A back-to-back stem and leaf diagram displays two data sets sharing one central stem, with each group's leaves spreading out on opposite sides. It is the clearest way to compare two distributions side by side, allowing medians, ranges and the overall shape to be read directly from the raw data.

What is a back-to-back stem and leaf diagram?

A back-to-back stem and leaf diagram (also called a back-to-back stemplot) places the shared stem values in a central column. One group's leaves extend to the left of the stem; the other group's leaves extend to the right.

Unlike a regular stem and leaf diagram, which shows one data set, a back-to-back version allows direct visual comparison of two groups: you can see at a glance whether one group's data is shifted higher or lower, and whether the spread is similar.

Each leaf represents the units digit of a data value; the stem represents the tens digit. For example, the entry 3 | 4 | 7 means the value 43 for the left group and 47 for the right group.

How do you draw a back-to-back stem and leaf diagram?

Data: Two classes took a test marked out of 70.

Class A scores: 23, 31, 35, 38, 42, 45, 47, 51, 54, 60 Class B scores: 29, 33, 36, 40, 43, 48, 52, 55, 58, 63

Step 1: Write the stems (tens digits) in a central column: 2, 3, 4, 5, 6.

Step 2: For Class A, write leaves to the LEFT of each stem in ascending order outward from the stem (largest leaf nearest the edge).

Step 3: For Class B, write leaves to the RIGHT in ascending order outward from the stem (smallest leaf nearest the stem).

Class A     |Stem|  Class B
          3 | 2  | 9
      8 5 1 | 3  | 3 6
      7 5 2 | 4  | 0 3 8
        4 1 | 5  | 2 5 8
          0 | 6  | 3

Key: For Class A, read leaves right to left from the stem: 3|2 = 23, 1|3 = 31, 5|3 = 35, etc. For Class B, read leaves left to right: 2|9 = 29, 3|3 = 33, 3|6 = 36, etc.

Always include a key showing how to read values from both sides.

How do you find the median from a back-to-back diagram?

For each group, count the total leaves, identify the middle position, then read off the value.

Class A — 10 values: median = average of 5th and 6th values.

Reading Class A from smallest to largest: 23, 31, 35, 38, 42, 45, 47, 51, 54, 60.

5th value = 42, 6th value = 45. Median of Class A = (42 + 45) / 2 = 43.5

Class B — 10 values: median = average of 5th and 6th values.

Reading Class B: 29, 33, 36, 40, 43, 48, 52, 55, 58, 63.

5th value = 43, 6th value = 48. Median of Class B = (43 + 48) / 2 = 45.5

How do you find the range from a back-to-back diagram?

Range = largest value − smallest value. Read the extreme leaves in each group.

Group Smallest Largest Range
Class A 23 60 37
Class B 29 63 34

How do you compare two distributions?

A good comparison uses both a measure of average (typically the median) and a measure of spread (typically the range or interquartile range), followed by a contextual statement.

Comparison for the example:

  • Class B has a higher median (45.5) than Class A (43.5), so Class B performed better on average.
  • Class A has a slightly larger range (37) than Class B (34), so Class A's scores were more spread out (less consistent).
  • Overall, Class B achieved higher and more consistent results.

What mistakes should you avoid?

Mistake 1 — Reading Class A's leaves in the wrong direction. Class A's leaves are read from the stem outward (right to left). The digit nearest the stem is the units digit of the smallest value in that row. Reading left to right reverses the order and gives incorrect values.

Mistake 2 — Omitting the key. A back-to-back stem and leaf diagram without a key is incomplete. The key must show how to read values from both sides.

Mistake 3 — Confusing median and mode. The median requires you to list all values in order for each group. The mode is simply the leaf that appears most often for a given stem. They can be very different.

Mistake 4 — Writing leaves in random order. Leaves should be arranged in ascending order away from the stem. Unordered leaves make it much harder to read off the median or quartiles.

Frequently asked questions

How do you find the lower and upper quartiles from a back-to-back diagram?

For n values, the lower quartile (Q1) is the median of the lower half and the upper quartile (Q3) is the median of the upper half. For 10 values, Q1 = average of 2nd and 3rd values; Q3 = average of 8th and 9th values. For Class A: Q1 = (31+35)/2 = 33; Q3 = (51+54)/2 = 52.5. IQR = 52.5 − 33 = 19.5.

When would you use a back-to-back stem and leaf diagram instead of a box plot?

A back-to-back stem and leaf diagram preserves the raw data, so individual values can be read off exactly. A box plot summarises the data (min, Q1, median, Q3, max) and is better for comparing large data sets where showing every value would be impractical. For data sets of about 10–30 values, the back-to-back diagram is often clearer.

Can a back-to-back diagram show more than two groups?

The standard back-to-back diagram compares exactly two groups — one on each side of the stem. If you need to compare three or more groups, use separate stem and leaf diagrams or a grouped bar chart.

What if both groups have different numbers of data values?

The construction method is the same — you still use a shared stem and write leaves on each side. Finding the median for each group uses that group's own count, so having different numbers on each side is not a problem. However, when comparing, be aware that range and median can be affected by sample size, and mention this in a written comparison.

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