Business Analytics I - Chapter 2: Displaying Data

Intro

• Technology aids in displaying statistics.
• The purpose of displaying data is to summarize information that isn't easily observable from a simple data table.

Intro

• Data summarization is used for:
  • Categorical variables (nominal or ordinal).
  • Quantitative variables (interval or ratio).
  • Two variables using tables.
  • Two variables using graphs.
• The chapter covers techniques for these cases.

Display: Categorical Variables

• Example: A survey of 25 JMU students regarding their state of residence.
  • Question: Categorical or Numerical?
  • Scale of measurement?

Display: Categorical Variables

• Summarizing data for categorical variables involves:
  • Frequency distributions (relative, percent).
  • Bar charts.
  • Pie charts.

Display: Categorical Variables - Frequency Distributions

• Frequency, relative frequency, and percent frequency:
  • A tabular summary of data showing frequencies of observations in each class.
  • Example:
    • Maryland: Frequency = 2, Relative Frequency = $2/25 = 0.08$, Percent Frequency = 8%
    • North Carolina: Frequency = 6, Relative Frequency = 0.24, Percent Frequency = 24%
    • South Carolina: Frequency = 4, Relative Frequency = 0.16, Percent Frequency = 16%
    • Virginia: Frequency = 12, Relative Frequency = 0.48, Percent Frequency = 48%
    • West Virginia: Frequency = 1, Relative Frequency = 0.04, Percent Frequency = 4%

Display: Categorical Variables - Bar Charts

• Bar charts display frequencies of categorical data, especially when numerical values are of interest.
• Example: A bar chart showing the count of students from each state of residence (Maryland, North Carolina, South Carolina, Virginia, West Virginia).
• Question: Is the data nominal or ordinal?

Display: Categorical Variables - Pie Charts

• Pie charts: Pie segments represent the percentage of each category frequency.
• Used when numerical values are NOT of interest.
• Example: A pie chart showing the percentage of students from each state of residence.

Display: Categorical Variables - Pie Chart vs. Bar Chart

• Pie charts are used for comparing relative sizes of all possible categories.
• Bar charts are used to highlight actual data values.

Display: Categorical Variables - Pareto Charts

• Pareto charts: A type of bar chart that shows frequencies of categories in decreasing order.
• Commonly used in manufacturing scenarios for investigating quality control issues.

Display: Quantitative Variables

• Summarizing data for quantitative variables involves:
  • Frequency distributions (relative, percent).
  • Histograms.
  • Ogive (cumulative).
  • Stem-and-leaf display.

Display: Quantitative Variables - Motivation

• Consider the question: How often did this store sell two iPads in one day? Three iPads in one day?
• It takes too long to deduce this information without summarized representations.

Display: Quantitative Variables - Frequency Distribution

• Frequency distribution: Shows the number of observations that fall into specific intervals.
• Allows for faster information retrieval.

Display: Quantitative Variables - Relative Frequency Distribution

• Relative frequency distribution: Shows the proportion of observations that fall into specific intervals.
• Example:
  • Number Sold Per Day: 0, Frequency: 5, Relative Frequency: $5/50 = 0.10$
  • Number Sold Per Day: 1, Frequency: 8, Relative Frequency: $8/50 = 0.16$
  • Number Sold Per Day: 2, Frequency: 14, Relative Frequency: $14/50 = 0.28$
  • Number Sold Per Day: 3, Frequency: 13, Relative Frequency: $13/50 = 0.26$
  • Number Sold Per Day: 4, Frequency: 6, Relative Frequency: $6/50 = 0.12$
  • Number Sold Per Day: 5, Frequency: 4, Relative Frequency: $4/50 = 0.08$
  • Total Frequency: 50, Total Relative Frequency: 1.00

Display: Quantitative Variables - Histogram

• Histogram: A graph showing the number of observations in each class of a frequency distribution.
• Question: Is the number of days and iPad sales per day discrete or continuous?

Display: Quantitative Variables - Ogive

• Ogive: A graph plotting the cumulative relative frequency distribution on the y-axis and bin endpoints on the x-axis.
• Question: Is the data discrete or continuous?

Display: Quantitative Variables - Stem-and-Leaf Display

• Stem-and-leaf display:
  • Splits data into stems (large place value) and leaves (small place value).
  • Lists leaves to the right of the stem values.
  • Provides a rotated histogram-like view of the distribution.

Display: Quantitative Variables - Stem-and-Leaf Display Example

• Example: Stem and Leaf display
  • 7 | 8 8 9 9 9
  • 8 | 0 0 0 0 1 1 2 3 3 4 4 4 5 6 7 8
  • 9 | 0 2 5
• Leaf unit = 1
• More detailed stems can be split in half (e.g., 70-74 and 75-79 instead of just 70s and 80s).

Display: Extra Credit Example

• Data: 11.3, 9.6, 10.4, 7.5, 8.3, 10.5, 10.0, 9.3, 8.1, 7.7, 7.5, 8.4, 6.3, 8.8
• Task: Construct a stem-and-leaf display for the data.
• Leaf unit = 0.1

Display: Two Variables - Tables

• Summarizing data for two variables using tables:
  • Contingency tables (displays data for one variable in rows and data for another variable in columns).

Display: Two Variables - Contingency Tables

• Contingency table: A cross-tabulation method to summarize two categorical and/or quantitative variables.
• Example: ABC Store Sales Details for August (count of customers)
  • 18 out of 200 customers were returning customers and paid using cash: $18/200 = 0.09$
  • Contingency Table:
    • Rows: Returning customers, First-time customers, Total
    • Columns: Cash, Credit, Payment app, Total
    • Example data points: Returning customers using Cash: 18 (count), 0.09 (relative frequency).

Display: Two Variables - Graphs

• Summarizing data for two variables using graphs:
  • Scatter plot.
  • Side-by-side bar chart.
  • Stacked bar chart.

Display: Two Variables - Graphs - Scatter Plot

• Scatter plot: Provides a picture of the relationship between two data points that are paired.
  • The dependent variable (y-axis) is influenced by changes in the independent variable (x-axis).
  • Example: Hours studied (x-axis) vs. Exam score (y-axis) for twenty students.

Display: Two Variables - Graphs - Scatter Plot Example

• Example: Scatter Plot of Exam Performance vs. Hours Studied
  • Illustrates the relationship between hours studied and exam scores.

Display: Two Variables - Graphs - Side-by-Side and Stacked Bar Charts

• Example: Restaurant data depicting average price per plate and quality rating.
  • Average price per plate: $10, $15, $20, $25
  • Quality rating: Poor, Average, Good, Excellent

Display: Two Variables - Graphs - Side-by-Side Bar Chart

• Side-by-side bar chart: A graphical display for depicting multiple bar charts on the same display.
  • Used to compare average price per plate vs. quality rating.

Display: Two Variables - Graphs - Stacked Bar Chart

• Stacked bar chart: A graphical display for depicting multiple variables on the same display.
  • Used to compare average price per plate vs. quality rating.

Display: Summary

• Distribution of data:
  • Bar chart: Frequencies for categorical data.
  • Pie chart: Relative frequencies for categorical data.
  • Histogram: Frequency distribution for quantitative data.
  • Ogive: Percent frequency distribution for quantitative data.
  • Stem-and-leaf display: Rank order and shape of distribution for quantitative data.
• Comparisons of data:
  • Side-by-side bar chart: Compare relative frequency of two variables.
  • Stacked bar chart: Compare relative frequency of two variables.
• Relationships of data:
  • Scatter plot: Shows the relationship between two quantitative variables.

Copyright

• Note: Select figures, tables, and examples were obtained from the course textbook publisher.
• Donnelly, Robert A. (2015) Business Statistics. Upper Saddle River, New Jersey: Pearson Education, Inc. (Custom edition for James Madison University) ISBN:978-1-269-88344-3.