Bias, Generalization, Association, and Causation

Objective: Identify a mismatch between a sample source and target population.

Prerequisite skills: study designs, sample selection, and uncertainty. A sampling frame can exclude or underrepresent important groups. Voluntary response can favor people with strong opinions. These mechanisms create bias that margin-of-error calculations based only on sample size do not remove. Common error: treating a large response count as proof of representativeness. Restrictions and interpretation: Limit conclusions to the target population and design; a strong association alone does not prove cause.

Objective: Explain how a third factor can produce an observed association.

A confounder is related to both the explanatory variable and outcome and can provide an alternative explanation for their association. Good randomized design reduces systematic confounding, whereas observational adjustment may still leave uncertainty. Common error: assuming an observed association must run directly from the named x to y. Restrictions and interpretation: Limit conclusions to the target population and design; a strong association alone does not prove cause.

Objective: Choose a conclusion consistent with sampling and treatment design.

State only what the design supports: an association or causal comparison, and a sample-specific or population-level conclusion. A statistically persuasive difference still does not guarantee an effect for every individual. Common error: generalizing a volunteer experiment to all students without random sampling. Restrictions and interpretation: Limit conclusions to the target population and design; a strong association alone does not prove cause. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.