Chapter 1: Data Collection and Introduction to Statistics

Statistics: Informed Decisions Using Data

Introduction to Statistics

Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. It also provides a measure of confidence in any conclusions. The information used in statistics is called data, which are facts or propositions used to draw conclusions or make decisions. Data describe characteristics of individuals and are subject to variability.

• Variability: Data vary among individuals and even within the same individual over time.

• Goal: One goal of statistics is to describe and understand sources of variability.

Types of Statistics

• Descriptive Statistics: Organizing and summarizing data using numerical summaries, tables, and graphs.

• Inferential Statistics: Methods that take results from a sample, extend them to the population, and measure the reliability of the result.

• Parameter: Numerical summary of a population.

• Statistic: Numerical summary based on a sample.

Example: Parameter vs. Statistic

• If 48.2% of all students own a car, this is a parameter.

• If 46% of a sample of 100 students own a car, this is a statistic.

The Process of Statistics

Steps in the Statistical Process

1. Identify the Research Objective: Clearly define the question and population to be studied.

2. Collect Data: Gather data needed to answer the question, often from a sample due to practical constraints.

3. Describe the Data: Use descriptive statistics to obtain an overview and determine appropriate methods.

4. Perform Inference: Apply techniques to extend sample results to the population and report reliability.

Example: Trust Your Neighbor Survey

• Population: Adult Americans

• Sample: 1628 surveyed, 847 trust all or most neighbors

• Descriptive Statistic: 52% trust all or most neighbors

• Inference: 95% certain the true percentage is between 49.5% and 54.5% (margin of error ±2.5%)

Variables and Data Types

Definition of Variables

Variables are characteristics of individuals within the population. Variables vary, and understanding this variability is a key research goal.

Types of Variables

• Quantitative Variables: Provide numerical measures; values can be meaningfully added or subtracted.

• Qualitative (Categorical) Variables: Allow classification based on attributes or characteristics.

Quantitative Variable Subtypes

• Discrete Variables: Finite or countable values (e.g., number of cars).

• Continuous Variables: Infinite values, measurable to any desired accuracy (e.g., distance, time).

Between any two values of a continuous variable, there are infinitely many possible values.

Examples: Distinguishing Variable Types

• Race: Qualitative

• Temperature: Quantitative

• Number of study days: Quantitative

• Zip code: Qualitative

Data Classification

• Qualitative Data: Observations from qualitative variables.

• Quantitative Data: Observations from quantitative variables.

• Discrete Data: Observations from discrete variables.

• Continuous Data: Observations from continuous variables.

Example: Parking Meter Data

Table 1 presents data from a sample of on-street parking meters in Seattle. The individuals are cars; variables include payment method, amount paid, duration, side of street, and parking space number.

• Qualitative variables: Payment method, side of street, parking space number

• Quantitative variables: Amount paid (discrete), duration (continuous)

Levels of Measurement

Nominal Level

Values name, label, or categorize; no ranked order.

Ordinal Level

Values can be ranked or ordered.

Interval Level

Ranked values; differences have meaning; zero does not mean absence of quantity.

Ratio Level

Ranked values; differences and ratios have meaning; zero means absence of quantity.

Examples: Levels of Measurement

• Gender: Nominal

• Temperature (°F): Interval

• Number of study days: Ratio

• Letter grade: Ordinal

• Years of cicada emergence: Interval

Observational Studies vs. Experiments

Observational Study

Measures the value of the response variable without influencing variables. Only association can be claimed, not causation.

Designed Experiment

Researcher manipulates explanatory variables and records response variable. Allows for causation claims.

Confounding and Lurking Variables

• Confounding Variable: Effects cannot be separated from other explanatory variables.

• Lurking Variable: Not considered in the study but affects the response variable.

Observational studies are subject to confounding and lurking variables, limiting causal inference.

Types of Observational Studies

• Cross-sectional: Information at a specific point in time.

• Case-control: Retrospective, comparing individuals with and without certain characteristics.

• Cohort: Prospective, following a group over time.

Sampling Methods

Random Sampling

Chance is used to select individuals, ensuring representativeness.

Simple Random Sampling

Every possible sample of size n has an equally likely chance of occurring.

Other Sampling Methods

• Systematic Sampling: Select every kth individual after a random start.

• Stratified Sampling: Divide population into homogeneous strata, sample from each.

• Cluster Sampling: Divide population into groups, randomly select groups, survey all individuals in selected groups.

• Convenience Sampling: Individuals easily obtained; results are suspect.

• Multistage Sampling: Combination of methods, often used in large-scale surveys.

Bias and Errors in Sampling

Types of Bias

• Sampling Bias: Technique favors one part of the population.

• Nonresponse Bias: Individuals who do not respond differ from those who do.

• Response Bias: Answers do not reflect true feelings due to interviewer error, misrepresented answers, wording, ordering, or question type.

Types of Errors

• Nonsampling Errors: Undercoverage, nonresponse, response bias, data-entry error.

• Sampling Error: Sample does not accurately represent the population.

Experimental Design

Key Concepts

• Experiment: Controlled study to determine effect of explanatory variables (factors) on response variable.

• Treatment: Combination of factor values applied to experimental units.

• Experimental Unit: Person, object, or item receiving treatment.

• Control Group: Baseline treatment for comparison.

• Placebo: Inert treatment resembling the experimental treatment.

• Blinding: Nondisclosure of treatment received; single-blind (subject unaware), double-blind (subject and researcher unaware).

Example: Lipitor Experiment

• Population: Individuals aged 40–75 with type 2 diabetes, no prior cardiovascular disease

• Sample: 2838 subjects

• Treatments: Lipitor 10 mg daily or placebo

• Response Variable: Occurrence of major cardiovascular event (qualitative)

• Results: Lipitor group had fewer major events and deaths compared to placebo group

Summary Table: Sampling Methods

Sampling Method	Description	Advantages	Disadvantages
Simple Random	Every sample equally likely	Unbiased, easy to analyze	May require complete list (frame)
Systematic	Select every kth individual	Easy to implement	May introduce periodic bias
Stratified	Divide into strata, sample from each	Ensures representation of subgroups	Requires knowledge of strata
Cluster	Divide into groups, sample groups	Efficient for large populations	May not represent entire population
Convenience	Sample easily obtained individuals	Quick, inexpensive	Results often biased

Key Formulas

• Margin of Error: $\text{Margin of Error} = z^* \cdot \frac{\sigma}{\sqrt{n}}$

• Sample Proportion: $\hat{p} = \frac{x}{n}$

Additional info: Academic context and examples were expanded for clarity and completeness. All images included are directly relevant to the explanation of the adjacent paragraphs.