Questions: Stem-and-Leaf Plots

Question 1

A data set contains the values {12, 15, 21, 22, 28, 34}. What does the row for stem '2' look like in a correctly constructed stem-and-leaf plot?

• A. 2 | 1 2 8
• B. 2 | 8 2 1
• C. 2 | 21 22 28
• D. 2 | 1 1 2 8

Answer: (A)

For two-digit numbers, the stem is the tens digit and the leaf is the units digit. The values 21, 22, and 28 all have stem 2 and leaves 1, 2, and 8 respectively. Leaves must be written in ascending order — so '2 | 1 2 8' is correct. Option B has unordered leaves (a common error). Option C writes full values rather than just the leaf digits. Option D duplicates the leaf '1' incorrectly.

Question 2

What is the key advantage of a stem-and-leaf plot over a histogram displaying the same data?

• A. It is easier to construct for data sets with hundreds of values
• B. It reveals the exact value of every individual data point, not just a count by interval
• C. It displays the mean more clearly than a histogram
• D. It eliminates the need to sort data before analysis

Answer: (B)

A histogram shows how many values fall within each interval (e.g., '6 values between 30 and 40') but discards the exact values. A stem-and-leaf plot shows that those 6 values are, say, 31, 34, 34, 38, 39, and 39 — you can reconstruct the original data set from the plot. This is the defining feature. The tradeoff is that stem-and-leaf plots become unwieldy with very large data sets, where histograms are more readable.

Question 3

In a stem-and-leaf plot of two-digit numbers, reading a stem and any one of its leaves together always recovers an original data value.

• T. True
• F. False

Answer: True

This is precisely what makes stem-and-leaf plots unique. The stem holds all digits except the last, and the leaf is the last digit — so stem 3 and leaf 7 means 37. Unlike a histogram, which groups values into intervals and loses the original numbers, a stem-and-leaf plot is reversible: every entry can be decoded back to the exact data point.

Question 4

A data set contains both 43 and 34. Because these values use the same two digits, they would appear in the same row of a stem-and-leaf plot.

• T. True
• F. False

Answer: False

The stem is determined by all digits except the last, which for two-digit numbers is the tens digit. The value 43 has stem 4 and leaf 3; the value 34 has stem 3 and leaf 4. They appear in different rows. This is a common confusion — it is the position of digits, not the set of digits, that determines placement. The key (e.g., '3 | 4 means 34') makes this explicit.

Question 5

A classmate constructs a stem-and-leaf plot but forgets to include a key. Explain why the key is essential, and give an example of the ambiguity that results without it.

Model answer: Without a key, a reader cannot determine what the stem and leaf values represent. For example, a stem of '1' and a leaf of '5' could mean 15, 150, 1.5, or 105 depending on the scale of the data. The key (e.g., '1 | 5 means 15') tells the reader exactly how to combine stem and leaf to recover the original value. Without it, the plot looks the same for data measured in ones, tens, or decimals.

The key is not optional decoration — it is the decoder ring for the entire display. This matters most when the data involves decimals (stem '3' leaf '2' means 3.2, not 32) or three-digit numbers (stem '12' leaf '4' means 124). Even for simple two-digit integers, without a key a reader is guessing the scale.