Stem-and-Leaf Plots: Displaying Quantitative Data

Stem-and-Leaf Plots (Stemplots)

Introduction to Stemplots

Stem-and-leaf plots are graphical tools used in statistics to display quantitative data. They show both the general shape of the data distribution and the specific values present in the dataset. Stemplots are especially useful for small to moderate-sized datasets, allowing for quick visual inspection and comparison.

• Definition: A stem-and-leaf plot (or stemplot) is a method of organizing numerical data by splitting each value into a "stem" (typically the leading digit(s)) and a "leaf" (typically the last digit).

• Purpose: To display the distribution and individual values of a dataset.

• Application: Useful for identifying patterns, outliers, and the shape of the data.

How to Construct a Stemplot

Follow these steps to create a stemplot from a set of quantitative data:

1. Order the Data: Arrange the data values in increasing order.

2. Draw a Vertical Line: Separate the stems from the leaves.

3. Identify Stems and Leaves:

   • The stem consists of all but the final digit of each data value.

   • The leaf is the last digit of each data value.

4. List Leaves: For each stem, list the corresponding leaves in increasing order to the right of the stem.

Example

Suppose the weekly study times (in minutes) for a group of students are: 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105.

• Stems: 2, 3, 4, 5, 6, 7, 8, 9, 10

• Leaves: For stem 2: 0, 5 (representing 20, 25); for stem 3: 0, 5 (representing 30, 35); and so on.

Practice Example

Given the stemplot for the row 700548589, the data points are: 70, 70, 54, 58, 58, 89.

Interpreting Stemplots

Stemplots allow for easy identification of the highest and lowest values, as well as the frequency of values within certain ranges.

• Comparing Distributions: By constructing stemplots for two or more groups, you can compare their distributions, centers, and spreads.

• Identifying Scores: For example, in test score stemplots, you can quickly see which scores are most common and which are rare.

Example: Comparing Two Classes

Consider the test scores of two classes:

Class A	Class B
6 | 7 8 8 9	6 | 3 4 4 5 9 9
7 | 2 3 5 5 7 9	7 | 3 4 4 5 5 8 9
8 | 2 2 4 5 6 8	8 | 2 2 4 5 6 8
9 | 0 3 5 5 6 8 9	9 | 1 4 4 5 6 8 9

• Highest and Lowest Scores: The highest score in both classes is 99; the lowest is 67 in Class A and 63 in Class B.

• Frequency of C Grades: If C is defined as scores between 70-79, count the leaves in the '7' stem row for each class.

• Distribution Comparison: Compare the spread and clustering of scores between the two classes.

Advantages and Limitations of Stemplots

• Advantages:

  • Displays actual data values.

  • Shows the shape of the distribution.

  • Easy to construct for small datasets.

• Limitations:

  • Not practical for large datasets.

  • Less effective for data with many digits or decimal places.

Summary Table: Stemplot Features

Feature	Description
Stem	All but the final digit of each data value
Leaf	Final digit of each data value
Shape	Shows distribution (e.g., symmetric, skewed)
Data Values	All individual values are visible

Key Terms

• Stem: The leading digit(s) of a number in a stemplot.

• Leaf: The last digit of a number in a stemplot.

• Distribution: The overall pattern of data values.

Additional info:

• Stemplots are closely related to histograms but retain the original data values.

• They are covered in introductory statistics courses under the topic of "Exploring Data With Graphs and Numerical Summaries."