Stem and leaf diagrams

A stem and leaf diagram is a way of showing numerical data in order while keeping every value. Each number is split into a stem, usually its tens digit, and a leaf, its units digit. The leaves are written in a row beside their stem, smallest first, and a key explains how to read them. From the finished diagram you can read off the median, mode, range and quartiles. In the US it is called a stem and leaf plot.

What is a stem and leaf diagram?

Here are the times, in seconds, that 15 students took to solve a puzzle:

12, 15, 18, 21, 23, 23, 27, 29, 30, 34, 35, 38, 41, 42, 47

As a stem and leaf diagram, the tens digits become the stems and the units digits become the leaves:

Key: 2 | 3 means 23 seconds

Read the second row as 21, 23, 23, 27 and 29. Turn the page on its side and the rows look like a bar chart, so you can also see the shape of the data: most times are in the 20s, with fewer at either end.

How to draw a stem and leaf diagram

1. Choose the stems. For two-digit data, use the tens digits. Write them in a column from smallest to largest, including any stem with no leaves.
2. Draw a vertical line to the right of the stems.
3. Write each leaf beside its stem. Doing it in the order the data comes is fine for a first draft.
4. Order the leaves in each row from smallest to largest, keeping them evenly spaced. Most exam questions want an ordered diagram.
5. Add a key, such as 2 | 3 means 23 seconds, with units.

Decimals and bigger numbers

The key tells the reader what the digits mean, so stems and leaves can stand for other place values. For lengths such as 4.2, 4.5, 5.1, 5.6 and 6.3 cm, the stems are the units and the leaves are the tenths, with the key 5 | 1 means 5.1 cm. For three-digit numbers such as 132 or 147, the stems can be 13 and 14, with the key 13 | 2 means 132.

Finding averages and spread from a stem and leaf diagram

Because the diagram is in order, the values you need are easy to count to. Use the puzzle times above.

Example 1: finding the median

With n = 15, the position of the median is (15 + 1) ÷ 2, which is the 8th value. Count along the leaves: the first row has 3, the second row brings the total to 8, and the 8th leaf is the last one in that row, the 9. With stem 2, that is 29 seconds.

With an even number of values, the median is halfway between the two middle values. For 16 values, it would be halfway between the 8th and 9th.

You can check the mean, median, mode and range of any list in the Mean, Median and Mode Calculator.

Back-to-back stem and leaf diagrams

To compare two sets of data, put them on either side of the same stems. The leaves on the left are read from right to left, so the smallest leaf sits next to the stem on both sides.

Here are the puzzle times for a second class of 13 students, Class B, beside Class A from above:

Key: Class B 6 | 2 means 26 seconds. Class A 2 | 3 means 23 seconds.

Example 2: comparing the two classes

• Class B median: 13 values, so the 7th value: 26 s. Class A median: 29 s.
• Class B range: 52 − 9 = 43 s. Class A range: 35 s.

A good comparison makes one point about the average and one about the spread, in context: "Class B had a lower median, so on average they solved the puzzle faster. Class B had a larger range, so their times were more spread out."

Check your reading of a stem and leaf diagram.

Advantages and disadvantages

For large data sets, a grouped frequency table, a histogram or a box plot works better. A stem and leaf diagram is ideal for about 10 to 50 values.

Common mistakes

• Forgetting the key. Without it, 3 | 4 could mean 34, 3.4 or 340. A key is usually worth a mark.
• Leaving the leaves unordered when the question asks for an ordered diagram.
• Missing out repeated values. Two 23s need two 3s in the 2 row.
• Skipping an empty stem. If no value is in the 40s, still write the 4 stem with no leaves, so the shape stays true.
• Writing the median as a leaf, such as 9 instead of 29. Put the stem back on.
• Reading the left side the wrong way in a back-to-back diagram. Those leaves go outward from the stem.

Practise with a short timed paper

Reading charts and finding averages comes up in every GCSE Maths course. The daily GCSE Maths Foundation and GCSE Maths Higher papers are short timed papers of 8 questions, with a new one every day.

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