On the Mathematics behind the CUSUM Control Charts

This paper describes the mathematics behind CUSUM control charts. An introduction to CUSUM charts is found in (Madsen 1998).CUSUM charts are well suited for checking a measuring system in operation for any departure from some target or specified values. In general they can be used for:- detecting a drift (or shift in the level) of the measuring system, and- detecting a change of the precision of the measuring system.In both cases the CUSUM procedure contains methods for estimating the shift such that maintenance (e.g. recalibration) can take place.The CUSUM procedure is described in ISO/CD 7871 from CEN-TC 69. However the presentation in ISO/CD 7871 concentrates on the level, whereas CUSUM charts for detecting changes in the precision are not discussed.This document outlines the theory of CUSUM control charts, and both CUSUM procedures for detecting a drift as well as CUSUM procedures for detecting a shift in the precision are outlined. In the later case a simple example is given.