STATS 1.1-1.3
Chapter 1: Statistics - Basic Terms and Concepts
1.1 Definition and Analysis of Statistics
As you study statistics, you will learn to read and analyze various statistical measures to arrive at appropriate conclusions.
Statistics is being viewed as:
A science:
It employs careful definitions of situations, gathers data, summarizes data accurately, and derives meaningful conclusions.
A field that includes diverse interpretations:
Some view it as overwhelming, while others recognize it as a method for collecting and displaying information, or making decisions under uncertainty.
Key Areas in Statistics
Statistics can be subdivided into two main areas:
Descriptive Statistics
Involves the collection, presentation, and description of data.
Inferential Statistics
Techniques for interpreting data and making predictions or decisions about population characteristics.
Applications of Statistics
Statistics inform a variety of fields:
Examples include education (test scores), science (experiment data), and government (statistical surveys).
The US government is one of the largest collectors of statistical data.
Terminology in Statistics
Population: A set of individuals, objects, or events whose properties are analyzed. It is the complete collection of individuals or objects relevant to the studies being conducted. For example:
Set of all students who ever attended a US college.
Collection can also include animals, manufactured objects, trees, etc.
Types of Populations:
Finite Population: Can be physically listed (e.g., books in a library).
Infinite Population: Membership is unlimited (e.g., potential aspirin users).
Sampling in Statistics
Sample: A subset selected from a population for analysis.
It consists of individuals, objects, or measurements from the population.
Variables: Characteristics of interest (e.g., age, height, weight).
Data Values: Values associated with individuals in a population/sample (e.g., a specific student’s age, height).
Data: The collection of values for all variables from each of the sample elements.
Experiments and Parameters
Experiment: A planned activity yielding data collected from a sample.
Parameter: A numerical summary describing an entire population (e.g., average age of college students).
Often represented by Greek letters in statistical notation.
Additional Insights
Sample data can illustrate broader statistical insights, leading to generalized conclusions about populations.
Importance of checking the source of statistical information for credibility and reliability e.g., reputable organizations conducting surveys.
Key Terminology Identified
Population
Definition: The population is the collection of all cars owned by all faculty members at the college.
Sample
Definition: A sample is any subset of the population.
Example: The cars owned by members of the mathematics department represent a sample.
Variable
Definition: A variable is the dollar value of each individual car.
Data Value
Definition: A data value is the dollar value of a specific car.
Example: Mr. Jones' car is valued at $9,400.50.
Data Set
Definition: The data are the set of values corresponding to the sample obtained.
Example Data: $9,400.50, $8,700.00, $15,950.00
Experiment
Definition: The experiment comprises the methods used to select the cars that form the sample and to determine each car's value.
Example Methods: Questioning members of the mathematics department or using other data collection methods.
Parameter
Definition: The parameter represents the average value of all cars in the population.
Statistic
Definition: The statistic describes the average value of the cars in the sample.
Notation: For every parameter, there exists a corresponding sample statistic that summarizes the sample data.
Example: The average height found using a set of 25 heights highlights a sample statistic.
Parameters vs. Statistics
Statistics vary in value depending on the sample, while parameters are fixed.
Example Scenario: If a different sample from the English department is taken, a different value for the statistic is expected; however, the average value of all faculty-owned cars remains constant.
Types of Variables
Qualitative vs. Quantitative Variables
Qualitative (Categorical) Variables
Definition: Variables that result in qualitative information describing an attribute of the population.
Quantitative (Numerical) Variables
Definition: Variables that quantify an element of the population.
Examples of Variables
Qualitative Variables
Hair color, hometown, satisfaction level in a hair salon survey.
Data Collected: Blonde, brown, black; Cities: Brighton, Columbus, Albany, Jacksonville; Satisfaction: Very satisfied, satisfied, somewhat satisfied, not satisfied.
Quantitative Variables
Total cost of textbooks purchased: Resulted data from the sample - $238.87, $94.57, $139.24.
Calculation of Average Cost: avg = {238.87 + 94.57 + 139.24}/{3} = 157.56
Important Note: Arithmetic operations like addition and averaging are meaningful for quantitative variables only.
Subcategories of Variables
Qualitative Variables
Nominal Variables
Definition: Qualitative variables that categorize without a specific order.
Example: Hair color, hometown.
Ordinal Variables
Definition: Qualitative variables that incorporate an order or ranking.
Example: Satisfaction level (very satisfied > satisfied > somewhat satisfied).
Quantitative Variables
Discrete Variables
Definition: Quantitative variables that take on countable values.
Example: Number of courses registered for (can only be whole numbers).
Continuous Variables
Definition: Quantitative variables that can take on any value within a range.
Example: Weight of books carried to class (can be measured in any fraction).
Understanding Measurement and Variability
Measurability and Variability
Within a set of measurement data, variability is always expected.
If little to no variability exists, the measuring device may not be accurately calibrated.
Example of Measurement Variability: Individual weights of candy bars may slightly differ when weighed on a precise scale.
Sampling and Data Collection Methods
Sampling Methods Overview
Sampling Method
Process for selecting items or events that will form a sample.
Bias Sampling Method
Produces data that systematically differ from the population.
Example: Surveying a convenience sample, where only easily accessible data is collected.
Unbiased Sampling Method
Produces a representative sample of the population.
Common Sampling Methods and Their Implications
Convenience Samples
Definition: Items chosen arbitrarily from a population without a structured method.
Example: Selecting a basket of fruit based solely on visible appearance, neglecting the rest.
Volunteer Samples
Definition: Results from individuals choosing to provide information voluntarily.
Implication: Represents a biased sample as typically only those with strong opinions are inclined to respond.
Data Collection Process
Steps in Data Collection
Define objectives (e.g., compare drug effectiveness).
Define variables and population of interest (e.g., length of recovery, total household income).
Define data collection and measuring schemes (e.g., questionnaires, sample size).
Collect sample data.
Review the sampling process upon completion.
Example of a Population and Variable in Investigation
Admissions dean aims to estimate average textbook costs for enrolled students. Population: Student body; Variable: Total spent on textbooks.
Types of Studies Comparing Experimental and Observational Methods
Experimental Studies
Definition: The investigator modifies conditions and observes effects.
Example: Testing doses of medication with controlled conditions.
Observational Studies
Definition: The investigator merely observes without modifying conditions.
Example: Conducting surveys to gather responses from a population.
Sampling Frame and Sample Design
Sampling Frame
Definition: A list of elements from the population from which the sample is chosen.
Importance: Ideally, it should contain every element of the population once and only once.
Sample Design
Types of Sampling Designs:
Judgment Sample: Based on judgment to represent typicality.
Probability Sample: Chosen basis on known probabilities, with each element having a chance of being selected.
Advanced Sampling Techniques
Single Stage Sampling
Each element has an equal chance of selection.
Systematic Sampling
An efficient method of drawing samples, selecting every K-th item after a random start.
Multistage Sampling
Involves dividing the population into strata and sampling from those strata sequentially, often using stratified random sampling.
Conclusion
Understanding statistics involves comprehensive knowledge of the definitions and applications of various components within statistical data collection, interpretation, and analysis.