medianum

Medianum is a theoretical generalization of the concept of central tendency in statistics. It introduces a parameter that selects a point near the center of a dataset, allowing a continuum of central values between the two central observations when the sample size is even. The term itself comes from the Latin medianus, meaning middle, and it appears in discussions that seek to interpolate between different central-measure ideas.

Definition: For a finite multiset S with n elements, sort the values so that x1 ≤ x2 ≤ ... ≤ xn. If n is odd, medianum(S) equals x_{(n+1)/2}. If n is even, medianum(S) is defined as x_{n/2}*(1−t) + x_{n/2+1}*t, where t is a fixed parameter in [0,1]. When t = 0.5, medianum reduces to the conventional median (the average of the two central values). When t = 0 or t = 1, it equals the lower or upper central value, respectively.

Properties: Medianum yields a family of estimators indexed by t. It is monotone in each data point and continuous in t. For odd sample sizes, medianum is independent of t and equals the middle observation. As t varies, the estimator moves within the central interval spanned by the two central values in even-sized samples, providing a tunable notion of central tendency.

Examples: For the dataset {1, 2, 3, 100}, the central values are 2 and 3. With t = 0, medianum = 2; with t = 0.5, medianum = 2.5; with t = 1, medianum = 3.

Applications: Medianum offers flexibility in exploratory data analysis and robust regression, enabling analysts to explore how shifting the central point within the core data affects conclusions. It is not a universally standardized statistic but serves as a conceptual bridge between lower, upper, and traditional medians.