Organizing and Visualizing Variables: Structured Study Notes for Business Statistics

Organizing and Visualizing Variables

Introduction

This chapter focuses on methods for organizing and visualizing both categorical and numerical variables, which are foundational skills in business statistics. Effective data representation is crucial for accurate analysis, interpretation, and decision-making.

Why Data Representation Is Important

Significance of Data Representation

• Condenses Raw Data: Summarizes large datasets into manageable forms for analysis.

• Facilitates Interpretation: Enables quick visual understanding of data patterns and distributions.

• Highlights Key Characteristics: Reveals where data are concentrated, clustered, or outlying.

Organizing Categorical Data

Summary Tables

A summary table displays the frequencies or percentages of items in each category, allowing for easy comparison between categories.

• Definition: Table listing counts or percentages for each category.

• Example: Devices Millennials Use to Watch Movies or TV Shows.

Devices Used To Watch Movies or TV Shows	Percent
Television Set	49%
Tablet	9%
Smartphone	10%
Laptop / Desktop	32%

Contingency Tables

Contingency tables organize data for two or more categorical variables, showing joint frequencies and revealing patterns or relationships.

• Rows: Categories of one variable

• Columns: Categories of another variable

	No Errors	Errors	Total
Small Amount	170	20	190
Medium Amount	100	40	140
Large Amount	65	5	70
Total	335	65	400

Contingency Table Percentages

• Overall Total: Percentage of each cell relative to the grand total.

• Row Total: Percentage of each cell relative to its row total.

• Column Total: Percentage of each cell relative to its column total.

Example Calculation: $\text{Row Percentage} = \frac{\text{Cell Value}}{\text{Row Total}} \times 100$

Organizing Numerical Data

Ordered Array

An ordered array is a sequence of data arranged from smallest to largest, useful for identifying the range and outliers.

Day Students	Night Students
16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 25, 27, 32, 38, 42	18, 18, 19, 19, 20, 21, 23, 28, 32, 33, 41, 42

Frequency Distribution

A frequency distribution is a summary table where data are grouped into numerically ordered classes.

• Class Interval: $\text{Class Interval} = \frac{\text{Range}}{\text{Number of Classes}}$

• Class Boundaries: Define the limits for each class grouping.

• Class Midpoints: Average of lower and upper class boundaries.

Class	Midpoints	Frequency
10 but less than 20	15	3
20 but less than 30	25	6
30 but less than 40	35	5
40 but less than 50	45	4
50 but less than 60	55	2
Total		20

Relative and Percent Frequency Distribution

• Relative Frequency: $\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total}}$

• Percentage: $\text{Percentage} = \text{Relative Frequency} \times 100$

Class	Frequency	Relative Frequency	Percentage
10 but less than 20	3	0.15	15%
20 but less than 30	6	0.30	30%
30 but less than 40	5	0.25	25%
40 but less than 50	4	0.20	20%
50 but less than 60	2	0.10	10%
Total	20	1.00	100%

Cumulative Frequency Distribution

• Cumulative Frequency: Sum of frequencies up to and including the current class.

• Cumulative Percentage: $\text{Cumulative Percentage} = \frac{\text{Cumulative Frequency}}{\text{Total}} \times 100$

Class	Frequency	Percentage	Cumulative Frequency	Cumulative Percentage
10 but less than 20	3	15%	3	15%
20 but less than 30	6	30%	9	45%
30 but less than 40	5	25%	14	70%
40 but less than 50	4	20%	18	90%
50 but less than 60	2	10%	20	100%

Visualizing Categorical Data

Bar Chart

• Definition: Visualizes a categorical variable as a series of bars, with length representing frequency or percentage.

• Application: Useful for comparing categories side by side.

Pie Chart

• Definition: Circle divided into slices representing categories; size of slice proportional to percentage.

• Application: Effective for showing parts of a whole.

Doughnut Chart

• Definition: Similar to pie chart but with a blank center; slices represent categories.

• Application: Used for visualizing proportions in a more modern format.

Pareto Chart

• Definition: Vertical bar chart with categories in descending order of frequency; includes cumulative line.

• Purpose: Separates the "vital few" from the "trivial many" (Pareto principle: 80/20 rule).

Cause	Frequency	Percent	Cumulative Percent
Warped card jammed	365	50.41%	50.41%
Card unreadable	234	32.32%	82.73%
ATM malfunctions	32	4.42%	87.15%
ATM out of cash	28	3.87%	91.02%
Invalid amount requested	23	3.18%	94.20%
Wrong keystroke	23	3.18%	97.38%
Lack of funds in account	19	2.62%	100.00%

Visualizing Numerical Data

Histogram

• Definition: Vertical bar chart of frequency distribution; no gaps between bars.

• Axes: Horizontal axis shows class boundaries or midpoints; vertical axis shows frequency, relative frequency, or percentage.

Stem-and-Leaf Display

• Definition: Data are split into "stems" (leading digits) and "leaves" (trailing digits) to show distribution and concentration.

• Application: Useful for small datasets and identifying outliers.

Visualizing Two Numerical Variables

Scatter Plot

• Definition: Plots paired observations from two numerical variables; X-axis for one variable, Y-axis for the other.

• Purpose: Examines relationships or correlations between variables.

Time Series Plot

• Definition: Plots values of a numeric variable over time; time on X-axis, variable on Y-axis.

• Purpose: Identifies trends, cycles, or patterns over time.

Summarizing a Mix of Variables

Multidimensional Contingency Tables

• Definition: Tallies responses for three or more categorical variables.

• Application: Reveals complex patterns and relationships in multidimensional data.

• Best Practice: Limit to three or four variables for clarity.

Avoiding Common Errors in Data Visualization

Common Graphical Errors

• Selective Summarization: Presenting only part of the data can mislead.

• Improperly Constructed Charts: Issues include missing labels, axes not starting at zero, or broken axes.

• Chartjunk: Unnecessary decorations or effects that distract from the data.

Best Practices for Constructing Visualizations

• Use the simplest possible visualization.

• Include a title and label all axes.

• Begin vertical axes at zero and use a constant scale.

• Avoid 3D effects and chartjunk.

• Use consistent coloring for comparable charts.

• Avoid uncommon chart types unless necessary.

Chapter Summary

• Organizing and visualizing categorical variables using tables and charts.

• Organizing and visualizing numerical variables using arrays and distributions.

• Summarizing a mix of variables with multidimensional tables.

• Avoiding common errors in data visualization for accurate interpretation.