Monitoring the Process Mean and Variance Using Individual Observations and Variable Sampling Interval - ASQ

Monitoring the Process Mean and Variance Using Individual Observations and Variable Sampling Interval

In this paper we investigate control charts for monitoring a process to detect changes in the mean and/or variance of a normal quality variable when an individual observation is taken at each sampling point. The traditional X chart and moving range (MR) chart are evaluated. Also evaluated are the exponentially weighted moving average (EWMA) chart of the observations and the EWMA chart of the squared deviations of the observations from the target. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts. The ability of charts to diagnose the type of parameter shift produced by a special cause is also investigated. It is shown that combinations involving the EWMA charts are just as effective at diagnosing the type of parameter shift as the traditional combination of the X and MR charts. The effect of adding the variable sampling interval (VSI) feature is also evaluated for some of the combinations of charts. The VSI feature allows the sampling interval to be varied as a function of the values of the statistics being plotted. It is shown that adding the VSI feature to the combinations of charts results in very substantial reductions in the expected time required to detect shifts in process parameters.

Keywords: Average Number of Observations to Signal, Average Time to Signal, Exponentially Weighted Moving Average Control Charts, Moving Range Control Charts, Shewhart Control Charts, Statistical Process Control, Steady State, Variable Sampling Interval Control Charts, X Control Charts.

by Marion R. Reynolds, JR. Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0439 and Zachary G. Stoumbos, Rutgers, The State University of New Jersey, Newark, NJ 07102-1895


A control chart is used to monitor a process for the purpose of detecting special causes of process variation that may result in lower-quality process output. The statistic plotted on a control chart is usually based on samples (or subgroups) of n observations that are taken at regular sampling intervals. For example, a sample of n = 4 observations might be taken every hour. There are, however, many applications in which the control chart is based on individual observations (n = 1) rather than samples of n > 1. In these applications, sampling may be expensive, destructive, and/or time consuming, and the sampling interval may be relatively long. It is not unusual to find industrial examples in which there is only one sample per shift. In some situations, for example when an instrument is being monitored for accuracy and precision, the observations obtained for monitoring may be days or even weeks apart.

In most situations in which a continuous quality variable is being measured, it is assumed that this variable, say X, has a normal distribution. If X is normal and a special cause affects the distribution of X, then the special cause will change the mean , the standard deviation , or both and . When samples of n > 1 are taken, the usual practice is to use two control charts, one for detecting changes in and the other for detecting changes in . The control chart for monitoring is usually based on the sample means. For example, the Shewhart chart is based on plotting the sample means. The control chart for is usually based on a measure of within-sample dispersion, such as the sample variance or sample range. For example, the Shewhart R chart is based on plotting the sample ranges.

When individual observations are taken, a chart for monitoring would use these individual observations in place of the sample means. For example, the Shewhart X chart would be used in place of the Shewhart chart. Monitoring is now more difficult, however, because the usual within-sample measures of dispersion no longer apply. Thus, there is the question of how to monitor when n = 1. A traditional choice is to base a chart on the moving range (MR), which is the range of successive individual observations, but there is now considerable evidence that there is little or no benefit to using the Shewhart MR chart when a Shewhart X chart for is also being used (see, e.g., Nelson (1982), Roes, Does, and Schurink (1993), Rigdon, Cruthis, and Champ (1994), Albin, Kang, and Shea (1997), and Stoumbos and Reynolds (2000)).

Shewhart charts are widely used for process monitoring, and they are effective for detecting large changes in process parameters; however, a Shewhart chart may take a very long time to detect a small persistent shift in a process parameter. The ability to detect smaller parameter shifts can be improved by using a chart based on a statistic that incorporates information from past samples in addition to current samples. For example, the exponentially weighted moving average (EWMA) chart is based on a weighted average of current and past sample statistics, and the cumulative sum (CUSUM) chart is based on a sum of current and past sample statistics. When individual observations are taken at widely spaced intervals, there is relatively little data from the process, so using a chart such as the EWMA chart or the CUSUM chart will provide efficient use of the scarce data.

A useful approach to improving the ability to detect process changes is to use a variable sampling interval (VSI) control chart instead of the traditional fixed sampling interval (FSI) control chart. In a VSI control chart, the sampling interval is varied as a function of the control statistic. A short sampling interval is used whenever there is some indication that a process parameter may have changed, and a long sampling interval is used if there is no indication of a parameter change. Most work on developing VSI control charts has been done for the problem of monitoring (see, e.g., Reynolds Amin, Arnold, and Nachlas (1988), Reynolds, Amin, and Arnold (1990), and Reynolds (1996a, b), and Stoumbos and Reynolds (1996, 1997, 2001)). Very little work has been done on VSI charts for monitoring and . Chengular, Arnold, and Reynolds (1989) considered VSI Shewhart charts for monitoring and when samples of n > 1 are taken. Shamma and Amin (1993) considered the performance of a single VSI EWMA chart that can be used to monitor and .

We have two main objectives in this paper. The first objective is to consider various combinations of Shewhart and EWMA charts in order to determine which combinations are most effective for detecting shifts in and/or when single observations are taken. We show that there are much better alternatives than the traditional approach of using the X and MR charts. The second objective is to show how the use of the VSI feature can offer substantial additional improvements in the ability to detect parameter shifts.

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