Organizing Data: Types of Variables and Frequency Distributions

Chapter 2: Organizing Data

Introduction

This chapter introduces fundamental concepts in statistics related to organizing and classifying data. It covers the definition and types of variables, distinctions between qualitative and quantitative data, and methods for summarizing categorical data using frequency tables and charts.

Variables and Data

Definition of Variables

• Variable: A characteristic or property that can take on different values for different individuals or objects. Examples include height, weight, number of siblings, sex, marital status, and zipcode.

• Data: The observed values of variables collected from individuals or objects.

Example Table: Types of Variables

height	weight	siblings	sex	marriage	zipcode
5	110	0	female	married	97219
6.3	180	4	male	married	97405
5.7	145	2	male	single	97219

Types of Variables

Qualitative vs. Quantitative Variables

• Qualitative (Categorical) Variables: Variables that describe qualities or categories and are non-numerically valued. Examples: sex, marital status, zipcode.

• Quantitative Variables: Variables that are numerically valued and represent counts or measurements. Examples: height, weight, number of siblings.

Subtypes of Quantitative Variables

• Discrete Variables: Quantitative variables that take on a countable number of distinct values, often representing counts. Example: number of siblings.

• Continuous Variables: Quantitative variables that can take on any value within a given range, typically representing measurements. Example: height, weight.

Classification Diagram

• Variable

  • Qualitative

  • Quantitative

    • Discrete

    • Continuous

Examples: Identifying Variable Types

• Number of people in your household: Quantitative, Discrete

• Height of waterfalls: Quantitative, Continuous

• Finishing time of marathon runners: Quantitative, Continuous

• Order of finish in a running competition: Qualitative (ordinal)

• Global Industry Classification Standard (GICS) code: Qualitative (categorical)

Organizing Qualitative Data

Frequency Distribution Table

A frequency distribution table summarizes categorical data by listing each category and the number of observations in each.

Example: Stocks Classified by GICS Sector

Stocks	GICS_Sector
Apple	Information Technology
Microsoft	Information Technology
Johnson & Johnson	Healthcare
Pfizer	Healthcare
Exxon Mobil	Energy
Chevron	Energy
Walmart	Consumer Staples
Procter & Gamble	Consumer Staples
Coca-Cola	Consumer Staples
Amazon	Consumer Discretionary
Tesla	Consumer Discretionary
Home Depot	Consumer Discretionary
JP Morgan Chase	Financials
Goldman Sachs	Financials
Visa	Financials
Berkshire Hathaway	Financials
AT&T	Communication Services
Verizon	Communication Services
Duke Energy	Utilities
American Electric Power	Utilities

Grouping Data by Category

GICS_Sector	Stocks
Communication Services	AT&T, Verizon
Consumer Discretionary	Amazon, Tesla, Home Depot
Consumer Staples	Walmart, Procter & Gamble, Coca-Cola
Energy	Exxon Mobil, Chevron
Financials	JP Morgan Chase, Goldman Sachs, Visa, Berkshire Hathaway
Healthcare	Johnson & Johnson, Pfizer
Information Technology	Apple, Microsoft
Utilities	Duke Energy, American Electric Power

Frequency Table: Counting Observations per Category

GICS_Sector	Frequency
Communication Services	2
Consumer Discretionary	3
Consumer Staples	3
Energy	2
Financials	4
Healthcare	2
Information Technology	2
Utilities	2

Relative Frequency Table

Relative frequency expresses the proportion of observations in each category, calculated as:

$\text{Relative Frequency} = \frac{\text{Frequency of Category}}{\text{Total Number of Observations}}$

GICS_Sector	Relative_Frequency
Communication Services	0.10
Consumer Discretionary	0.15
Consumer Staples	0.15
Energy	0.10
Financials	0.20
Healthcare	0.10
Information Technology	0.10
Utilities	0.10

Pie Chart Representation

• A pie chart visually represents categorical data, dividing a circle into wedge-shaped pieces proportional to the relative frequencies of each category.

• Each sector's wedge size reflects its proportion in the dataset.

Summary Table: Types of Variables

Type	Description	Examples
Qualitative	Non-numerical, describes categories or qualities	Sex, Marital Status, Zipcode, GICS Sector
Quantitative - Discrete	Numerical, countable values	Number of siblings, Number of people in household
Quantitative - Continuous	Numerical, measurable values within a range	Height, Weight, Marathon finishing time

Key Points

• Variables are classified as qualitative or quantitative, with quantitative variables further divided into discrete and continuous types.

• Frequency and relative frequency tables are essential tools for summarizing categorical data.

• Pie charts provide a visual summary of the distribution of categorical data.

Example Application

In portfolio analysis, stocks can be grouped by sector, and the frequency and relative frequency of each sector can be calculated to understand the composition of the portfolio.

Additional info: Relative frequency values in the table are inferred based on a total of 20 stocks. The summary table of variable types is added for academic completeness.