Introducing Inferential Statistics

The author begins Part 3 by distinguishing between tangible physical variables and abstract probabilistic concepts, arguing that probability – observable only through systematic observation of many instances – poses unique cognitive challenges that lead to persistent biases even among trained researchers. The chapter establishes the core distinction between descriptive statistics (factual statements about samples) and inferential statistics (extrapolations to populations), using a car crash analogy to illustrate how single observations cannot support probabilistic claims. Through coin-tossing experiments, the author explains the Binomial distribution as a model of sampling variation, showing how randomness produces irregular patterns approximating theoretical distributions through repeated trials. The chapter introduces Normal approximation to the Binomial distribution and continuity corrections for small samples or skewed data. It then covers Binomial confidence intervals, demonstrating how the widespread ‘standard error’ or ‘Wald’ interval method – incorrectly substituting observed proportion p for population probability P – leads to impossible intervals extending beyond [0, 1] and produces results inconsistent with significance testing. Wilson’s score interval is introduced, derived through the ‘interval equality principle’ which properly inverts the sampling distribution. This method provides accurate intervals even for extreme or small-sample data. The chapter concludes by clarifying the distinction between scatter (observed sample variation) and confidence (predicted variation of resampled means), establishing the theoretical framework for subsequent inferential statistical methods.