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Control charts, one of the powerful tools of the Statistical Process Control (SPC)
techniques, are widely used in industry for process improvement and for estimating
parameters or monitoring the variability of a given process. Shewhart X charts are
originally developed by Shewart (1931). The limits of the chart are known as Upper
Control Limit (UCL) and Lower Control Limit (LCL). He suggested two control
limits, i.e., UCL = X ? 3 n and LCL = X ? 3 n , where X is the target
mean, ’ is the population standard deviation and n is the sample size. The width of
control limits is usually taken as 3.
The process is assumed to be out of control when the sample average falls beyond
these limits. The Shewhart X chart is more useful to detect large shifts (> 2) in
process mean. One of the assumptions of implementing the X chart is that the process outputs must be IID, but usually there is some correlation among the data. When this
correlation builds up automatically in the entire process, this phenomenon is called
autocorrelation. The observations from the process output are usually positively
correlated in most of the cases. In this case, if the current observation is on one side
of the mean, the next observation will most likely be found on the same side of the
mean. Positively correlated data are characterized by runs above and below the mean.
Positive correlation is more often encountered in practice than negative
autocorrelation. The observations shown in Figure 1 are positively correlated.
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