The Cumulative Distribution Function for a finite data set

In this work, the Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) are examined for a data set of finite elements. The CDF and the PDF are valid only for the theoretical asymptotes when the number of elements in the set approaches infinity. The equivalent functions defined for a finite set are currently unknown. In various fields, especially in signal processing, data size is usually statistically limited and more accurate analysis is often required for the validation of new algorithms. In this work, discontinuous CDF (DCDF) is defined and proposed for measuring the `statistical distance´ and the `statistical error´. These new definitions enable comparisons of different data sets with each other and with the theoretical asymptotic function CDF. The proposed statistical functions are illustrated on Gaussian distributed data.