Introduction to Statistics: Populations, Samples, and Variables

Introduction to Statistics

What is Statistics?

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It provides a collection of tools to answer questions about data, especially for making inferences about a larger group based on a smaller, representative sample.

• Definition: Statistics is the art or science of collecting and analyzing data, especially for the purpose of inferring properties about a population from a representative sample.

• Purpose: To make conclusions about a large group (population) by studying a smaller group (sample).

• Data: Information gathered from experiments and surveys.

Populations and Samples

Definitions

• Population: The set of all subjects or items of interest in a study. Keywords include "all" or "every" (e.g., "all SC residents").

• Sample: A subset of the population for which data is actually collected, often selected randomly. Keyphrases include "a sample of..." or "a survey of..." (e.g., "a sample of 1,000 SC residents").

• Parameter: A numerical summary describing a characteristic of a population (e.g., the mean age of all residents).

• Statistic: A numerical summary describing a characteristic of a sample (e.g., the mean age of sampled residents).

Example

• If a company has a database with records for every employee, you are working with a population. The median income calculated is a parameter.

• If the database contains records for only a subset of employees, you are working with a sample. The median income calculated is a statistic, and you use inference to predict the parameter for all employees.

Identifying Population, Sample, Statistic, and Parameter

Given a scenario, you should be able to identify these four elements:

• Population: All state residents

• Sample: A random sample of 1,000 state residents

• Statistic: 20% of those sampled are in favor of the change

• Parameter: 15% of all residents are in favor of the change

Descriptive vs. Inferential Statistics

Descriptive Statistics

Descriptive statistics provide useful summaries and help you understand the data you collected. They include methods for summarizing data from a sample or population.

• Examples: Mean, median, standard deviation, frequency tables

Inferential Statistics

Inferential statistics help you make predictions or decisions about a population based on data from a sample. This includes hypothesis testing and estimation.

• Examples: Confidence intervals, hypothesis tests

Sampling

Why Sample?

It is often impractical or impossible to collect data from every item in a population due to constraints such as time, cost, and feasibility. Sampling allows us to study a manageable subset and make inferences about the whole population.

• Studying every item takes more time

• Costs more money

• Makes it harder to manage data collection

Variables in Statistics

Definition of Variables

A variable is a characteristic or property that can take on different values for different individuals or items in a study.

• Examples: Number of absences, commute time, age

Types of Variables

• Quantitative Variable: Represents a measured quantity and is expressed as numbers. Can be further classified as:

  • Discrete Quantitative Variable: Represents counts (e.g., number of absences).

  • Continuous Quantitative Variable: Represents measurements that can take any value within a range (e.g., commute time in hours).

• Categorical Variable (Qualitative): Consists of values that represent categories or groups (e.g., gender, type of car).

Identifying Variable Types

• Ask: Are the results numbers?

• If yes, are they counts (discrete) or measurements (continuous)?

• If not, the variable is categorical.

Examples of Variable Types

Variable	Type
Number of Absences	Discrete Quantitative
Commute Time (in hours)	Continuous Quantitative
Age (in years)	Discrete Quantitative (if treated as integers)
Gender	Categorical

Summary Table: Key Terms

Term	Definition	Example
Population	All subjects of interest	All SC residents
Sample	Subset of the population	1,000 SC residents surveyed
Parameter	Numerical summary of a population	Mean age of all SC residents
Statistic	Numerical summary of a sample	Mean age of sampled residents

Key Formulas

• Sample Mean: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Population Mean: $\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$