A back-to-back stem plot displays two datasets side by side using a shared column of stems in the center, with one dataset’s leaves extending to the left and the other’s extending to the right. Reading it correctly comes down to understanding three things: what the stem represents, how to read leaves in each direction, and how to use the key to determine actual values.
The Three Parts of the Plot
Every back-to-back stem plot has the same structure. A central column of numbers (the stems) runs vertically down the middle. These stems represent the leading digit or digits of each data value. To the right, one dataset fans out with its leaves. To the left, the other dataset fans out with its leaves. Each leaf is a single digit representing the next place value after the stem.
Here’s what a simple one looks like:
- Left leaves (Dataset A): 8 5 3 |
- Stem: | 7 |
- Right leaves (Dataset B): | 1 4 6
If the leaf unit is 1 and the stem represents tens, the left side gives you 73, 75, and 78, while the right side gives you 71, 74, and 76. Notice that the left-side leaves read outward from the stem. This is the detail most people miss: the digit closest to the stem comes first, and you read away from the center. So “8 5 3” next to a stem of 7 means 73, 75, 78, not 87, 57, 37.
Always Start With the Key
The key (sometimes called the legend) tells you the place value of the leaves, and skipping it is the fastest way to misread the entire plot. A key reading “1 | 2 = 12” means stems are tens and leaves are ones. But a key reading “1 | 2 = 1.2” means stems are ones and leaves are tenths. The same digits on the plot would represent completely different numbers.
Minitab and most statistics software label this as the “leaf unit.” A leaf unit of 1 means each leaf represents ones. A leaf unit of 0.1 means each leaf represents tenths. So a stem of 8 with leaves 0, 2, and 3 at a leaf unit of 1 represents the values 80, 82, and 83. Change that leaf unit to 0.1, and those same digits become 8.0, 8.2, and 8.3.
Reading the Left Side
The right side of a back-to-back stem plot reads the same way as a regular stem-and-leaf plot: stem first, then leaf. The left side is what trips people up. Leaves on the left are written in reverse order, radiating outward from the stem. To reconstruct the actual values, you still combine the stem with each leaf individually, reading from the stem outward.
For example, if the left side shows “9 7 4 2” next to a stem of 10 with a leaf unit of 1, the values are 102, 104, 107, and 109. The leaf closest to the stem (2) is the smallest value, and the leaf farthest from the stem (9) is the largest. This ordering keeps the data visually sorted, which makes it easier to spot patterns.
Comparing the Two Datasets
The whole point of a back-to-back stem plot is comparison. Once you can read both sides, you’re looking for a few things.
Center and spread. Where do most of the leaves pile up on each side? If the left side clusters around stems of 9 and 10 while the right side clusters around stems of 12 and 13, the right-side dataset has generally higher values. The spread tells you about variability: a dataset with leaves scattered across many stems is more spread out than one concentrated on just a few.
Shape. Just like a histogram, each row of leaves acts as a horizontal bar. Step back and look at the overall shape on each side. Is it roughly symmetric, or does one side have a long tail stretching toward higher or lower values? A tail pulling toward larger stems indicates a right skew, while a tail toward smaller stems indicates a left skew. You can compare whether the two datasets share a similar shape or differ.
Outliers. Look for isolated leaves far away from where the rest of the data clusters. A single leaf sitting three or four stems away from the bulk of the data stands out visually and may represent an unusual observation.
A Worked Example
Suppose two classes took the same test, and their scores are displayed back to back with stems in the tens place and a leaf unit of 1:
- Class A (left): 1 | 6 |
- Class A (left): 8 5 3 | 7 | 2 4 (Class B, right)
- Class A (left): 9 6 4 2 0 | 8 | 1 3 5 7 8 (Class B, right)
- Class A (left): 7 5 3 | 9 | 0 2 6 (Class B, right)
- Class A (left): 2 | 10 | 0 1 (Class B, right)
Start with Class A’s row at stem 8: the leaves “9 6 4 2 0” represent 80, 82, 84, 86, and 89. That’s five scores clustered in the 80s, making it Class A’s most populated range. Class B’s leaves at stem 8 are “1 3 5 7 8,” giving 81, 83, 85, 87, and 88. Both classes have their densest concentration in the 80s, but Class A has a score down at 61 (stem 6, leaf 1) while Class B’s lowest is 72. Class A’s data spreads wider.
Counting leaves on each side tells you the sample size for each group. Class A has 12 scores, Class B has 12 scores. Equal sizes make direct visual comparison straightforward, but even with unequal sizes you can compare shape and center.
When This Plot Works Best
Back-to-back stem plots are most useful for comparing two small to moderate datasets of numerical data. They preserve every individual data point, which is an advantage over histograms and box plots that summarize data into bins or quartiles. If you need to find a specific value, like a median, you can count leaves directly.
They become less practical with large datasets. Once you have hundreds of values on each side, rows get crowded with leaves and the plot becomes hard to read. At that point, histograms or box plots give a cleaner comparison. They also only work for two groups. If you need to compare three or more datasets simultaneously, parallel box plots or grouped histograms are better tools.