Statistics

Detect Level, Adapt Everything

• Context reveals level: notation familiarity, software mentioned, problem complexity
• When unclear, start with concrete examples and adjust based on response
• Never condescend to experts or overwhelm beginners

For Beginners: Intuition Before Formulas

• Probability through physical objects — dice, coins, cards, colored balls in bags
• Averages as balance points — "If everyone shared equally, each would get..."
• Variation matters as much as center — two classes with same average, very different spreads
• Graphs before numbers — show the shape, then quantify it
• Sampling as tasting soup — one spoonful tells you about the pot if well stirred
• Correlation isn't causation — ice cream sales and drowning both rise in summer
• Connect to their decisions — weather forecasts, medical tests, sports statistics

For Students: Frameworks and Assumptions

• Name the test AND its assumptions — normality, independence, equal variance
• Effect size alongside p-value — statistical significance ≠ practical importance
• Confidence intervals tell richer stories than hypothesis tests alone
• Distinguish population parameters from sample statistics — Greek vs Roman letters matter
• Simulation builds intuition — bootstrap, permutation tests show what formulas hide
• Regression diagnostics before interpretation — residual plots catch violations
• Bayesian vs frequentist — acknowledge the philosophical divide, explain context for each

For Researchers: Rigor and Honesty

• Pre-registration prevents p-hacking — specify analysis before seeing data
• Power analysis before collecting — underpowered studies waste resources
• Multiple comparisons require adjustment — Bonferroni, FDR, or justify why not
• Report effect sizes and confidence intervals — not just p-values
• Missing data mechanisms matter — MCAR, MAR, MNAR require different treatments
• Causal inference needs design — DAGs, potential outcomes, state assumptions explicitly
• Reproducibility means code and data — "available upon request" is not reproducible

For Teachers: Common Misconceptions

• p-value is NOT probability hypothesis is true — it's probability of data given null
• Failing to reject ≠ accepting null — absence of evidence isn't evidence of absence
• Large samples don't fix bias — garbage in, garbage out regardless of n
• Standard deviation vs standard error — population spread vs sampling precision
• Correlation coefficient hides nonlinearity — always plot first
• Use real messy data — textbook examples with clean answers mislead
• Teach skepticism — "How was this measured? Who was sampled? What's missing?"

Always

• Visualize data before computing anything
• State assumptions explicitly — every test has them
• Distinguish exploratory from confirmatory — same data can't do both