Stem and Leaf Diagram — Definition, Formula & Examples
A stem and leaf diagram is a way to display numerical data by splitting each value into two parts: the stem (all digits except the last) and the leaf (the last digit). The stems are listed vertically, and the leaves are written in order next to their matching stem.
A stem and leaf diagram is a statistical display in which each data value is decomposed into a leading portion (the stem) and a trailing digit (the leaf). Stems are arranged in a vertical column in ascending order, and the corresponding leaves are listed horizontally beside each stem, also in ascending order, preserving the original data values while revealing the shape of the distribution.
How It Works
To build a stem and leaf diagram, first decide how to split each number. For two-digit numbers, the tens digit becomes the stem and the ones digit becomes the leaf. Write all unique stems in a vertical column from smallest to largest, then draw a vertical line to the right. For each data value, place its leaf digit next to the correct stem. Finally, reorder the leaves from smallest to largest within each row. You can read back any original value by combining a stem with one of its leaves.
Worked Example
Problem: Display the following test scores in a stem and leaf diagram: 42, 45, 51, 53, 53, 58, 62, 67, 68, 71, 75, 80.
Step 1: Identify the stems (tens digits). The data ranges from 42 to 80, so the stems are 4, 5, 6, 7, and 8.
Step 2: Place each leaf (ones digit) next to its stem, in order:
Step 3: Write the completed diagram: 4 | 2 5, 5 | 1 3 3 8, 6 | 2 7 8, 7 | 1 5, 8 | 0. A key should state: 4 | 2 means 42.
Answer: The stem and leaf diagram has five rows. The row with stem 5 has the most leaves (4 values), showing that the 50s is the most common score range. Every original data value can be recovered from the diagram.
Why It Matters
Stem and leaf diagrams let you see the shape of a distribution (clusters, gaps, outliers) while keeping every original data value visible — something a histogram cannot do. They appear frequently in middle-school and introductory statistics courses and serve as a quick way to organize small data sets by hand.
Common Mistakes
Mistake: Forgetting to include a key (legend) that explains what the stem and leaf represent.
Correction: Always write a key such as '5 | 3 means 53' so the reader knows the place value of each part.