Back-to-Back Stemplots Comparison

Two Groups, One Shared Set of Stems

A back-to-back stemplot displays two groups on opposite sides of the same stem column. Like the displays in Comparing Two Dotplots, it helps you compare distributions. Unlike a dotplot, it also organizes the observations by place value: a stem gives the leading digit or digits, and each leaf gives a final digit. Most importantly, the display preserves the exact observations, so you can recover individual values rather than estimate them from intervals.

A back-to-back stemplot does not combine the groups into one data set. The left leaves belong to one group, the right leaves belong to the other, and the shared stems simply provide a common numerical scale. Always read the key before decoding values. For a two-digit variable, a key might say \(4|2=42\). For a decimal variable, it might say \(12|4=12.4\).

Leaves on the right are usually listed in increasing order as you read from the stem outward. Leaves on the left are usually listed in decreasing order as you read from left to right, so the smallest leaf is nearest the stem. This convention keeps the values in order when you read outward from the stem on either side. The left-to-right order of printed leaves is not the order you use to decode a value: pair each leaf with its stem, then follow the key.

A Reliable Way to Read the Plot

Before comparing groups, check the title or labels to identify the variable, units, and group names. Then locate the key and confirm what one stem and one leaf represent. Scan the stems from smallest to largest. For each stem, pair every leaf on each side with that stem. A repeated leaf represents a repeated observation, not a printing mistake. Count each leaf once when finding the group’s sample size.

A useful check is to count the leaves on each side and compare that count with the stated sample size, if one is provided. You can also write the decoded values in increasing order to verify that no leaf was skipped, duplicated, or assigned to the wrong group. As in Comparing Distributions Needs All Four Features, a complete comparison may consider shape, center, spread, and unusual features. A question about a particular exact value, however, should be answered directly rather than replaced by a general description.

Worked Example: Comparing Quiz Scores

Repeated Leaves and Exact Values

Each leaf stands for one observation. Thus, two leaves with the same digit on the same stem indicate two observations with the same value. Keep both when you count the sample size or find an ordered position such as the median. A repeated value can occur on either side, and matching values across the two sides still belong to separate groups.

The sides can also make it easy to locate particular values. For example, if a question asks how many Group A observations are in the 50s, count Group A’s leaves beside stem 5. If it asks whether a particular value appears, look for its stem and then its leaf on the specified side. The key determines the exact value; the visual position of a leaf does not change its meaning.

Worked Example: Comparing Travel Times

Reading a Different Place-Value Key

Not every key uses a ones digit as the leaf. A plot might show tenths, hundredths, or another place-value convention. Never assume that \(12|4\) means 124, 12.4, or 12 and 4 tenths without checking the key. The key defines the scale for the entire plot, including both groups.

When leaves are decimals, their order still follows the numerical order within the stem. Repeated decimal leaves still represent repeated exact measurements. The same comparison habits apply: decode each side, count its leaves, and then describe a feature that answers the question.

Worked Example: Comparing Seedling Heights

Common Mistakes and AP Exam Tips

• Ignoring the key. A stem and leaf do not have a universal scale. A full-credit response decodes values using the stated key and includes units when appropriate.
• Reading left-side leaves in the wrong direction. The left leaves are commonly printed in descending order from left to right. Pair each leaf with its stem and list the resulting values in increasing order before calculating a median.
• Dropping repeated leaves. Every leaf is an observation. Keep repeated values when counting data or finding the median.
• Mixing up the groups. A leaf belongs to the side on which it appears. The common stem column does not make observations shared between groups.
• Treating aligned positions as pairs. Opposite-side leaves are not matched observations. The plot compares two groups; it does not indicate that the first value on one side corresponds to the first value on the other.
• Giving a vague comparison. Name the groups, variable, feature, and units. If you compare medians or ranges, show how you obtained them and state the difference in context.

A strong answer uses exact values where the plot allows them, but does not claim more than it shows. For instance, state that a group has a higher median when the calculated medians support that statement; do not conclude that all its observations are higher. As in Comparing Parallel Boxplots for Several Groups, be clear about which feature is being compared. A stemplot adds the advantage of preserving the individual values behind the comparison.