sqrtmeanytrue

sqrtmeanytrue is a mathematical transformation used in statistics and data analysis to summarize a binary-valued sample. It is defined as the square root of the mean of a sequence of boolean or binary outcomes.

Formally, for a vector y = (y1, ..., yn) with yi ∈ {0,1} (or {False, True}), sqrtmeanytrue(y) = sqrt( (1/n) Σ_i yi ). The domain requires n ≥ 1, and the result lies in the interval [0,1].

Properties of sqrtmeanytrue include monotonicity: if any component yi increases from 0 to 1, the value does not decrease. Since the square root function is increasing, sqrtmeanytrue preserves the order of the ordinary mean but applies a concave transformation, which dampens large proportions and accentuates small ones relative to a linear mean. The standard definition assumes binary inputs; generalizations may allow yi ∈ [0,1] and apply the transformation accordingly.

Examples help illustrate the measure. If y = [True, True, False, True], the mean is 3/4, and sqrtmeanytrue(y) = sqrt(0.75) ≈ 0.866. If all values are False, the result is 0; if all values are True, the result is 1.

Usage and interpretation: sqrtmeanytrue provides a compact, nonlinear descriptor of the prevalence of True outcomes. It can be used in simulations, scoring rules, or simple models where a nonlinear emphasis on proportions is desirable. It should be used with care in contexts requiring straightforward interpretability, as the square-root transformation changes the scale relative to the plain mean.