Splitting Stems for Dense Stemplots

Why Split a Stem?

A stemplot keeps the individual data values visible, but that advantage can become harder to appreciate when many leaves crowd into the same row. A row with a long run of digits is difficult to scan, and it may be hard to distinguish repeated leaves or see where the values fall within that stem. Splitting a stem gives its leaves more room without grouping the observations into intervals or discarding their exact values.

The decision is about readability, not about changing the data. Each observation must still appear exactly once, and the stem-and-leaf key must still make it possible to recover the original value. The main choice is how to divide the possible leaves into smaller ranges and how consistently to apply that division.

For example, if the key says \(7|3=73\), then a leaf 3 beside stem 7 represents a score of 73. In a split display using the 0–4 and 5–9 ranges, that leaf belongs in the lower row for stem 7. A leaf 8 beside stem 7 represents 78 and belongs in the upper row. The split changes the placement of the leaves, not the values they encode.

Compare an Unsplit and Split Stemplot

Suppose a hypothetical instructor recorded these 26 exam scores, in points:

52, 54, 57, 58, 61, 62, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 72, 73, 74, 75, 76, 77, 78, 79, 79.

Use the tens digit as the stem and the units digit as the leaf. In an unsplit display, each score from 60 through 69 goes on stem 6, and each score from 70 through 79 goes on stem 7.

The row for stem 7 contains 12 leaves, so it is relatively crowded. The row for stem 6 contains 10. The leaves are correctly ordered, but a reader scanning either row has to follow a long string of digits. Splitting the stems creates shorter rows:

The repeated stem labels are intentional. The first row for stem 6 represents scores from 60 to 64; the second represents scores from 65 to 69. For example, the two leaves 2 on the lower row encode two scores of 62, while the leaf 5 on the upper row encodes 65. Repeated observations remain repeated leaves.

A Reliable Construction Method

A consistent method helps prevent misplaced leaves. First choose the stem-and-leaf place values and state a key with units. Then decide on the leaf ranges before distributing any observations. With the two-row convention, assign leaves 0–4 to the lower row and 5–9 to the upper row. Sort leaves within each row from smallest to largest, retaining duplicates. Finally, check the number of leaves against the number of observations.

The rule can be justified by noting that the two ranges partition all possible unit digits: every digit from 0 through 9 belongs to one range and only one. In symbols,

Consequently, each original leaf has exactly one permitted row. If every observation is assigned by its stem and leaf, the split display preserves the data. This argument also shows why it matters to state the split rule: without it, a repeated stem label alone does not say which row should receive a particular leaf.

How Much Should a Stemplot Be Split?

Two rows per stem are a useful standard, but they are not an automatic improvement for every data set. If most stems have only one or two leaves, splitting every stem may create many short or empty rows and make the display taller without making it clearer. If one or two stems are crowded, a display may split only those stems, provided its key or labels clearly identify the leaf ranges and distinguish split from unsplit rows. For a beginner constructing a display, applying one rule consistently is often easier to read and check.

The goal is not to make every row the same length. It is to provide enough room to see the individual leaves while keeping the structure understandable. A reader should be able to tell, without guessing, what values each row can contain. If a display uses a less uniform choice, label the rows explicitly rather than relying on position or an unstated convention.

Common Mistakes and Why the Choice Matters

Splitting is useful because it can make individual observations easier to locate in a dense row. However, a neat-looking display is not necessarily a correct one. The key, row labels, ordering, and leaf count all need to agree with the actual data.

• Changing the stem's meaning. A repeated stem does not represent a new place value. Both rows for stem 6 still represent scores in the 60s; the leaf range distinguishes the rows.
• Putting a boundary leaf in the wrong row. Under the 0–4 and 5–9 rule, leaf 4 belongs in the lower row and leaf 5 in the upper row. State the convention clearly.
• Removing duplicate leaves. Two observations with the same value require two leaves. Repeated digits are not a formatting error.
• Dropping zeros or other values that look unusual. With the key \(9|0=90\), a zero leaf represents a real score and must be shown.
• Assuming a split display changes the observations. It only changes the layout. Each value must still be recoverable from its stem and leaf.
• Splitting without making the ranges clear. A reader may not know which repeated-stem row contains a particular leaf. Include a key, labels, or an explicit statement of the row ranges.

A common mistaken argument is, “The score 65 is close to 64, so either row is acceptable.” That is incorrect under the stated split rule: the row is determined by the leaf digit, not by which other score seems closest. Since 5 is in the 5–9 range, 65 belongs in stem 6's upper row. Applying the rule mechanically protects the values and makes the display interpretable.