Statistics: Sampling Methods, Data Types, and Experimental Design

Sampling Methods

• Simple Random Sample (SRS): sample of size $n$ from a population of size $N$ where every collection of $n$ population items is equally likely to form the sample.
  • Notation: $n$ = sample size; $N$ = population size; sample is chosen by a random process to avoid bias.
• Sample of Convenience: selected without a well-defined random process; often feasible but can differ systematically from the population, leading to bias.
• Stratified Sampling: divide population into strata (groups) that differ; apply SRS within each stratum; can ensure representation from each group.
• Cluster Sampling: divide population into clusters (groups) and sample entire clusters; useful when population is large and spread out.
• Systematic Sampling: order items and select every $k$-th item after a random start; simple and often used in quality checks.
• Voluntary Response Sampling: participants choose to respond; tends to bias toward strong opinions and is generally unreliable.

Statistics and Parameters

• Population vs Sample:
  • Population: entire collection of individuals of interest.
  • Sample: a subset of the population observed.
• Parameter vs Statistic:
  • Parameter: numeric summary describing the population.
  • Statistic: numeric summary describing the sample.
• Notation (helpful): statistic ↔ sample, parameter ↔ population; statistic uses $s$, parameter uses $p$ in many contexts.
• Quick example distinctions:
  • If 57% of teachers at a specific school are female, that describes the population of teachers at that school → parameter. If you only sampled teachers and found 57% female in the sample, that would be a statistic.

Investigative Process of Statistics

• Steps:
  • Formulate questions.
  • Collect data (via survey or other methods).
  • Describe the data (numerical and graphical summaries).
  • Draw conclusions using appropriate methods (to be covered in Unit 3).
• Population data vs sample data: collect to form summaries and draw inferences.

Data, Variables, and Dataset Structure

• Structure:
  • Individuals → Variables → Data → Dataset.
• Variables types:
  • Qualitative (Categorical): categories or labels.
  • Quantitative (Numerical): numeric values.
• Five-question survey example (students’ version): five variables (qualitative or quantitative); Data from all respondents forms the dataset for the project.

Qualitative vs Quantitative; Discrete vs Continuous

• Qualitative (Categorical): quality or category; usually words. Example: place of birth, jersey color.
• Quantitative (Numerical): numeric values; can be used to compute averages, etc.
  • Discrete: countable, isolated values (e.g., number of siblings).
  • Continuous: any value in an interval (e.g., height, time).
• Quick checks:
  • Age on last birthday is typically discrete (whole years) but can be treated as continuous if measured more finely (months, days).
  • Height is continuous; number of siblings is discrete.

Experimental Design and Observational Studies

• Experimental designs aim to assess causality by applying treatments.
• Randomized Experiment: units randomly assigned to treatment groups; ideal is that groups differ only by treatment.
  • Randomization helps attribute observed differences to the treatments rather than other factors.
  • Placebo group: inactive treatment used to control expectations; placebo effect occurs when subjects improve due to belief in treatment.
  • Blinding:
  • Single blind: participants do not know which group they are in.
  • Double blind: neither participants nor researchers know group assignments.
• Observational Studies: treatments are not randomly assigned; researchers observe natural variation.
  • Less reliable due to potential confounding variables.
  • Confounder: a variable related to both the treatment and the outcome, which can bias conclusions.

Randomized Block Design and Matched Pairs

• Randomized Block Design: partition units into homogeneous blocks (e.g., by grade, sex, age) and randomize treatments within each block to reduce variability.
• Matched Pairs Design: blocks consist of pairs of similar units; pair members receive different treatments (or one receives treatment and the other placebo).
  • Examples: twins, pre/post tests, or comparing left vs right hand in the same person.

Quick Reference Points

• Always distinguish between population (parameter) and sample (statistic).
• When designing sampling, aim for randomness to reduce bias; beware convenience and voluntary response.
• Know the main data types and when to treat data as discrete or continuous.
• In experiments, randomization and control groups help identify treatment effects; in observational studies, be cautious of confounding.
• Blinding and placebo controls strengthen experimental validity.

$ext{SRS: every subset of size } n ext{ from a population of size } N ext{ is equally likely.}$
$n = ext{sample size}, \, N = ext{population size}.$