Shewhart X-Charts (Average X) with Estimated Process Variance.
Abstract
Properties of the Shewart X-chart (average x) for controlling the mean of a process with a normal distribution are investigated for the situation where the process variance sigma-squared must be estimated from initial sample data. The control limits of the X-chart (average x) depend on the estimate of sigma-squared and thus, unlike the case when sigma-squared is known, the X-chart (average x) is not equivalent to a sequence of independent tests. When sigma-squared is estimated the distribution of the run length is not geometric and cannot be characterized simply in terms of the probability of a signal at a given point. The average run length (ARL) for the X-chart (average x) is expressed in terms of an integral involving the normal cdf, and it is shown that the chart signals with probability one, but the ARL may not be finite if the size of the sample used to estimate sigma-squared is sufficiently small. In addition, certain bounds for the ARL are also derived. Numerical integration is used to show that the effect of using small sample sizes in estimating sigma-squared is to increase the ARL and the variance of the run-length distribution. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1980
- Accession Number
- ADA083045
Entities
People
- B. K. Ghosh
- Marian R. Reynolds
- Yer Van Hui
Organizations
- Virginia Tech