Data and Variables

Module 1 - Section 2: Data and Variables

Introduction

• Presenter: Rosana Fok

Key Concepts

Population vs. Sample

• Population: The entire collection of individuals or items of interest in a statistical study.

• Sample: A subset of the population selected for study in a specified manner.

• Parameter: A numerical summary that describes a characteristic of the population; often unknown.

• Statistic: A numerical summary that describes a characteristic of the sample; known once data are observed.

• Notation:

  • Greek letters denote parameters, e.g., mean $bc$, standard deviation $𝜎$.

  • Latin letters denote statistics, e.g., Mean $𝑦!$, Standard deviation $s$.

Commonly Used Parameters and Statistics

• Summary of key statistics:

  • Mean:

  • Statistic: $𝑦!$

  • Parameter: $𝜇$ (mu)

  • Standard Deviation:

  • Statistic: $s$

  • Parameter: $𝜎$ (sigma)

  • Correlation:

  • Statistic: $r$

  • Parameter: $𝜌$ (rho)

  • Regression Coefficient:

  • Statistic: $b$

  • Parameter: $𝛽$ (beta)

  • Proportion:

  • Statistic: $p$

  • Parameter: $p$

Problem of Data Collection

• It is generally impossible to gather all observations from a population.

• Population Data: Parameter -> Statistics -> Sampling -> Descriptive Statistics -> Inference

Practical Examples

Estimating Proportions

1. Example 1: Proportion of male students in a Statistics class.

   • Population of Interest: All students in the Statistics Class.

   • Sample: 20 randomly selected students.

   • Parameter: Proportion of male students in the class.

   • Statistic: Proportion of male students in the sample.

2. Example 2: Average study time of female students in Statistics.

   • Population of Interest: All female students in Statistics class.

   • Sample: 20 randomly selected female students.

   • Parameter: Average study time of female students.

   • Statistic: Average study time of the selected 20 female students.

Understanding Data

Definition of Data

• Data can be numbers, record names, or other labels. Context is provided by addressing:

  • 5 W's: Who, What, When, Where, Why

  • 1 H: How

Who: Subject Identification

• Definition: The individuals or subjects from whom data is collected.

  • Subjects: Individuals in an experiment.

  • Population: Entire set of subjects.

  • Sample: Observed subset of subjects.

  • Respondents: Individuals who respond to surveys.

  • Experimental Units: Includes animals, plants, and inanimate objects.

What: Measuring Characteristics

• Definition: The characteristics or variables being measured.

• Variables: Characteristics recorded about each individual, which can take different values.

• Measurement Units Examples:

  • Distance: Meter

  • Mass: Kilogram

  • Time: Second

  • Electric Current: Ampere

  • Temperature: Kelvin

  • Amount of Substance: Mole

Why: Purpose of Data Collection

• Importance: Understanding why data is collected shapes the analysis method and variable treatment.

• Key reminder: Need the Who, What, and Why to analyze data.

When and Where: Context of Data Collection

• Knowing when and where data is collected provides contextual information that influences interpretation.

• Example:

  • Comparing average price of a pen in the 1960s at the University of Alberta to present values.

  • Salary comparisons between Canada and third-world countries.

How: Data Collection Methodology

• Definition: The methods used for data collection can significantly impact results.

• Importance: Proper data gathering is crucial for valid analysis and conclusions.

• Potential issues with data collection:

  • Improper methods lead to incorrect conclusions.

  • Example: Voluntary internet surveys often yield unreliable results.

Case Study: Monitoring the Future (MTF) Project

• Purpose: Study changes in beliefs, attitudes, and behaviors of young Americans.

  • Who: 8th, 10th, and 12th graders.

  • What: Alcohol, illegal drug, and cigarette use.

  • Why: To study changes in beliefs, attitudes, and behaviors.

  • When: Spring 2004.

  • Where: United States.

  • How: Surveys administered to randomly selected students.

Types of Variables

Definition of Variables

• Variables are characteristics recorded about each individual.

Types of Variables

1. Categorical (Qualitative) Variables:

   • Define categories or groups within subjects.

   • Count cases in each category.

   • Types:

     • Nominal: Levels without a specific order (e.g., hair color).

     • Ordinal: Levels with a specific order (e.g., educational levels).

   • Example Variables:

     • Gender (Male, Female)

     • Hair color (blonde, brown, black, etc.)

     • Grade (A+, A, A-, etc.)

     • Car manufacturer (Ford, Honda, etc.)

2. Numerical (Quantitative) Variables:

   • Measure a numerical quantity in each subject.

   • Types:

     • Discrete: Distinct values only (e.g., count of siblings).

     • Continuous: Any value in an interval (e.g., height).

   • Clarification: Not all data represented by numbers are numerical (e.g., coding schemes like 1 = male, 2 = female).

Example of Variable Classification

• Given the following variables from a medical study:

  • Age (years) – Quantitative

  • Smoker (yes or no) – Categorical

  • Systolic blood pressure (mmHg) – Quantitative

  • Level of calcium in the blood (micrograms/mL) – Quantitative

  • Drug effectiveness (scale of 1-5) – Categorical

Data Table Example

• Each student's individual characteristics displayed:

  • Attributes Include: Age, Student ID, Height, Gender, Grade.

  • Context Provided:

  • Who (row titles) and What (column titles) indicated in the data table.

  • Note: Student ID is an identifier variable, categorical, with no quantitative units.

Conclusion

• Acknowledgment of the audience for their attention and participation in the discussion on data and variables.