A stem-and-leaf diagram retains each original value. Its key tells you how stem and leaf combine; \(\displaystyle 2|7=27\) is different from \(\displaystyle 2|7=2.7\). Arrange leaves in ascending order, including repeated leaves for repeated observations.
Count leaves to find the sample size, and read ordered values to locate a median. For an even number of observations, average the two middle values. A back-to-back diagram compares two groups using the same stems and key.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Stem-and-leaf diagrams
Stem-and-leaf data:
Stem
Leaves
4
1 3 3 7
5
2 5
Key: 4|1 means 41. Every leaf represents one observation.
Find the smallest value.
Find the largest value.
Find the median.
Find the mode.
Show the reasoning
(a)
Combine the first stem and leaf using the supplied key.
The first leaf gives 41.
Answer: \(41\).
(b)
Use the final stem and leaf, not just the leaf digit.
The last leaf gives 55.
Answer: \(55\).
(c)
Locate both middle observations in the ordered leaves.
There are six values; average the third and fourth, 43 and 47, giving \(45\).
Answer: \(45\).
(d)
Identify the most frequent complete value, retaining repeated leaves.
The repeated value 43 occurs twice; every other value occurs once.
Answer: \(43\).
Example 2 · Stem-and-leaf diagrams
Stem-and-leaf data:
Stem
Leaves
5
1 3 3 7
6
2 5
Key: 5|1 means 51. Every leaf represents one observation.
Find the smallest value.
Find the largest value.
Find the median.
Find the mode.
Show the reasoning
(a)
Combine the first stem and leaf using the supplied key.
The first leaf gives 51.
Answer: \(51\).
(b)
Use the final stem and leaf, not just the leaf digit.
The last leaf gives 65.
Answer: \(65\).
(c)
Locate both middle observations in the ordered leaves.
There are six values; average the third and fourth, 53 and 57, giving \(55\).
Answer: \(55\).
(d)
Identify the most frequent complete value, retaining repeated leaves.
The repeated value 53 occurs twice; every other value occurs once.
Answer: \(53\).
Example 3 · Stem-and-leaf diagrams
Stem-and-leaf data:
Stem
Leaves
6
1 3 3 7
7
2 5
Key: 6|1 means 61. Every leaf represents one observation.
Find the smallest value.
Find the largest value.
Find the median.
Find the mode.
Show the reasoning
(a)
Combine the first stem and leaf using the supplied key.
The first leaf gives 61.
Answer: \(61\).
(b)
Use the final stem and leaf, not just the leaf digit.
The last leaf gives 75.
Answer: \(75\).
(c)
Locate both middle observations in the ordered leaves.
There are six values; average the third and fourth, 63 and 67, giving \(65\).
Answer: \(65\).
(d)
Identify the most frequent complete value, retaining repeated leaves.
The repeated value 63 occurs twice; every other value occurs once.
Answer: \(63\).
Common mistakes
Keep intermediate values exact until the final step.