Variation, inference and screening calculations
Begin statistical interpretation with the type of variable, sampling process and unit of analysis. A nominal category has no inherent order; an ordinal scale has order without necessarily equal spacing. A continuous measurement may be approximately symmetric or strongly skewed. A mean and standard deviation answer different descriptive questions from a median and interquartile range. A geometric mean can be useful for positive measurements with an appropriate multiplicative or log-normal structure.
Standard deviation describes spread among observations. Standard error describes the sampling variability of an estimator and generally depends on sample size as well as underlying variation. A reference interval for individuals is therefore not a confidence interval for a population mean. Neither is automatically a treatment threshold. An interval should be described with its model and intended interpretation rather than read as a guarantee for a particular person.
Choose an inferential procedure using the design and assumptions. Paired observations require attention to within-person differences, while independent groups require an independent-sample approach. A pooled Student test assumes a common variance model; Welch methods address a different variance assumption. Rank methods do not eliminate all assumptions and do not always test an isolated median difference. Sparse contingency tables may justify exact methods. Historical sample-size rules for continuity corrections are not universal mathematical mandates.
A null hypothesis is a specified model or parameter claim. A type I error rejects a true null, while power concerns detecting a specified alternative. A small p-value does not measure clinical importance or prove the proposed causal explanation. Correlation is an association summary, and a fitted coefficient of determination is not the fraction of disease caused by one exposure.
For screening, sensitivity uses people with the condition and specificity uses people without it. Predictive values instead condition on the observed test result and depend strongly on prevalence. Parallel and serial rules change the classification logic; actual conditional dependence also matters. Always construct the relevant table, keep percentages and probabilities consistent, and distinguish a positive screen from a confirmed diagnosis or a beneficial screening programme.