A Brief Note on Process Capability

Abstract

Process capability, as revolutionized by Motorola, can be defined as the extent to which a process's natural variation falls within specifications. This note is a succinct introduction to the process capability ratio, which compares set specifications against the three standard deviations on either side of the process mean--six in total, referred to as Six Sigma. This introduction assumes a normal distribution of observations and a process that is performing predictably over time. Excerpt UVA-OM-1439 Rev. Sept. 19, 2011 A BRIEF NOTE ON PROCESS CAPABILITY Process behavior charts provide us with a way to determine whether a process is performing consistently and predictably over time and identify and eliminate special causes of abnormal variation. A process behavior chart does not provide us, however, with information about how well a process is performing relative to customer requirements. These charts offer evidence that a process is or is not performing consistently over time. To answer the question of how well a process is working relative to requirements, we need to have explicit understanding of what makes customers satisfied. Customer expectations guide a firm in setting specifications for performance. Specifications, also known as tolerances, may be based on government regulations, as with CAFE requirements for fuel emission per gallon in the auto industry or on engineering requirements, as with thread widths of mating components in an assembled product. If the threads of a particular screw must be within 8 and 12 millimeters, 8 millimeters is the lower specification limit or LSL, and 12 millimeters is the upper specification limit or USL. A service example would be a bank determining through market research that customers consider acceptable a transaction time of three minutes or less. Calculating the likelihood that a process meets specifications is one way to measure process capability. Motorola popularized another simple approach: comparing the spread of the natural variation in a process against the amount of variation that can be tolerated based on the requirements. Consider the screw example above. Suppose the screw manufacturing process in question produces screws with a width Y that is normally distributed around a mean of 10 millimeters, with a standard deviation of 0.5 millimeters. The process capability ratio, Cp, may be defined as follows: (1) Since width Y is normally distributed with a standard deviation of 0.5 millimeters, the “natural spread” of the process is defined as 3 millimeters (three standard deviations on either side of the mean). Also, given a USL of 12 millimeters and an LSL of 8 millimeters, the tolerable spread is 4 millimeters. Thus Cp = 4/3 = 1.33 for the screw manufacturing process. In this example, the fact that Cp > 1 suggests that the corresponding process largely meets specifications. A value of Cp = 1 would correspond to 3 defects per 1,000 units produced, since in this example the probability of a defect would equal the probability that Y 12, which is 0.27% (for the normally distributed variable). . . .