# 11. Statistical Process Control

Sven Knoth
July 9, 2002

Statistical Process Control (SPC) is the misleading title of the area of statistics which is concerned with the statistical monitoring of sequentially observed data. Together with the theory of sampling plans, capability analysis and similar topics it forms the field of Statistical Quality Control. SPC started in the 1930s with the pioneering work of Shewhart (1931). Then, SPC became very popular with the introduction of new quality policies in the industries of Japan and of the USA. Nowadays, SPC methods are considered not only in industrial statistics. In finance, medicine, environmental statistics, and in other fields of applications practitioners and statisticians use and investigate SPC methods.

A SPC scheme - in industry mostly called control chart - is a sequential scheme for detecting the so called change point in the sequence of observed data. Here, we consider the most simple case. All observations are independent, normally distributed with known variance . Up to an unknown time point the expectation of the is equal to , starting with the change point the expectation is switched to . While both expectation values are known, the change point is unknown. Now, based on the sequentially observed data the SPC scheme has to detect whether a change occurred.

SPC schemes can be described by a stopping time - known as run length - which is adapted to the sequence of sigma algebras . The performance or power of these schemes is usually measured by the Average Run Length (ARL), the expectation of . The ARL denotes the average number of observations until the SPC scheme signals. We distinguish false alarms - the scheme signals before , i.e. before the change actually took place - and right ones. A suitable scheme provides large ARLs for and small ARLs for . In case of one has to consider further performance measures. In the case of the oldest schemes - the Shewhart charts - the typical inference characteristics like the error probabilities were firstly used.

The chapter is organized as follows. In Section 11.1 the charts in consideration are introduced and their graphical representation is demonstrated. In the Section 11.2 the most popular chart characteristics are described. First, the characteristics as the ARL and the Average Delay (AD) are defined. These performance measures are used for the setup of the applied SPC scheme. Then, the three subsections of Section 11.2 are concerned with the usage of the SPC routines for determination of the ARL, the AD, and the probability mass function (PMF) of the run length. In Section 11.3 some results of two papers are reproduced with the corresponding XploRe quantlets.