Chapter 1: Data Collection

Section 1.1: Introduction to the Practice of Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting information to draw conclusions or answer questions. It also provides a measure of uncertainty in any conclusions. The information used in statistics is called data, which describes characteristics of individuals and exhibits variability. Understanding and describing sources of variability is a fundamental goal of statistics.

Key Definitions

• Statistics: The science of collecting, organizing, analyzing, and interpreting data.

• Data: Facts or propositions used to draw conclusions or make decisions; describes characteristics of individuals.

• Variability: The tendency of data to differ among individuals or over time.

Population, Sample, and Individual

• Population: The entire group of individuals to be studied.

• Sample: A subset of the population that is being studied.

• Individual: A single member of the population.

Descriptive vs. Inferential 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 reliability.

• Parameter: Numerical summary of a population.

• Statistic: Numerical summary of a sample.

The Process of Statistics

1. Identify the research objective.

2. Collect the data needed to answer the question(s).

3. Describe the data using descriptive statistics.

4. Perform inference to draw conclusions and report reliability.

Section 1.1: Types of Variables

Variables are characteristics of individuals within the population. They can be classified as qualitative or quantitative, and quantitative variables can be further classified as discrete or continuous.

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

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

• Discrete Variables: Quantitative variables with a finite or countable number of possible values (e.g., number of students).

• Continuous Variables: Quantitative variables with an infinite number of possible values, not countable (e.g., height, weight).

Examples

• Education level – Qualitative

• Today's high temperature – Quantitative (Continuous)

• Daily intake of whole grains (grams per day) – Quantitative (Continuous)

• Number of vending machines at a school – Quantitative (Discrete)

• Whether or not a student is prepared for class – Qualitative

Section 1.1: Levels of Measurement

The level of measurement of a variable determines the type of statistical analysis that can be performed.

• Nominal: Values name, label, or categorize; no order.

• Ordinal: Values can be ranked or ordered.

• Interval: Differences between values have meaning; zero does not mean absence of quantity.

• Ratio: Ratios of values have meaning; zero means absence of quantity.

Examples

• Temperature – Interval

• Income (in dollars) – Ratio

• Letter grade earned in Algebra – Ordinal

• Response to survey (e.g., strongly agree, agree, disagree) – Nominal

• Number of students in a classroom – Ratio

Section 1.2: Observational Studies vs. Designed Experiments

Research studies can be classified as observational or experimental, depending on whether the researcher manipulates variables.

• Observational Study: Measures the value of the response variable without influencing explanatory variables.

• Designed Experiment: Researcher manipulates explanatory variables and controls other variables, then records the response.

Confounding and Lurking Variables

• Confounding Variable: An explanatory variable whose effect cannot be distinguished from another variable.

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

Observational studies allow for association, not causation. Designed experiments can help establish causation but may still be affected by confounding variables.

Types of Observational Studies

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

• Case-control: Retrospective; compares individuals with and without certain characteristics.

• Cohort: Prospective; follows a group over time.

Section 1.3: Simple Random Sampling

Random sampling uses chance to select individuals from a population. A simple random sample ensures every possible sample of size n has an equally likely chance of occurring.

1. Obtain a frame listing all individuals in the population.

2. Use a random number table or software to generate n random numbers.

Section 1.4: Other Effective Sampling Methods

Several sampling methods are used to obtain representative samples from populations.

• Stratified Sampling: Divide population into homogeneous groups (strata), then take a simple random sample from each stratum.

• Systematic Sampling: Select every kth individual from the population, starting at a random number between 1 and k.

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

• Convenience Sampling: Individuals are easily obtained, not based on randomness; results are often biased.

Comparison of Sampling Methods

Section 1.5: Bias in Sampling

If a sample is not representative of the population, it is biased. There are several sources 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, question wording, etc.

• Data-entry Error: Mistakes in entering data can lead to inaccurate results.

Nonsampling Errors: Errors from bias or data-entry, can occur even in a census. Sampling Error: Error from using a sample to estimate population information; occurs because a sample gives incomplete information.

Section 1.6: The Design of Experiments

Experiments are characterized by random assignment, manipulation of explanatory variables, and control of other variables. Common designs include completely randomized, matched-pairs, and randomized block designs. (Content omitted; refer to textbook for details.)

Additional info:

Web scraping is a modern method for obtaining data from the internet, but it raises ethical concerns. Convenience samples and web scraping may introduce bias and are not recommended for rigorous statistical studies.