# Mean and Standard Deviation

A ‘statistic’ refers to a single number measure of a sample.  Statistics relevant to uncertainty are mean value, total covariance, total deviation, standard covariance and standard deviation.  The formulas needed to calculate these with traditional arithmetic are known.  However, the theory does not present error propagation.   A comprehensive theory should be valid for choices of datum and standardization scaling.  Therefore, when all numbers are formatted with error, the theory must be revisited and updated.   Thus ‘statistics’ is an application of duals arithmetic following both conversion of existing numbers and formulas or by re-derivation of theory.  The goal of this project is to obtain new formula or procedures for calculating ‘mean with error’ and ‘standard deviation with error,’ in support of the Uncertainty Project.

This project has progressed from casting each instance as a dual of a scaled point and an error vector, to casting a general datum as a dual, to formulating deviation duals and then using matrix-of-duals functions to calculate total covariance.  Next the special case of ‘most central datum’ creates a uniform reference to error forming the basis to determine the ‘mean with error’ by minimizing total variance.  Recycling this result specializes the datum to obtain ‘deviation with error’ and ‘covariance matrix with error’.

Standardization is the co-scaling of a ‘total with error’ such that the effect of growth with sample size is remediated.  This has been completed to find ‘standard covariance with error.’  Using the square-root-of-dual operation (currently a 1D error vector), the ‘standard deviation with error’ is found.  As the theory matures, the ‘with error’ label will be eliminated as all numbers will have error.