Stem-and-Leaf Plots

For Elementary Students

What is a Stem-and-Leaf Plot?

A stem-and-leaf plot is a special way to organize numbers so you can see patterns AND still see all the actual numbers!

Think about it like this: Imagine you're organizing books on shelves. The stem is the shelf number, and the leaves are the individual books on that shelf!

Numbers: 23, 28, 31, 35

Think of it like:
Shelf 2 (20s): books numbered 3 and 8 → 23, 28
Shelf 3 (30s): books numbered 1 and 5 → 31, 35

The Two Parts

Stem: The first part of the number (usually the tens place)

Leaf: The last digit of the number (the ones place)

Number: 47
Stem: 4 (the tens digit)
Leaf: 7 (the ones digit)

A Simple Example

Problem: Show these test scores: 82, 78, 85, 79, 88

Step 1: Find the stems (tens digits)

• Stems are: 7 and 8

Step 2: Draw the plot

Stem | Leaf
-----|-----
  7  | 8 9
  8  | 2 5 8

Step 3: Read it

• Key: 7|8 means 78
• Numbers in the 70s: 78, 79
• Numbers in the 80s: 82, 85, 88

All the numbers are still there! That's the cool part!

Making Your Own

Example: Show these ages: 12, 15, 18, 23, 27, 21

Step 1: Sort them (makes it easier) 12, 15, 18, 21, 23, 27

Step 2: Group by tens

• 10s: 12, 15, 18
• 20s: 21, 23, 27

Step 3: Make the plot

Stem | Leaf
-----|----------
  1  | 2 5 8
  2  | 1 3 7

Key: 1|2 means 12

Done! Now you can see there are 3 people in their teens and 3 in their twenties!

What Can You See?

Example:

Stem | Leaf
-----|----------
  6  | 2 5 8
  7  | 1 3 3 7 9
  8  | 0 2

What this tells you:

• Most numbers are in the 70s (5 numbers!)
• Smallest number: 62
• Largest number: 82
• The number 73 appears twice (two 3s in the 7 row)

Why Use Stem-and-Leaf Plots?

You can see the shape: Are most numbers high or low? Clustered together or spread out?

You keep all the data: Unlike a graph where you might lose exact numbers!

Easy to find the middle: Just count to the middle leaf!

Real-Life Example: Video Game Scores

Stem | Leaf
-----|----------
  3  | 2 5 9
  4  | 1 1 6 8
  5  | 0 3

Key: 3|2 means 32 points

Scores: 32, 35, 39, 41, 41, 46, 48, 50, 53

You can see:

• Most scores in the 40s
• Lowest score: 32
• Highest score: 53
• Two people got 41 points!

Memory Trick

"The stem holds up the leaves!"

Just like a flower: the stem (tens) holds up all the leaves (ones)!

Quick Tips

Tip 1: Always put leaves in order from smallest to largest

Tip 2: Don't forget the key! (It tells people what the numbers mean)

Tip 3: The stem is the tens place, the leaf is the ones place

Tip 4: Each leaf is just ONE digit

Tip 5: You can see all your original numbers by reading stem + leaf!

For Junior High Students

Definition and Structure

A stem-and-leaf plot (also called a stem plot) is a data display that organizes numerical data while preserving individual values, allowing visualization of distribution and shape.

Components:

Stem | Leaf
-----|----------
  s₁ | l₁ l₂ l₃ ...
  s₂ | l₁ l₂ ...

• Stem: Leading digit(s) representing the place value (typically tens)
• Leaf: Trailing digit representing the ones place
• Key: Notation explaining how to read stem|leaf combinations

Mathematical representation: For number n = 10s + l, stem = s and leaf = l

Construction Algorithm

Procedure:

1. Determine stems: Identify range and appropriate stem values
2. List stems vertically: In ascending order
3. Assign leaves: For each data value, place its ones digit next to appropriate stem
4. Order leaves: Arrange leaves in ascending order for each stem
5. Add key: Include interpretation guide

Example 1: Two-Digit Data

Data set: 23, 31, 28, 35, 22, 39, 24, 31, 27

Step 1: Identify range and stems

• Minimum: 22, Maximum: 39
• Stems: 2, 3

Step 2: Create framework

Stem | Leaf
-----|-----
  2  |
  3  |

Step 3: Add leaves (unsorted)

• 23 → stem 2, leaf 3
• 31 → stem 3, leaf 1
• 28 → stem 2, leaf 8
• etc.

Step 4: Sort leaves

Stem | Leaf
-----|----------
  2  | 2 3 4 7 8
  3  | 1 1 5 9

Key: 2|3 = 23

Interpretation:

• Values in twenties: 22, 23, 24, 27, 28 (n = 5)
• Values in thirties: 31, 31, 35, 39 (n = 4)
• Total observations: 9

Example 2: Finding Statistics

Data display:

Stem | Leaf
-----|----------
  4  | 2 5 7
  5  | 1 1 4 8
  6  | 0 3

Key: 4|2 = 42

Data reconstruction: 42, 45, 47, 51, 51, 54, 58, 60, 63

Statistical measures:

Count (n): 9 values

Range: Max − Min = 63 − 42 = 21

Mode: 51 (appears twice; leaf 1 appears twice in stem 5)

Median: Middle value = 5th position = 51

Mean:

Sum = 42 + 45 + 47 + 51 + 51 + 54 + 58 + 60 + 63 = 471
Mean = 471 ÷ 9 ≈ 52.33

Three-Digit Numbers

For larger numbers, use first two digits as stem.

Example: Data: 112, 125, 138, 141, 143, 155

Stem | Leaf
-----|-----
 11  | 2
 12  | 5
 13  | 8
 14  | 1 3
 15  | 5

Key: 11|2 = 112

Advantage: Maintains clarity while displaying larger values

Split Stems

When data clusters excessively in one stem, split stems improve distribution visibility.

Convention:

• First occurrence: leaves 0-4
• Second occurrence (marked with *): leaves 5-9

Example: Data: 41, 43, 44, 45, 48, 49

Without split:

Stem | Leaf
-----|----------
  4  | 1 3 4 5 8 9

With split:

Stem | Leaf
-----|-----
  4  | 1 3 4
  4* | 5 8 9

Result: Better visualization of distribution within the 40s range

Back-to-Back Stem-and-Leaf Plots

Purpose: Compare two data sets using shared stems

Structure:

Data Set A    Stem | Data Set B
         ...    s  | ...

Example: Test scores comparison

Boys: 65, 72, 68, 75, 71 Girls: 78, 82, 75, 79, 85

    Boys  Stem | Girls
     5 8     6 |
   5 2 1     7 | 5 8 9
             8 | 2 5

Key: 5|6 = 65 (boys), 7|5 = 75 (girls)

Analysis:

• Boys' distribution: 60s and 70s (range: 65-75)
• Girls' distribution: 70s and 80s (range: 75-85)
• Girls' scores generally higher (right-skewed for girls)

Distribution Analysis

Stem-and-leaf plots reveal distribution characteristics:

Symmetric distribution:

Stem | Leaf
-----|----------
  5  | 2 8
  6  | 1 4 6 9
  7  | 0 3 5 7
  8  | 2 9

Roughly balanced around center stems

Right-skewed (positively skewed):

Stem | Leaf
-----|----------
  2  | 1 3 5 6 8 9
  3  | 2 4 7
  4  | 1
  5  | 6

Tail extends toward higher values

Left-skewed (negatively skewed):

Stem | Leaf
-----|----------
  5  | 2
  6  | 4
  7  | 3 5 8
  8  | 1 2 4 6 7 9

Tail extends toward lower values

Advantages vs. Other Displays

Optimal use: Small to moderate data sets (n < 50) where preserving individual values is important

Applications

Education: Display test scores

• Quickly identify mode, median
• See distribution and outliers
• Preserve individual student scores

Science: Record measurements

• Maintain data precision
• Visualize experimental results
• Identify clustering patterns

Quality control: Monitor manufacturing data

• Track product specifications
• Detect abnormal values
• Analyze process consistency

Reading Complex Plots

Example: Population ages in a community sample

Stem | Leaf
-----|-------------------------
  0  | 5 7 9
  1  | 2 4 4 6 8 9
  2  | 1 3 5 7
  3  | 0 2 8
  4  | 5 5 5 9
  5  | 1 6
  6  | 2 8
  7  | 3

Key: 0|5 = 5 years old

Observations:

• Multimodal: clusters around teens (10-19) and mid-40s (45)
• Age range: 5 to 73 years
• Most common age: 45 (appears three times)
• Total individuals: 28

Demographic insight: Mixture suggests families with teenagers and middle-aged parents

Common Errors

Error 1: Incorrect leaf ordering

❌ Stem | Leaf
     3  | 5 1 9 3 (unordered)

✓ Stem | Leaf
    3  | 1 3 5 9 (ordered ascending)

Error 2: Multi-digit leaves

❌ Stem | Leaf
     2  | 13 8 (leaf should be single digit)

✓ Stem | Leaf
     2  | 3 8
     3  | 1

Error 3: Missing key

Always include interpretation guide: "Key: 3|4 = 34"

Error 4: Inconsistent stem intervals

Maintain uniform stem increments (e.g., all by 1s or all by 10s)

Tips for Success

Tip 1: Always sort leaves in ascending order within each stem

Tip 2: Include a clear, unambiguous key

Tip 3: Choose appropriate stem units based on data range

Tip 4: Use split stems for clustered data

Tip 5: Verify data count matches original data set

Tip 6: Orient plot vertically for better visualization

Tip 7: For back-to-back plots, mirror leaves outward from shared stem

Summary

Definition: Data display preserving individual values while showing distribution

Structure:

Stem | Leaf (sorted ascending)

Key properties:

• Retains original data values
• Shows distribution shape
• Facilitates quick statistical calculations
• Best for small-to-moderate data sets

Statistical uses:

• Finding median (middle leaf)
• Identifying mode (repeated leaves)
• Calculating range (max stem|leaf − min stem|leaf)
• Detecting outliers (isolated leaves)

Practice