Centilegr

Centilegr is a statistical index proposed to summarize an observation’s position within a reference distribution while also reflecting local density around that position. It combines a traditional percentile with a gradient component that captures how rapidly the distribution changes near the value. The resulting centilegr is a dimensionless score that can be rescaled to a 0–100 range, where higher values indicate higher relative standing in the reference population.

Definition and calculation: For a value x drawn from a continuous variable with reference distribution F, the centilegr CG(x) may be described as CG(x) = p(x) + α · g(x), where p(x) = F(x) is the percentile and g(x) is a normalized proxy for the local slope of F at x, such as g(x) = f(x)/max f, with f the density. α is a calibration constant chosen so that CG ∈ [0, 100]. In practice, smoothing, bootstrapping, and standardization are used to estimate p and g from sample data.

Applications: centilegr has been proposed for growth monitoring, educational assessment, and clinical risk stratification, offering a single metric that preserves percentile interpretation while incorporating local distribution shape. It is intended to complement existing metrics like percentile ranks and z-scores, not replace them.

Limitations: Its value depends on the choice of reference population and the estimation of F and f; interpreters must avoid comparing centilegrs across different reference standards; operational definitions vary by domain.