Stem-and-Leaf Plots - Sixth Grade Mathematics
A stem-and-leaf plot is a compact statistical display that organizes numerical data in order of place value, retaining every original data value while presenting the distribution's shape.
The Structure of a Stem-and-Leaf Display
In a stem-and-leaf plot, each raw data value is split into two components:
1. The Stem: The leading digit or digits representing higher place values (tens, hundreds, or whole units).
2. The Leaf: The final trailing digit representing the lowest place value (usually ones or tenths).
The Anatomy of a Stem-and-Leaf Plot:
Stems | Leaves
-------+----------------------
2 | 1 4 4 8
3 | 0 2 5 7 9
4 | 3 3 6
5 | 2 8
Key: 3 | 2 represents 32
The Key is an indispensable component of every stem-and-leaf plot. Without a key, an entry of 3 | 2 could represent 32, 3.2, 320, or 0.32.
Step-by-Step Construction of an Ordered Plot
To construct an accurate stem-and-leaf plot from raw data: Step 1: Identify the minimum and maximum values to determine the range of stems. Step 2: List all stems vertically in numerical order from smallest to largest, without skipping any intermediate numbers even if a stem has zero leaves! Draw a vertical dividing line to their right. Step 3: Write down the leaf for each data value next to its corresponding stem. Step 4: Reorder the leaves on each row in ascending numerical order from left to right. Step 5: Provide an explicit Key explaining the place value interpretation.
Worked Example:
Raw Data (Math Quiz Scores out of 50):
35, 42, 28, 45, 31, 24, 42, 50, 39, 42, 38, 29, 48
Stems range from 2 (for 20s) to 5 (for 50s).
Ordered Stem-and-Leaf Plot:
Stem | Leaves
------+-------------------
2 | 4 8 9
3 | 1 5 8 9
4 | 2 2 2 5 8
5 | 0
Key: 4 | 2 = 42 points
Notice that if a score appears multiple times (like 42 appearing three times), its leaf (2) must be written three times! Every single observation in the data set must have its own leaf.
Decimals in Stem-and-Leaf Plots
Stem-and-leaf plots are equally powerful for displaying decimal numbers. Whole number units serve as the stem, and decimal tenths serve as the leaves.
Runner Sprint Times in Seconds:
12.4, 11.8, 12.1, 13.0, 11.5, 12.4, 11.9, 12.7
Ordered Plot:
Stem | Leaves
------+---------------
11 | 5 8 9
12 | 1 4 4 7
13 | 0
Key: 12 | 4 = 12.4 seconds
Back-to-Back Stem-and-Leaf Plots
To compare two distinct data sets (such as Class A vs. Class B test scores), researchers use a back-to-back stem-and-leaf plot sharing a common central stem.
Back-to-Back Comparison (Class A vs. Class B):
Class A Leaves | Stem | Class B Leaves
----------------+------+----------------
8 4 1 | 2 | 5 9
9 7 5 2 | 3 | 0 2 6 8
6 3 1 | 4 | 2 2 5 7 9
0 | 5 | 0 0
Key: Class A: 1 | 2 = 21 points; Class B: 2 | 5 = 25 points
In the left display (Class A), leaves are read backwards from the center stem moving outward to the left (21, 24, 28).
Exercises
Exercise 1. Given the stem-and-leaf plot: Stem | Leaves ------+----------------- 4 | 2 5 7 5 | 0 3 3 8 9 6 | 1 4 6 7 | 2 8 Key: 5 | 3 = 53 miles per hour. (a) How many total vehicles were observed? (b) What was the slowest speed recorded? (c) What was the fastest speed recorded? (d) What speed was recorded most frequently? (e) How many vehicles traveled at 60 mph or faster?
Exercise 2. Construct an ordered stem-and-leaf plot including a key for the following set of student heights in inches: 58, 62, 55, 60, 64, 59, 61, 68, 62, 65, 57, 63.
Exercise 3. Explain why the key is essential on a stem-and-leaf plot by giving two entirely different values that the plot row 7 | 4 could represent depending on the key.
Solutions
Solution to Exercise 1. For (a), count the total number of leaves: 3 + 5 + 3 + 2 = 13 vehicles. For (b), the smallest stem with smallest leaf is 4 | 2, which equals 42 mph. For (c), the largest stem with largest leaf is 7 | 8, which equals 78 mph. For (d), the leaf 3 appears twice under stem 5, so 53 mph occurred most frequently. For (e), count all leaves in rows 6 and 7: 3 leaves in row 6 + 2 leaves in row 7 = 5 vehicles traveled at 60 mph or faster.
Solution to Exercise 2. Stems represent tens (5 and 6). Leaves represent ones. Ordered Stem-and-Leaf Plot: Stem | Leaves ------+----------------------- 5 | 5 7 8 9 6 | 0 1 2 2 3 4 5 8 Key: 6 | 1 = 61 inches.
Solution to Exercise 3. Without a key, the reader cannot determine the decimal place value intended by the author. For example, if the key specifies 7 | 4 = 74, the value is a whole two-digit integer like 74 degrees. But if the key specifies 7 | 4 = 7.4, the exact same symbols represent a decimal measurement such as 7.4 centimeters.