Organizing Quantitative Data: Stem-and-Leaf Diagrams

Section 2.3 – Organizing Quantitative Data

Stem-and-Leaf Diagrams

Stem-and-leaf diagrams are a graphical method for displaying quantitative data, particularly useful for small to moderate-sized datasets. They allow us to visualize the distribution of data while preserving the original values.

• Definition: A stem-and-leaf diagram separates each data value into two parts: the stem (all but the rightmost digit) and the leaf (the rightmost digit).

• Purpose: This method helps in organizing data to observe its shape, detect outliers, and identify clusters or gaps.

• Application: Commonly used in exploratory data analysis for small datasets, such as test scores, investment maturities, or growth measurements.

Steps to Construct a Stem-and-Leaf Diagram

Follow these steps to create a stem-and-leaf diagram from a set of quantitative data:

1. Step 1: Think of each observation as a stem (all but the rightmost digit) and a leaf (the rightmost digit).

2. Step 2: Write the stems from smallest to largest in a vertical column, and draw a vertical rule to the right of the stems.

3. Step 3: Write each leaf to the right of the vertical rule in the row that corresponds to its stem.

4. Step 4: Arrange the leaves in each row in ascending order.

Example: Maturity of Short-Term Investments

Consider the following dataset representing the time to maturity (in days) for 40 short-term investments:

Data Values
55, 64, 89, 87, 65, 70, 68, 95, 86, 99, 68, 95, 86, 70, 55, 81, 80, 98, 36, 55, 81, 79, 83, 80, 98, 51, 99, 68, 95, 86, 70, 55, 81, 80, 98, 36, 55, 81, 79, 83

To construct the stem-and-leaf diagram:

• Stems: The tens digit (e.g., 3 for 36, 5 for 55, 6 for 68, etc.)

• Leaves: The units digit (e.g., 6 for 36, 5 for 55, 8 for 68, etc.)

For example, the number 68 would be split into stem 6 and leaf 8.

Sample Stem-and-Leaf Diagram

Stem	Leaves
3	6, 6
5	1, 5, 5, 5, 5
6	4, 5, 8, 8, 8, 8
7	0, 0, 0, 9, 9
8	0, 0, 1, 1, 3, 6, 7
9	5, 5, 8, 8, 9, 9

Additional info: The above table is inferred from the provided data and may not include all values due to partial visibility in the source material.

Advantages of Stem-and-Leaf Diagrams

• Preserves Original Data: Unlike histograms, stem-and-leaf diagrams retain the actual data values.

• Quick Visualization: Allows for rapid assessment of data distribution, central tendency, and spread.

• Easy to Construct: Particularly suitable for hand calculations and small datasets.

Comparison: Stem-and-Leaf Diagram vs. Histogram

Feature	Stem-and-Leaf Diagram	Histogram
Data Preservation	Retains original values	Shows frequency, not individual values
Best for	Small datasets	Large datasets
Construction	Manual, quick	Requires binning

Key Formula

While stem-and-leaf diagrams do not require a specific formula, they are often used alongside measures of central tendency and spread, such as:

• Mean: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Median: The middle value when data are ordered

• Range: $\text{Range} = x_{\text{max}} - x_{\text{min}}$