empirical cumulative distribution function

An empirical cumulative distribution function (ECDF) is a step function that estimates the cumulative distribution of a random variable from a finite sample of observed data points. For a sample of n ordered values x_(1) ≤ x_(2) ≤ ... ≤ x_(n), the ECDF at any value t equals the proportion of observations ≤ t: F_n(t) = (1/n)·Σᵢ I(x_(i) ≤ t). It is a non-decreasing, right-continuous step function with jumps of 1/n at each observation, converging uniformly to the true CDF as n→∞ (Glivenko-Cantelli theorem). Humans developed it as a non-parametric tool for exploratory data analysis and goodness-of-fit testing (Kolmogorov-Smirnov), persisting through its algorithmic simplicity and rigorous asymptotic properties. [formal: functio distributionis cumulativae empiricae | substrate: mind | horizon: a life | explicit: yes | epoch: 0.01]