A stem and leaf diagram is a simple but powerful way to display data while keeping every individual value visible. Unlike a bar chart, it shows the actual numbers in order, making it easy to find the median, mode, and range at a glance. The defining rule: always order the leaves from smallest to largest.
What is a stem and leaf diagram?
A stem and leaf diagram splits each number into two parts:
- The stem — the leading digit or digits (usually the tens digit for two-digit numbers)
- The leaf — the final digit (usually the units digit)
For example, the number 37 has a stem of 3 and a leaf of 7.
Every diagram must include a key, which tells the reader what the stem and leaf represent. A typical key looks like:
Key: 3 | 7 means 37
Without a key, the diagram is ambiguous and will lose marks in an examination.
How do I draw a stem and leaf diagram?
Follow these steps carefully.
Step 1: Identify the stems. List the tens digits that appear in the data, in order, down the left-hand side of the diagram.
Step 2: Add the leaves. For each data value, write its units digit next to the correct stem. At this stage, do not worry about order — just get all the leaves in place.
Step 3: Order the leaves. Rewrite the diagram so that the leaves on each row run from smallest to largest, reading left to right.
Step 4: Write the key.
Worked Example — Draw a stem and leaf diagram for this data set:
23, 35, 17, 42, 38, 51, 29, 35, 44, 17, 26, 53
Step 1 — identify stems: The values range from 17 to 53, so the stems are 1, 2, 3, 4, 5.
Step 2 — place all leaves (unordered):
| Stem | Leaves |
|---|---|
| 1 | 7 7 |
| 2 | 3 9 6 |
| 3 | 5 8 5 |
| 4 | 2 4 |
| 5 | 1 3 |
Step 3 — order the leaves:
| Stem | Leaves (ordered) |
|---|---|
| 1 | 7 7 |
| 2 | 3 6 9 |
| 3 | 5 5 8 |
| 4 | 2 4 |
| 5 | 1 3 |
Key: 1 | 7 means 17
The complete ordered data set can be read off directly: 17, 17, 23, 26, 29, 35, 35, 38, 42, 44, 51, 53.
How do I find the median from a stem and leaf diagram?
The median is the middle value when all data is arranged in order. Because a stem and leaf diagram already puts values in order, finding the median is straightforward.
For n values:
- If n is odd, the median is the ½(n + 1)th value.
- If n is even, the median is the mean (average) of the (n/2)th and (n/2 + 1)th values.
Using the worked example above with n = 12 values:
- Median = average of the 6th and 7th values.
- Reading from the diagram in order: 17, 17, 23, 26, 29, 35, 35, 38, 42, 44, 51, 53.
- The 6th value is 35 and the 7th value is 35.
- Median = (35 + 35) ÷ 2 = 35
You can also read off:
- Mode = the most frequently occurring value(s). Here, 17 appears twice and 35 appears twice, so the data is bimodal: mode = 17 and 35.
- Range = largest value − smallest value = 53 − 17 = 36
What is a back-to-back stem and leaf diagram?
A back-to-back stem and leaf diagram compares two data sets using a shared stem column. One set's leaves extend to the left, the other's extend to the right.
Worked Example — Two classes sat a maths test. Their scores were:
- Class A: 47, 53, 58, 61, 65, 72, 76, 84
- Class B: 42, 49, 56, 63, 69, 71, 78, 83
Back-to-back stem and leaf diagram:
Class A (leaves) | Stem | Class B (leaves)
7 | 4 | 2 9
8 3 | 5 | 6
5 1 | 6 | 3 9
6 2 | 7 | 1 8
4 | 8 | 3
Key: For Class A, 4 | 7 means 47. For Class B, 4 | 2 means 42.
Note that Class A's leaves are written with the smallest closest to the stem — so row 5 reads, from the stem outward: 3, 8 (representing 53 and 58). Class B's leaves read left to right from the stem in the usual way.
From this diagram you can compare the two classes at a glance. Class B has slightly higher scores in the 40s and 60s ranges, while both classes are spread across all five stems.
When should I use a stem and leaf diagram?
A stem and leaf diagram is most useful when:
- You have a small to medium data set (roughly 5 to 30 values). For very large data sets, grouped frequency tables or histograms are more practical.
- You want to keep the original data visible rather than grouping it into classes.
- You need to find the median, mode, or range quickly from a raw data list.
- You are comparing two data sets (use a back-to-back diagram).
It is less suitable when the data values span a very wide range, because you end up with many stems that have no leaves.
What are the most common mistakes?
Mistake 1: Not ordering the leaves.
Leaving the leaves in the order you wrote them (for example, 3 9 6 instead of 3 6 9) makes it impossible to read off the median correctly and suggests you have not understood the diagram. Always rewrite with leaves ordered.
Mistake 2: Forgetting the key.
Without a key, the reader cannot tell whether 3 | 7 means 37, 3.7, or 370. The key is required in every stem and leaf diagram, including in examinations.
Mistake 3: Using the tens digit as the leaf and the units digit as the stem.
The stem is the leading (tens) digit and the leaf is the trailing (units) digit. Reversing them scrambles the entire diagram.
Mistake 4: Miscounting to find the median.
Count carefully from the smallest value, crossing off values one at a time if necessary. With 12 values, the median is the average of the 6th and 7th — not the 6th alone or the midpoint of the range.
Frequently asked questions
Does every leaf have to be a single digit?
Yes — each leaf is always a single digit (0 through 9), the units digit of the value. If your data contains three-digit numbers such as 137, the stem would be 13 (the hundreds and tens digits) and the leaf would be 7. Always define this clearly in your key.
Can I use a stem and leaf diagram for decimal data?
Yes. For data such as 3.2, 4.5, 3.7, 4.1, you would use the units digit as the stem and the tenths digit as the leaf. The key would read, for example, 3 | 2 means 3.2. The method is identical; only the key changes to reflect what the digits represent.
How do I find the mean from a stem and leaf diagram?
Add up all the values shown in the diagram (reading each stem–leaf combination as a number) and divide by the total count. For the worked example: 17 + 17 + 23 + 26 + 29 + 35 + 35 + 38 + 42 + 44 + 51 + 53 = 410. Mean = 410 ÷ 12 ≈ 34.2. The diagram makes it easy to list every value without missing any.
What is the difference between a stem and leaf diagram and a bar chart?
A bar chart groups data into categories or class intervals and shows only the frequency (count) of each group — the original values are lost. A stem and leaf diagram preserves every individual value, so you can calculate exact averages and the range. The trade-off is that stem and leaf diagrams become impractical for very large data sets, whereas bar charts scale well.
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