Mathematics

What Is a Stem-and-Leaf Diagram?

Fifteen test marks turned into a stem-and-leaf diagram, why the key is worth a mark on its own, and how the median, mode and range come straight off it.

Stem-and-leaf diagram

A stem-and-leaf diagram displays data in order by splitting each value into a stem, usually the leading digits, and a leaf, usually the final digit.

Also called: Stem-and-leaf plot, Stemplot
Where students meet it: Grade 6 or 7 statistics, when a class first has to put a list of raw results in order before finding a median.

The short answer
A stem-and-leaf diagram splits each value into a stem, usually the tens digit, and a leaf, the units digit, so 27 is written as 2 | 7. Because the leaves are in order, the median, mode and range can be read straight off it.

An example

3 | 0 3 4 6 8 8 (Key: 3 | 0 means 30 marks)

This is one row of a diagram built from test marks. The stem 3 stands for thirty, and each leaf beside it completes a value: 30, 33, 34, 36, 38 and 38 again. Six students scored in the thirties, and every one of their marks is still recoverable from the row. That is the property a bar chart gives up and this diagram keeps.

Building one from raw marks

Take fifteen marks out of 50, recorded in the order the papers were handed in: 34, 27, 41, 38, 27, 45, 33, 29, 41, 36, 27, 42, 30, 38, 45. The tens digits present are 2, 3 and 4, so those are the stems. Every units digit goes beside its stem, written in order, with repeats repeated.

Count the leaves: four, six and five, making fifteen. That check should be automatic, because a dropped leaf silently changes every average you take afterwards.

It is worth being clear about what this diagram is. It is not a picture of the data in the way a bar chart is a picture. It is the data, rearranged — every original mark can still be read back out — and that is exactly why the averages come off it so easily.

2 | 7 7 7 9
3 | 0 3 4 6 8 8
4 | 1 1 2 5 5
Key: 2 | 7 means 27 marks

The key is not decoration

Without a key, the row 2 | 7 could mean 27, or 2.7, or 270. Nothing in the diagram itself decides, which is why a missing key costs a mark of its own and why it is written even when the meaning seems obvious.

The key also tells you what the split was, and the split does not have to be tens and units. For data from 1.2 to 4.9 the stem is the units digit and the leaf is the first decimal place. For data from 120 to 490 the stem is the hundreds and tens together, so 3 | 6 would read as 360. Choose the split that gives you roughly five to ten stems: two stems make an unreadable clump, twenty make a list.

Reading the averages straight off

This is the whole point of the diagram. Because the leaves are already sorted, no further ordering is needed and every summary statistic is a matter of counting positions.

With fifteen values, the median sits in position (15 + 1) ÷ 2 = 8. Counting leaves from the start — 27, 27, 27, 29, 30, 33, 34 — the eighth is 36. The mode is 27, the only value appearing three times, and it is obvious because three identical leaves sit together. The range is 45 − 27 = 18, taken from the first and last leaves. The quartiles fall at positions 4 and 12, giving 29 and 41, so the interquartile range is 12.

A back-to-back diagram extends the idea to two groups: one group's leaves are written to the left of the stem, reading outwards, and the other's to the right. It puts both distributions on one shared scale, which makes comparing them a matter of looking rather than calculating.

Common questions

What is the key for on a stem-and-leaf diagram?

It states what one entry means, such as "2 | 7 means 27 marks", so the reader knows whether the values are tens and units, units and tenths, or something else. The diagram is ambiguous without it, and in an exam the key is usually worth a mark by itself.

How do you find the median from a stem-and-leaf diagram?

Count the leaves to get n, then find position (n + 1) ÷ 2 and count in from the start. With 15 values the median is the 8th leaf. Because the leaves are already in order, no sorting is needed — which is the main reason the diagram is drawn at all.

What do you do if a value appears more than once?

Write the leaf again. Three students scoring 27 gives three sevens on the 2 row. Removing duplicates would break the count and make the median and mode wrong, and repeated leaves sitting side by side are what makes the mode visible at a glance.

Can you use a stem-and-leaf diagram for three-digit numbers?

Yes, by splitting differently. For values from 120 to 490, use the hundreds and tens as the stem and the units as the leaf, so 3 | 6 means 360. Aim for roughly five to ten stems; the key then tells the reader which split you chose.

What is a back-to-back stem-and-leaf diagram?

One diagram comparing two groups. A single column of stems runs down the middle, with one group's leaves written to the left, increasing outwards from the stem, and the other's to the right. Both sets share a scale, so differences in centre and spread are visible without any calculation.

Last updated August 21, 2026