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The arithmetic of inferential statistics is usually handled by software; the logic is where errors occur, from overstated findings to real effects dismissed because a p-value fell just above .05. This article defines inferential statistics, contrasts them with descriptive statistics, and explains sampling error and the central limit theorem as the basis of all inference. It covers estimation with confidence intervals and hypothesis testing with p-values, and it addresses the two most frequent misinterpretations of p-values.

Five common procedures (t-tests, analysis of variance, chi-square tests, correlation and regression) are each illustrated with a worked or simulated example, and doctoral extracts show how results are reported. The article consistently distinguishes statistical evidence from claims that depend on research design. It is suited to research-methods and introductory statistics courses at master's and doctoral level. The article is the formal edition of a piece first published by Grad Coach and is released under CC BY 4.0.

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