Note: See Computing Average Run Lengths for s Charts in the SAS/QC Sample Library.
This example illustrates how you can compute the average run length of an s chart. The data used here are the power measurements in the data set Turbine
, which is introduced in Creating Standard Deviation Charts from Raw Data.
The in-control average run length of a Shewhart chart is , where p is the probability that a single point exceeds its control limits. Since this probability is saved as the value of the variable
_ALPHA_
in an OUTLIMITS= data set, you can compute ARL for an s chart as follows:
title 'Average In-Control Run Length'; proc shewhart data=Turbine; schart KWatts*Day / outlimits=Turblim nochart; data ARLcomp; keep _var_ _sigmas_ _alpha_ arl; set Turblim; arl = 1 / _alpha_; run;
The data set ARLcomp
is listed in Output 17.30.1, which shows that the ARL is equal to 358.
Output 17.30.1: The Data Set ARLcomp
Average In-Control Run Length |
_VAR_ | _ALPHA_ | _SIGMAS_ | arl |
---|---|---|---|
KWatts | .002792725 | 3 | 358.073 |
To compute out-of-control average run lengths, define f as the slippage factor for the process standard deviation , where f > 1. In other words, the “shifted” standard deviation to be detected by the chart is . The following statements compute the ARL as a function of f:
data ARLshift; keep f f_std p arl_f; set Turblim; df = _limitn_ - 1; do f = 1 to 1.5 by 0.05; f_std = f * _stddev_; low = df * ( _lcls_ / f_std )**2; upp = df * ( _ucls_ / f_std )**2; p = probchi( low, df ) + 1 - probchi( upp, df ); arl_f = 1 / p; output; end; run;
The data set ARLshift
is listed in Output 17.30.2. For example, on average, 53 samples are required to detect a ten percent increase in (a shifted standard deviation of approximately 219). The computations use the fact that has a distribution with degrees of freedom, assuming that the measurements are normally distributed.
Output 17.30.2: The Data Set ARLshift
Average Run Length Analysis |
f | f_std | p | arl_f |
---|---|---|---|
1.00 | 198.996 | 0.00279 | 358.073 |
1.05 | 208.945 | 0.00758 | 131.922 |
1.10 | 218.895 | 0.01875 | 53.322 |
1.15 | 228.845 | 0.03984 | 25.102 |
1.20 | 238.795 | 0.07388 | 13.535 |
1.25 | 248.745 | 0.12239 | 8.171 |
1.30 | 258.694 | 0.18475 | 5.413 |
1.35 | 268.644 | 0.25834 | 3.871 |
1.40 | 278.594 | 0.33923 | 2.948 |
1.45 | 288.544 | 0.42298 | 2.364 |
1.50 | 298.494 | 0.50546 | 1.978 |