Stem-and-Leaf Plots and Pie Charts in Descriptive Statistics

Stem-and-Leaf Plots

Definition and Purpose

A stem-and-leaf plot is a graphical method used in descriptive statistics to organize and display quantitative data. Each data value is split into a "stem" (typically the leading digit or digits) and a "leaf" (the last digit). This format preserves the original data values while showing the distribution and shape of the data set.

• Stem: The leftmost digit(s) of each data value.

• Leaf: The rightmost digit of each data value.

• Key: A notation that explains how to interpret the stems and leaves (e.g., "2 | 7 = 27").

Constructing a Stem-and-Leaf Plot

1. Identify the minimum and maximum data values to determine the range of stems.

2. List stems in ascending order, each representing a group of data values.

3. For each data value, write its leaf next to the appropriate stem.

4. Include a key to clarify the representation.

Example: Exam Scores

Given exam scores: 75, 85, 90, 87, 67, 82, 88, 96, 91, 75, 80, 69

Stem-and-leaf plot:

Key: 3|0 = 30 Exam Scores 6 | 7 9 7 | 5 5 8 7 8 | 0 2 5 7 9 9 | 0 1 6

• Interpretation: Most scores are in the 80s and 90s, indicating a high-performing class.

Example: Hours Worked by Nurses

Given hours: 40, 45, 32, 40, 39, 32, 40, 32, 29

Key: 3|3 = 33 Hours Worked 2 | 9 3 | 2 2 2 9 4 | 0 0 0 5

• Interpretation: Most nurses work between 30 and 40 hours per week, inclusive.

Example: Income Distribution

Given incomes (in millions): 34, 38, 53, 73, 54, 36, 49, 53, 41, 42, 45, 33

Key: 3|3 = 33 Income ($ millions) 3 | 3 4 6 8 4 | 1 2 5 9 5 | 3 3 4 | 7

• Interpretation: Most incomes are less than 60 million dollars.

Advantages and Applications

• Preserves actual data values for further analysis.

• Quickly reveals the shape, spread, and central tendency of the data.

• Useful for small to moderately sized data sets.

Pie Charts

Definition and Purpose

A pie chart is a circular statistical graphic divided into slices to illustrate numerical proportions. Each slice represents a category's relative frequency or percentage of the total.

• Sector: A slice of the pie chart representing a category.

• Central Angle: The angle of each sector, proportional to the category's frequency.

Constructing a Pie Chart

1. Calculate the relative frequency of each category: $\text{Relative Frequency} = \frac{\text{Category Frequency}}{\text{Total Frequency}}$

2. Determine the central angle for each sector: $\text{Central Angle} = \text{Relative Frequency} \times 360^\circ$

3. Draw the circle and divide it into sectors according to the calculated angles.

Example: Student Loan Balances

Balance Category	Number of Borrowers (millions)	Relative Frequency	Central Angle
$1 to $10,000	10.0	0.312	112.3°
$10,001 to $25,000	12.7	0.40	145.8°
$25,001 to $50,000	12.2	0.385	138.6°
$50,001+	4.8	0.15	54.3°

• Interpretation: Most student loan balances are $25,001 or less.

Example: Beverage Size Preferences

Size	Percentage
Medium	49%
Large	31%
Small	15%
XL	5%

• Interpretation: The medium size is the most popular, chosen by 49% of customers.

Advantages and Applications

• Effective for displaying categorical data proportions.

• Visually highlights the largest and smallest categories.

• Commonly used in business, media, and survey reporting.

Comparing Stem-and-Leaf Plots and Pie Charts

Feature	Stem-and-Leaf Plot	Pie Chart
Data Type	Quantitative	Categorical
Preserves Data Values	Yes	No
Shows Distribution Shape	Yes	No
Best For	Small/medium data sets	Proportions of categories

Summary

• Stem-and-leaf plots are useful for displaying the distribution and actual values of quantitative data.

• Pie charts are ideal for showing the relative proportions of categorical data.

• Both tools are foundational in descriptive statistics and data visualization.

Additional info: Academic context and examples have been expanded for clarity and completeness.