Average Run Length (ARL) - Introduction

Average Run Length (ARL) - Introduction

Average Run Length (ARL) characteristics are very useful to compare the performance of control charts and determine optimal parameter settings for EWMA and CUSUM time weighted control charts.

The ARL value for a shift in mean = 0 sigma is the in-control run length and is denoted as ARL₀. ARL₀ is 1/α, where α is the type I false alarm probability, so this should be as large as possible minimizing the likelihood that an out-of-control signal is a false alarm. In a Shewhart Individuals Control Chart, ARL₀ = 1/α = 1/(0.00135*2) = 370.4. On average, we will see a false alarm once every 370 observations. Note that this is a mean of a geometric distribution, so in practice the actual ARL₀ will vary widely with the standard deviation approximately equal to the mean value.

When we have a sustained shift in mean > 0, the ARL value is the "out-of-control" run length and is denoted as ARL₁. ARL₁ = 1/(1-β), where β is the type II miss probability and (1-β) is the power to detect. This should be as small as possible so that a shift in process mean is quickly detected.

The calculations for ARL are quite complex, involving either Exact, Markov Chain approximation or Monte Carlo simulation to solve. The SigmaXL ARL Templates take care of these calculations and are easy to use. If Monte Carlo simulation is used, additional Run Length standard deviation and percentile statistics are reported. Monte Carlo results will take some time and vary slightly every time they are run. With 1000 (1e3) replications it will be fast, approx. 10 seconds, but will have an ARL₀ error of approximately +/- 10%; 10,000 (1e4) replications will take about a minute, with an ARL₀ error of +/- 3.2%; 100,000 (1e5) replications will take about ten minutes, with an ARL₀ error = +/- 1%. More than 1e5 replications will be very time consuming and may run into memory limitations, so is not recommended.

When using EWMA or CUSUM charts, we typically set the parameters to minimize the ARL₁ to give rapid detection for a small shift in mean of 1 sigma. Shewhart charts are typically used when trying to rapidly detect a large shift in mean of >= 3 sigma. Tests for special causes may be used with Shewhart to improve the small shift performance, but they give poor ARL₀ performance resulting in frequent false alarms.

Subgroups improve the small shift performance of a Shewhart chart without impacting the ARL₀ rate and, if possible, should be used. The ARL for subgroup averages is adjusted by using the sigma of averages, sigma/$\sqrt{\mathrm{n}}$. For example, with a subgroup size of 4, the ARL₁ values at shift in mean = 1 will match the ARL performance of an Individuals chart with shift in mean = 2 sigma.

Note that subgroups for CUSUM and EWMA are not available in SigmaXL.

The problem of robustness to nonnormality can also be considered by using the Pearson family to specify any value of skewness and kurtosis and estimate the ARLs. For further details, see Appendix: Pearson Family of Distributions.

The average run length calculators are for two-sided charts with zero-state, i.e., the shift is assumed to occur at the start. The parameters (mean, standard deviation and proportion) are also assumed to be known. This will not likely be the case in use, but is still useful for determining parameter settings and comparison of ARL across chart types.

Due to the complexity of calculations, SigmaXL must be loaded and appear on the menu in order for the ARL templates to function. Do not add or delete rows or columns in these templates.