Stem-and-Leaf Plots: Fifth Grade Mathematics

Stem-and-leaf plots organize numerical data by splitting each data value into a leading "stem" and a trailing "leaf" to show data distribution while preserving exact values.

The Structure of a Stem-and-Leaf Plot

In a stem-and-leaf plot:
- The Stem represents the leading digit or digits (typically the tens place).

- The Leaf represents the trailing final digit (the ones place).

- A vertical line separates the stems on the left from the leaves on the right.

- The Key explains what the digits represent (e.g., "5 | 2 means 52 points").

+--------------------------------------------------------------------------+
|                  TEST SCORES STEM-AND-LEAF PLOT                          |
|                                                                          |
|       Data Set: 65, 72, 78, 80, 84, 84, 88, 91, 95, 95, 95, 98          |
|                                                                          |
|       Stem | Leaves                                                      |
|       -----+-----------------                                            |
|         6  | 5                                                           |
|         7  | 2  8                                                        |
|         8  | 0  4  4  8                                                  |
|         9  | 1  5  5  5  8                                               |
|                                                                          |
|       Key: 8 | 4 = 84 points                                             |
+--------------------------------------------------------------------------+

Critical Construction Rules

1. Order the Leaves: Leaves on each stem line must be written in ascending order from least to greatest.
2. Repeated Values: If a number occurs multiple times, its leaf must be written every time it appears. In the plot above, the score 95 appears three times, so the digit 5 is written three times (5 5 5).
3. Continuous Stems: If there is a stem with no data points in the middle of a range, the stem digit must still be written, with its leaf space left completely blank.

Reading Distribution from the Plot

A stem-and-leaf plot behaves like a horizontal bar graph turned on its side:
- The longest row of leaves shows where data clusters most heavily (e.g., the 90s stem has 5 scores).

- The minimum score is found at the top left leaf: 65.

- The maximum score is found at the bottom right leaf: 98.

Exercises and Solutions

Exercise 1: Constructing a Stem-and-Leaf Plot. Create a stem-and-leaf plot for the following list of student quiz scores: 42, 55, 38, 45, 51, 42, 39, 48, 55, 59. Include a key.

Solution 1: First, order the raw scores from least to greatest: 38, 39, 42, 42, 45, 48, 51, 55, 55, 59. The tens digits range from 3 to 5, forming stems 3, 4, and 5. For stem 3, write leaves: 8, 9. For stem 4, write leaves: 2, 2, 5, 8. For stem 5, write leaves: 1, 5, 5, 9. The completed plot is: Stem 3 | Leaves: 8 9; Stem 4 | Leaves: 2 2 5 8; Stem 5 | Leaves: 1 5 5 9. Key: 4 | 2 = 42 points.

Exercise 2: Analyzing Data from a Plot. Look at the "Test Scores" plot above (Stem 6 to 9). Part A: How many total students took the test? Part B: How many students scored 84 or higher? Part C: What was the score range (highest minus lowest)?

Solution 2: Part A: Count the total number of leaves across all stems: 1 + 2 + 4 + 5 = 12 students took the test. Part B: Locate scores 84 or higher: on stem 8, the leaves are 4, 4, 8 (3 scores); on stem 9, all 5 leaves are higher than 84 (5 scores). Total students scoring 84 or higher = 3 + 5 = 8 students. Part C: Highest score is 98 (stem 9, leaf 8); lowest score is 65 (stem 6, leaf 5). Range = 98 - 65 = 33 points.