Mathematics - Applied & Stats: Big Ideas

The three most consequential errors in applied statistics — treating p-values as posterior probabilities, treating correlation as causation, and treating model outputs as reality — are not errors of calculation. They are errors of interpretation that arise from removing the conditional qualifiers that statistical procedures require.

Each error produces confident wrong conclusions from correctly executed calculations. The correction requires reinstating the qualifier: p measures surprise given H₀, correlation measures co-movement not mechanism, and the model describes fit to observed data not the underlying process.

The Central Limit Theorem is not a claim about populations — it is a claim about averages. Whatever the population distribution, the mean of a large sample is approximately normally distributed. This single fact is the foundation of most standard statistical inference.

Without the CLT, every t-test and confidence interval would require the population to be normal — an unverifiable assumption. With it, these methods are valid for any population with finite variance, whenever the sample is large enough. The CLT converts a restrictive population assumption into a broad result about how averaging behaves.