Review of Basic Definitions and Terminology in Statistics

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Introduction to Statistics

Definition and Scope

Statistics is a branch of mathematical science focused on methods to collect, summarize, organize, analyze, present, describe, explain, interpret data, and draw conclusions. It involves making inferences, forecasting, and decision-making based on data. Statistics is divided into two main branches: descriptive statistics and inferential statistics.

Descriptive Statistics: Summarizes and describes features of a dataset, presenting data for easy comprehension.
Inferential Statistics: Uses sample data to make generalizations or predictions about a population.

Example: Forecasting election results using a sample of voters to predict the outcome for the entire population.

Basic Statistical Terms

Elements, Observations, Experimental Units, Observational Units, Subjects

These terms refer to the individual items or entities about which data are collected in a statistical study. When discussing the objects of study, we use terms such as elements, experimental units, observational units, or subjects.

Population and Sample

Population: The complete set of all data to be studied or analyzed. The frame is the complete set of all objects (elements, subjects, etc.) from which the sample is drawn.
Sample: A subset of objects from the target population being studied. Samples are used to make inferences about the population.

Example: Surveying 1,000 students from a university to estimate the average GPA of all students.

Statistical Study (Statistical Analysis)

The application of statistical methods to study, analyze, and learn about particular characteristics, properties, or attributes of a specific target population.

Census

The process of collecting data from every member of the population (or frame). Often impractical for large populations, so samples are used instead.

Steps of a Statistical Study

State the specific purpose of the study and identify the target population.
Decide whether a census is practical or suitable.
If not, collect data from a sample and summarize, organize, present, and interpret the data.
Draw conclusions and make inferences about the population.

Representative Sample

A sample that has relevant characteristics of the target population, selected using a representative sampling method.

Voluntary Response Sample (Self-Selected Sample)

A sample in which subjects themselves decide whether to be included. Common examples include internet polls and mail-in polls. These samples are often biased.

Statistical Significance vs. Practical Significance

Statistical Significance: Results are unlikely to occur by chance alone, as determined by formal statistical analysis.
Practical Significance: Results are meaningful or useful in real-world terms, regardless of statistical significance.

Example: A weight-loss program shows a statistically significant average loss of 3 pounds, but this may not be practically significant for participants.

Parameters vs. Statistics

Parameter: A measure that describes characteristics of a population.
Statistic: A measure that describes characteristics of a sample.

Example: The mean GPA of all students at a university is a parameter; the mean GPA of a sample of students is a statistic.

Variables and Constants

Definitions

Variable: A symbol that may assume multiple values, either numerical or non-numerical.
Constant: A symbol that always assumes exactly one value.

Example: The boiling point of water (100°C) is a constant; the height of students in a class is a variable.

Classification of Variables (Data)

Qualitative (Categorical) Variables: Non-numerical values representing characteristics or categories (e.g., gender, color, type).
Quantitative Variables: Numerical values representing counts or measurements (e.g., age, GPA, number of children).

Types of Measurement

Nominal: Categories without a natural order (e.g., gender, political affiliation).
Ordinal: Categories with a natural order (e.g., class rank, letter grades).
Interval: Numerical values with meaningful differences but no true zero (e.g., temperature in Celsius).
Ratio: Numerical values with meaningful differences and a true zero (e.g., height, weight).

Types of Statistical Studies

Observational Study: The researcher observes and records data without influencing the subjects.
Experimental Study: The researcher applies treatments and observes effects, often using control and treatment groups.

Example: Testing the effectiveness of a new drug by randomly assigning subjects to treatment and control groups.

Confounding

Occurs when it is difficult to distinguish the effects of different variables or factors in an experiment.

Sampling Methods

Probability Sampling

Simple Random Sampling: Every member of the population has an equal chance of being selected.
Systematic Random Sampling: Select every k-th element from a list after a random start.
Stratified Random Sampling: Divide the population into strata and randomly sample from each stratum.
Cluster Sampling: Divide the population into clusters, randomly select clusters, and sample all or some elements within selected clusters.

Non-Probability Sampling

Convenience Sampling: Select elements that are easiest to access.
Judgment Sampling: Select elements based on the researcher's judgment.

Statistical Thinking

Statistical thinking involves understanding the context, source, and methods of data collection, and critically evaluating the validity and reliability of statistical studies.

Key Formulas

Mean (Average):
Proportion:

Summary Table: Types of Variables

Type	Description	Examples
Nominal	Categories, no order	Gender, Political Affiliation
Ordinal	Ordered categories	Class Rank, Letter Grade
Interval	Numerical, no true zero	Temperature (°C)
Ratio	Numerical, true zero	Height, Weight

Summary Table: Sampling Methods

Method	Description
Simple Random	Equal chance for all members
Systematic	Select every k-th member
Stratified	Divide into strata, sample from each
Cluster	Divide into clusters, sample clusters
Convenience	Sample easiest to access
Judgment	Sample based on judgment

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