Process Analysis/ From Search on May 18, 2011 by Margith Strand

Process Analysis

Sampling Plans
General Purpose
Computational Approach
Means for H0 and H1
Alpha and Beta Error Probabilities
Fixed Sampling Plans
Sequential Sampling Plans
Summary
Process (Machine) Capability Analysis
Introductory Overview
Computational Approach
Process Capability Indices
Process Performance vs. Process Capability
Using Experiments to Improve Process Capability
Testing the Normality Assumption
Tolerance Limits
Gage Repeatability and Reproducibility
Introductory Overview
Computational Approach
Plots of Repeatability and Reproducibility
Components of Variance
Summary
Non-Normal Distributions
Introductory Overview
Fitting Distributions by Moments
Assessing the Fit: Quantile and Probability Plots
Non-Normal Process Capability Indices (Percentile Method)
Weibull and Reliability/Failure Time Analysis
General Purpose
The Weibull Distribution
Censored Observations
Two- and three-parameter Weibull Distribution
Parameter Estimation
Goodness of Fit Indices
Interpreting Results
Grouped Data
Modified Failure Order for Multiple-Censored Data
Weibull CDF, Reliability, and Hazard Functions

Sampling plans are discussed in detail in Duncan (1974) and Montgomery (1985); most process capability procedures (and indices) were only recently introduced to the US from Japan (Kane, 1986), however, they are discussed in three excellent recent hands-on books by Bohte (1988), Hart and Hart (1989), and Pyzdek (1989); detailed discussions of these methods can also be found in Montgomery (1991).

Step-by-step instructions for the computation and interpretation of capability indices are also provided in the Fundamental Statistical Process Control Reference Manual published by the ASQC (American Society for Quality Control) and AIAG (Automotive Industry Action Group, 1991; referenced as ASQC/AIAG, 1991). Repeatability and reproducibility (R & R) methods are discussed in Grant and Leavenworth (1980), Pyzdek (1989) and Montgomery (1991); a more detailed discussion of the subject (of variance estimation) is also provided in Duncan (1974).

Step-by-step instructions on how to conduct and analyze R & R experiments are presented in the Measurement Systems Analysis Reference Manual published by ASQC/AIAG (1990). In the following topics, we will briefly introduce the purpose and logic of each of these procedures. For more information on analyzing designs with random effects and for estimating components of variance, see Variance Components.
Sampling Plans
General Purpose
Computational Approach
Means for H0 and H1
Alpha and Beta Error Probabilities
Fixed Sampling Plans
Sequential Sampling Plans
Summary
General Purpose
A common question that quality control engineers face is to determine how many items from a batch (e.g., shipment from a supplier) to inspect in order to ensure that the items (products) in that batch are of acceptable quality. For example, suppose we have a supplier of piston rings for small automotive engines that our company produces, and our goal is to establish a sampling procedure (of piston rings from the delivered batches) that ensures a specified quality. In principle, this problem is similar to that of on-line quality control discussed in Quality Control. In fact, you may want to read that section at this point to familiarize yourself with the issues involved in industrial statistical quality control.
Acceptance sampling. The procedures described here are useful whenever we need to decide whether or not a batch or lot of items complies with specifications, without having to inspect 100% of the items in the batch. Because of the nature of the problem – whether to accept a batch – these methods are also sometimes discussed under the heading of acceptance sampling.

Advantages over 100% inspection. An obvious advantage of acceptance sampling over 100% inspection of the batch or lot is that reviewing only a sample requires less time, effort, and money. In some cases, inspection of an item is destructive (e.g., stress testing of steel), and testing 100% would destroy the entire batch. Finally, from a managerial standpoint, rejecting an entire batch or shipment (based on acceptance sampling) from a supplier, rather than just a certain percent of defective items (based on 100% inspection) often provides a stronger incentive to the supplier to adhere to quality standards.

Computational Approach
In principle, the computational approach to the question of how large a sample to take is straightforward. Elementary Concepts discusses the concept of the sampling distribution. Briefly, if we were to take repeated samples of a particular size from a population of, for example, piston rings and compute their average diameters, then the distribution of those averages (means) would approach the normal distribution with a particular mean and standard deviation (or standard error; in sampling distributions the term standard error is preferred, in order to distinguish the variability of the means from the variability of the items in the population). Fortunately, we do not need to take repeated samples from the population in order to estimate the location (mean) and variability (standard error) of the sampling distribution. If we have a good idea (estimate) of what the variability (standard deviation or sigma) is in the population, then we can infer the sampling distribution of the mean. In principle, this information is sufficient to estimate the sample size that is needed in order to detect a certain change in quality (from target specifications). Without going into the details about the computational procedures involved, let us next review the particular information that the engineer must supply in order to estimate required sample sizes.