Stem & Leaf Diagrams

Purpose of Stem & Leaf Diagrams

Stem & leaf diagrams are a way to organise and display numerical data clearly. They help to:

• Show the distribution of data values
• Retain the original data values, unlike grouped frequency tables
• Provide a quick visual summary to identify patterns such as clusters or gaps

They are especially useful for small to medium-sized datasets where you want to see the shape of the data and exact values at the same time.

Constructing Stem & Leaf Diagrams

To construct a stem & leaf diagram:

1. Identify the stem: These are the leading digits of the data values. For example, for the number 47, the stem is 4.
2. List the leaves: These are the trailing digits of the data values. For 47, the leaf is 7.
3. Group the leaves: Write the stems in a vertical column and list the leaves next to each stem.
4. Order the leaves: Arrange the leaves in ascending order for each stem to make the diagram easier to read.
5. Include a key or legend: This explains how to read the diagram, for example, “4 | 7 means 47”.

Stem & leaf diagrams can be extended to include decimals or larger numbers by adjusting the stem and leaf definitions accordingly.

For instance, if you have the data set: 12, 15, 17, 21, 23, 24, 31, 33, 35, the stem & leaf diagram would be:

Stem | Leaf
1 | 2 5 7
2 | 1 3 4
3 | 1 3 5
4 | 7 means 47 (example key)

Interpreting Stem & Leaf Diagrams

You can extract useful information from stem & leaf diagrams:

• Read individual data values: Combine the stem and leaf to get the original numbers.
• Find the median: Count the total number of data points, then find the middle value(s) in order.
• Identify the mode: Look for the data value(s) that appear most frequently by counting repeated leaves within stems.
• Determine the range: Subtract the smallest data value from the largest.
• Observe the spread and shape: Notice if data is clustered, skewed, or evenly spread.

For example, given the stem & leaf diagram:

3 | 2 4 5 7
4 | 1 3 3 6
5 | 0 2 8

The data values are 32, 34, 35, 37, 41, 43, 43, 46, 50, 52, 58.

The median is the 6th value (since there are 11 values), which is 43.

The mode is 43 (appears twice).

The range is $58 - 32 = 26$.

Advantages and Limitations

Stem & leaf diagrams have several advantages:

• They provide a quick and clear visual summary of data.
• They retain original data values, unlike grouped frequency tables.
• They show the shape and spread of the data, helping to identify clusters, gaps, and outliers.

However, there are limitations:

• They become less useful for very large datasets as the diagram can get cluttered.
• They are not suitable for categorical data or data with many decimal places unless rounded.

Example: Constructing and Interpreting a Stem & Leaf Diagram

Data: Test scores out of 50 for 15 students:

38, 42, 45, 37, 40, 44, 41, 39, 36, 43, 47, 35, 38, 40, 39

Step 1: Identify stems (tens digit) and leaves (units digit).

Step 2: Group leaves by stems:

3 | 5 6 7 8 8 9 9
4 | 0 0 1 2 3 4 5 7

Step 3: Order leaves (already in ascending order).

Step 4: Add a key: “3 | 5 means 35”.

Step 5: Interpret:

• Median: 39 (the 8th value when ordered)
• Mode: 38 and 39 (both appear twice)
• Range: $47 - 35 = 12$