Foundations of Statistics: Populations, Samples, Data Types, and Sampling Methods

Parameters vs. Statistics

Introduction to Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make informed decisions. Understanding the distinction between populations and samples, as well as parameters and statistics, is fundamental to statistical analysis.

• Data: Information gathered from counting, measuring, or collecting responses.

• Population: The entire set containing all data points ("every," "each").

• Sample: A subset of the population, representing only part of the whole.

• Parameter: A numerical value that describes a characteristic of a population.

• Statistic: A numerical value that describes a characteristic of a sample.

Example

• Population: All employees at a marketing firm.

• Sample: 12 out of 100 employees at the firm.

• Parameter: The average salary of all employees ($41,000).

• Statistic: The average salary of the sample ($58,000).

Term	Definition	Example
Population	Entire group of interest	All employees at a firm
Sample	Subset of the population	12 employees from the firm
Parameter	Numerical summary of a population	Average salary of all employees
Statistic	Numerical summary of a sample	Average salary of 12 employees

Types of Data

Qualitative vs. Quantitative Data

Data can be categorized as qualitative or quantitative, each with distinct characteristics and uses in statistical analysis.

• Qualitative Data: Describes qualities or categories (e.g., favorite color, eye color).

• Quantitative Data: Describes quantities or amounts and can be further divided into discrete and continuous types.

Type	Description	Examples
Qualitative	Qualities, categories	Favorite color, eye color
Quantitative: Discrete	Countable quantities	Dice roll, number of students
Quantitative: Continuous	Measurable quantities	Time, temperature

Example

• Surveying nationalities: Qualitative

• Measuring distances walked: Quantitative, Continuous

Intro to Collecting Data

Methods of Data Collection

There are two main ways to collect data: experiments and observational studies. The choice of method affects whether causation can be inferred.

• Experiment: Apply a treatment and measure its effects; can assume causation.

• Observational Study: Observe characteristics without intervention; cannot assume causation.

Example

• Testing medication: Experiment, causation possible.

• Surveying sleep habits: Observational Study, causation not assumed.

• Comparing dice rolls: Experiment if dice are manipulated, otherwise observational.

Simple Random Sampling

Sampling Methods

Sampling is the process of selecting a smaller group (sample) from a larger group (population). The goal is to obtain a representative sample that reflects the characteristics of the population.

• Representative Sample: Made up of equal proportions of characteristics as the original population.

• Simple Random Sampling (SRS): Each subject has an equal chance of being selected.

Example

• Selecting marbles at random: Simple Random Sample if each marble has an equal chance.

• Surveying students with proportional representation: Representative Sample.

Sampling Method	Description	Example
Simple Random Sampling	Equal chance for each subject	Randomly selecting 5 out of 20 students
Representative Sample	Reflects population characteristics	Surveying equal numbers from each group

Generating a Simple Random Sample

• Assign each member of the population a unique number.

• Use a random number generator to select the desired sample size.

• Ensure each member has an equal chance of selection.

Additional info: In practice, random sampling can be performed using computer software, random number tables, or drawing lots.