Abstract
<p>Reported summary statistics are not free to vary independently of one another. The sample size, the mean, and a scale''s minimum and maximum possible values jointly constrain the values that the sample standard deviation (SD) can take. A reported SD that falls outside these bounds is impossible, and signals an error or worse. Because such checks require only the numbers printed in an article, they are practical tools for error detection and trustworthiness assessment even when the underlying data are unavailable. The mathematics involved is nearly a century old but badly scattered. We document at least eight literatures that independently derived, and repeatedly rediscovered, partial versions of the same results, including the little-known fact that Popoviciu''s 1935 paper already contained the sharp odd-n bound usually credited to later authors. We organize these results into a single framework of nested constraints, from the scale limits alone through sample size, mean, integer-valued data, and reported internal consistency (Cronbach''s alpha). Along the way we sharpen several published bounds and derive novel ones, including the first that use a reported reliability coefficient. We name the resulting checks BRIM and BRIMMER (Bounds-Related Inconsistency of Means, and of Means and Errors Reported), by analogy with the granularity tests GRIM and GRIMMER. Both are implemented, with rounding handled explicitly and validation against exhaustive enumeration, in the R package {strait} and in a point-and-click web application (https://errors.shinyapps.io/brimmer).</p>