Section 1.1: An Overview of Statistics

Section 1.1: An Overview of Statistics

Objectives

This section introduces the foundational concepts of statistics, including the distinction between populations and samples, parameters and statistics, and the two main branches of statistics: descriptive and inferential.

• Define statistics and its purpose.

• Distinguish between a population and a sample, and between a parameter and a statistic.

• Differentiate between descriptive statistics and inferential statistics.

What is Data?

Data refers to pieces of information collected from observations, counts, measurements, or responses. Data is the raw material for statistical analysis and can be quantitative (numerical) or qualitative (categorical).

• Examples: Survey responses, test scores, ages, heights, or opinions.

• Application: According to surveys, more than 7 in 10 Americans believe being a doctor is a prestigious occupation. Social media studies show children create accounts at an average age of 11.4 years.

What is Statistics?

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make decisions. It provides methods for understanding and drawing conclusions from data.

• Key Steps:

  1. Collect data

  2. Organize and summarize data

  3. Analyze data

  4. Interpret results

Data Sets

A data set is a collection of outcomes, responses, measurements, or counts that are of interest in a study. Data sets can represent information from an entire population or a sample.

• Population: The complete set of individuals, items, or data under consideration.

• Sample: A subset of the population, selected for analysis.

Identifying Data Sets

It is important to distinguish between the population and the sample in any statistical study.

• Example: In a survey of 834 employees, the population is all employees, while the sample consists of the 834 surveyed employees. If 517 answered 'yes' and 317 'no', these are sample data.

• Exercise: In a survey of 1,228 people in the United States about health, the population is all people in the U.S., and the sample is the 1,228 surveyed individuals.

Parameter and Statistic

A parameter is a numerical description of a population characteristic, while a statistic is a numerical description of a sample characteristic.

• Parameter: Describes the entire population (e.g., average income of all U.S. residents).

• Statistic: Describes a sample (e.g., average income of 1,000 surveyed U.S. residents).

• Note: A parameter is constant for a population, but a statistic can vary between samples.

Distinguishing Parameter and Statistic

To determine whether a number is a parameter or a statistic, consider whether it describes a whole population or just a sample.

• Example 1: If the average time spent on athletics by student-athletes is calculated from several hundred surveyed athletes, it is a statistic. If it is calculated from all athletes, it is a parameter.

• Example 2: If a university class has an average SAT math score of 514, and this is for the entire class, it is a parameter. If it is for a subset, it is a statistic.

• Example 3: A small company spent a total of $5,150,694 on employees’ salaries. If this is for all employees, it is a parameter.

Branches of Statistics

Statistics is divided into two main branches: descriptive statistics and inferential statistics.

Branch	Description	Purpose
Descriptive Statistics	Organizes, summarizes, and displays data.	To describe and present data in a meaningful way.
Inferential Statistics	Uses sample data to draw conclusions about a population.	To make predictions or generalizations beyond the data collected.

Descriptive and Inferential Statistics

Descriptive statistics summarize data from a sample or population, while inferential statistics use sample data to make generalizations about a population.

• Example: A survey of 2,560 U.S. adults found that 23% of adults not using the Internet are from households earning less than $30,000 annually. The population is all U.S. adults; the sample is the 2,560 surveyed adults.

• Descriptive Statistic: The statement "23% of U.S. adults not using the Internet are from households earning less than $30,000 annually" describes the sample.

• Inferential Conclusion: The study may infer that lower-income households cannot afford access to the Internet.

Additional info: In practice, inferential statistics often involve hypothesis testing, confidence intervals, and regression analysis to make predictions or decisions based on sample data.