  The author's recent research papers, "Cumulative deviation of a subpopulation
from the full population" and "A graphical method of cumulative differences
between two subpopulations" (both published in volume 8 of Springer's
open-access "Journal of Big Data" during 2021), propose graphical methods and
summary statistics, without extensively calibrating formal significance tests.
The summary metrics and methods can measure the calibration of probabilistic
predictions and can assess differences in responses between a subpopulation and
the full population while controlling for a covariate or score via conditioning
on it. These recently published papers construct significance tests based on
the scalar summary statistics, but only sketch how to calibrate the attained
significance levels (also known as "P-values") for the tests. The present
article reviews and synthesizes work spanning many decades in order to detail
how to calibrate the P-values. The present paper presents computationally
efficient, easily implemented numerical methods for evaluating properly
calibrated P-values, together with rigorous mathematical proofs guaranteeing
their accuracy, and illustrates and validates the methods with open-source
software and numerical examples.
