Stem-and-Leaf Displays University Study Guide

Definition: A stem-and-leaf display (also known as a "Stemplot") is a method for summarizing the distribution of a quantitative variable. Unlike histograms, which only show the distribution shape, stem-and-leaf displays allow the viewer to see the individual data values.
Historical Context: This data visualization technique was devised by John W. Tukey, who was one of the most prominent statisticians of the $20^{th}$ century. While some modern computer programs and textbooks refer to these as "stemplots," Tukey's original terminology, "Stem-and-Leaf display," remains preferred by many scholars.
Comparison to Histograms: * Histograms: Provide an easy-to-understand summary of a distribution but fail to show the actual data values themselves. * Stem-and-Leaf Displays: Function similarly to a histogram in showing the overall shape and span of the data but retain the specificity of individual values. They are also considered the easier of the two to create manually by hand.

Manual Construction: Stem-and-leaf displays are designed to be easily made by hand, especially for batches containing fewer than a few hundred data values.
Spatial Consistency: In a proper display, each digit must be given an equal amount of space to ensure the visual length of the "row" accurately represents the number of data points.
No Punctuation: There should be no commas placed between the digits in the leaf section of the display.
Leaf Digit Limitations: A leaf must always consist of only single digits ( $0$ through $9$ ).
Numerical Ordering: The leaves must be arranged in increasing numerical value as they move away from the stem.
Orientation and Sorting: While higher values can be placed at the bottom, it typically makes more sense for higher values to be positioned at the top of the display.

The Stem: Represents the "major units" of the data. For pulse rates or similar two-digit numbers, the stem is typically the tens place ( $10s$ ).
The Leaf: Represents the "next major unit" or the ones place ( $1s$ ). If a pulse rate is $68$ , the leaf is the digit $8$ .
The Context Key: It is mandatory to include a key to explain the data's scale. For example, a key might state that ( $3|8$ ) means $38$ . In the specific example provided in the text, $6|8$ means a pulse of $68$ beats/min.
Bins: Stems act as bins. In granular displays, stems may be split to represent smaller ranges. For example: * A stem of $8$ containing values from $85-89$ . * A second stem of $8$ containing values from $80-84$ .

Data Source: Pulse rates taken from $24$ women by a researcher at a health clinic.
Data Interpretation from Histogram: Initial visual analysis of the histogram ( $Pulse\,Rate\,(beats\,per\,minute)$ ) shows the span and a typical pulse rate, but individual rates are obscured.
Individual Values from Stem-and-Leaf Display: By analyzing the pulse rates for the $24$ women in the provided display: * Top Row ( $85-89$ range): $8|8$ signifies one pulse of $88\,bpm$ . * Second Row ( $80-84$ range): $8|000044$ shows four instances of $80\,bpm$ and two instances of $84\,bpm$ . * Third Row ( $75-79$ range): $7|6666$ shows four instances of $76\,bpm$ . * Fourth Row ( $70-74$ range): $7|2222$ shows four instances of $72\,bpm$ . * Fifth Row ( $65-69$ range): $6|8888$ shows four instances of $68\,bpm$ . * Sixth Row ( $60-64$ range): $6|0444$ shows one instance of $60\,bpm$ and three instances of $64\,bpm$ .
Structural Note: Because the display is laid out vertically with the stem, if one turns the display on its side (or turns their head to the right), the shape mirrors that of a histogram. However, the higher values appear on the left in this orientation, effectively making it a "backwards" version of the histogram's shape.

Batch Size Reference: These displays are particularly useful for data batches with fewer than a few hundred data points.
Efficiency: They are described as a quick way to both display and record numbers simultaneously.
Granular Insight: Because the leaves preserve the individual values, researchers can often identify additional patterns or characteristics in the data that are not visible through the simple shape of a distribution.