Stem and Leaf displays

Stem and Leaf Display Overview

  • Definition: A stem and leaf display is a graphical method of displaying data that is particularly useful when the dataset is not overly large.

  • Purpose: It provides an intuitive way to visualize the distribution, shape, and individual values of a dataset.

Construction and Interpretation of a Stem and Leaf Display

  • Initial Example: The example used is based on the number of touchdown passes thrown by the 31 NFL teams in the 2000 season.

    • Display Components:

    • Stems: The left portion contains the stems which represent the tens digits. These are arranged in a column on the left. In this example, the stems are:

      • 3 (representing 30-39)

      • 2 (representing 20-29)

      • 1 (representing 10-19)

      • 0 (representing 0-9)

    • Leaves: The leaves on the right side of the stems represent the ones digits, contributing to identifying the exact values in the dataset.

  • Example Interpretation:

    • Top Row: Stem of 3 corresponds to leaves 2, 3, 3, 7. Thus,

    • Values represented are 32, 33, 33, 37 touchdowns for the first four teams.

    • Second Row: Stem of 2 with 12 leaves representing:

    • 2 occurrences of 20 touchdowns

    • 3 occurrences of 21 touchdowns

    • 3 occurrences of 22 touchdowns

    • 1 occurrence of 23 touchdowns

    • 2 occurrences of 28 touchdowns

    • 1 occurrence of 29 touchdowns.

    • Third Row: To be interpreted by students as a task.

    • Fourth Row: Stem of 0 with leaves 9 and 6 representing the last two entries (09 and 06 touchdowns).

  • Key Observations:

    • A stem and leaf display clarifies the shape of data distributions effectively.

    • It allows viewers to readily identify ranges, trends, and counts.

    • Example conclusions drawn include:

    • Most teams scored between 10 and 29 passing touchdowns, with fewer teams having scores higher or lower than this range.

Advanced Construction Techniques

  • Splitting Stems:

    • This technique is employed to make clearer graphs when single stems contain multiple values.

    • The enhanced display divides the figures into smaller segments.

    • Example: The range 35-39 is shown separately to highlight specific data points.

    • Effectiveness: Splitting stems can lead to more intelligible displays as it prevents excessive data from being lumped into one category.

  • Back-to-Back Stem and Leaf Displays:

    • This variation allows for comparison between two distributions by placing them along a common column of stems.

    • Example Used: Comparing touchdown passes from 1998 and 2000 seasons.

    • Each stem serves as the reference with leaves on either side showing historical data.

    • Specific observations can be drawn, detailing performance changes between the seasons.

Practical Considerations for Data Representation

  • Characteristics of Data for Suitable Stem and Leaf Displays:

    • Whole numbers are preferred, ideally allowing representation with one-digit stems and leaves.

    • All values should be positive to maintain the format.

    • If decimal points or large numbers are present, data should be suitably rounded (to two-digit accuracy preferred).

Additional Examples

  • Data on Aggressive Thinking:

    • Context: Study on the speed of naming aggressive words when primed by either a weapon or non-weapon word.

    • Result Interpretation:

    • Positive differences indicate faster pronunciation with weapon words.

    • Negative differences indicate slower pronunciation.

    • Example Values: Range from 43.2 milliseconds faster to -27.4 milliseconds slower, illustrating the apparent differences in speed.

  • Negative and Zero Handling:

    • Examples derived from the aggressive thinking study utilize negative stems for interpreting negative values.

    • Special zero handling allows value distributions to be depicted correctly between zero and negative numbers:

    • Zero stem for numbers 0-9.

    • Negative zero stem for numbers between 0 and -9.

Limitations and Applications of Stem and Leaf Displays

  • Data Size:

    • Optimal for datasets of up to 200 observations.

  • Population Dataset Example:

    • Observations of populations from 185 US cities in 1998, rounded to the nearest 10,000 residents, plotted appropriately.

  • Judgment in Graphing Choice:

    • Assessing whether the dataset can be aptly represented in stem and leaf format is crucial. Some datasets may lose important details if rounded excessively.

    • The effectiveness of a statistical representation relies on good judgment and understanding individual datasets' nature.