on both sides must be the smallest in their row. Many students put the smallest leaves furthest from the stem on the left side, which makes the diagram impossible to read correctly.
错误四：在背靠背图中叶子未对齐。两侧最靠近茎的叶子必须都是该行最小的。许多学生会把左侧最小的叶子放在离茎最远处，这导致图无法正确阅读。
12. Summary | 总结
Stem-and-leaf diagrams give you a neat way to keep all original data visible while organising it into a shape that reveals the median, mode and range quickly. They work for whole numbers, decimals and comparative data sets with back-to-back layouts. Remember the four key rules: identify the stem unit, record every leaf, order the leaves, and always give a key. Practice with small data sets first, then build up to comparing two distributions. Master this skill, and you will find interpreting data much easier in your Cambridge Checkpoint and beyond.
茎叶图为你提供了一种简洁的方式，既保留所有原始数据，又将其整理成一种能迅速揭示中位数、众数和极差的形状。它们适用于整数、小数以及用背靠背布局比较的数据集。记住四条关键规则：确定茎的单位，记录每一片叶，排列叶子顺序，并始终提供图例。先从小型数据集练起，再扩展到比较两个分布。掌握这一技巧后，你将在 Cambridge Checkpoint 乃至更远的数学学习中发现解读数据变得容易得多。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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