Notes on Sampling and Bias

Purpose of a Sample: To learn about an entire population without surveying everyone, which is often impractical or impossible.
Opinion Polls: Designed to gather data from a smaller group (sample) to infer about the larger population.
Representation: Pollsters must ensure that a sample accurately reflects the population to avoid misleading results.

Definition: Bias occurs when methods used to select a sample favor certain parts of the population.
Consequences of Bias: Leads to flawed conclusions that cannot be rectified post-sampling.
Example of Bias: The 1936 Literary Digest Poll predicted Alf Landon would win the presidency based on biased sampling, leading to incorrect results.
- They used a list from phone books, which omitted poorer demographics who were more likely to vote for Roosevelt.
- Result: Incorrectly indicated Landon would win with 57% of the vote, while Roosevelt actually won with 62%.

Concept: Randomization ensures each individual has an equal chance of being selected, which helps achieve representativeness.
Analogy: Sampling soup without stirring can lead to incorrect perceptions (top vs bottom sampling). Randomization similarly mixes the population, creating a more accurate sample.
Protection Against Unknowns: Randomness guards against unnoticed factors that may skew results.

Key Point: The size of the sample (number of participants) is more important than the overall size of the population.
Example: A sample of 100 students (regardless of total student population size) provides insights representative of the entire body.

Census: Involves surveying the entire population; however, it is often impractical as populations change frequently.
Sampling: More feasible and can yield accurate insights without needing to survey everyone.

Parameter: A value that provides information about a population (e.g., population mean).
Statistic: A summary measure from a sample used to estimate a population parameter.
- Example: If 21.7% of surveyed teens report not wearing seat belts, this is a statistic that estimates a population parameter.

Criteria for Representativity: Samples must accurately reflect the population’s characteristics.
Notation: Use $p$ for population parameters and statistics for sample estimates.

Simple Random Sample (SRS): Each individual has an equal chance of being selected. Consider the sampling frame from which it’s drawn.
Stratified Sampling: The population is divided into subgroups (strata) before sampling, enhancing representation of varied groups.
- Example: Ensuring gender balance when surveying opinion on funding by selecting equal numbers of men and women.
Cluster Sampling: Selecting random clusters and surveying all individuals within those clusters, making logistics simpler.
Multistage Sampling: Combining multiple methods to create a sample (e.g., stratifying by chapter and clustering by pages).
Systematic Sampling: Involves selecting every nth individual after a random start.

Voluntary Response Samples: Bias arises due to self-selection, as those with strong opinions are more likely to participate.
Convenience Sampling: Often yields non-representative samples based on accessibility.
Undercoverage: Parts of the population may be completely excluded or underrepresented in the sample.
Nonresponse Bias: Differences in responses between those who participate and those who do not can skew findings.
Influence of Survey Design: Question wording and context can affect responses, introducing bias.

Strategies: Identify and reduce biases in sampling methods, pilot-test surveys, and detail sampling methods for transparency.