6.
Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts 6.3.2. What are Variables Control Charts?


Samples are Individual Measurements  
Moving range used to derive upper and lower limits 
Control charts for individual measurements, e.g., the sample
size = 1, use the moving range of two successive observations
to measure the process variability.
The moving range is defined as $$ MR_i = x_i  x_{i1} \, , $$ which is the absolute value of the first difference (e.g., the difference between two consecutive data points) of the data. Analogous to the Shewhart control chart, one can plot both the data (which are the individuals) and the moving range. 

Individuals control limits for an observation  For the control chart for individual measurements, the lines plotted are: $$ \begin{eqnarray} UCL & = & \bar{x} + 3\frac{\overline{MR}}{1.128} \\ \mbox{Center Line} & = & \bar{x} \\ LCL & = & \bar{x}  3\frac{\overline{MR}}{1.128} \, , \end{eqnarray} $$ where \(\bar{x}\) is the average of all the individuals and \(\overline{MR}\) is the average of all the moving ranges of two observations. Keep in mind that either or both averages may be replaced by a standard or target, if available. (Note that 1.128 is the value of \(d_2\) for \(n = 2\).  
Example of moving range 
The following example illustrates the control chart for individual
observations. A new process was studied in order to monitor flow
rate. The first 10 batches resulted in


Limits for the moving range chart  This yields the parameters below. $$ \begin{eqnarray} UCL & = & \bar{x} + 3\frac{\overline{MR}}{1.128} = 50.81 + 3\frac{1.8778}{1.128} = 55.8041 \\ & & \\ & & \\ \mbox{Center Line} & = & \bar{x} = 50.81 \\ & & \\ LCL & = & \bar{x}  3\frac{\overline{MR}}{1.128} = 50.81  3\frac{1.8778}{1.128} = 45.8159 \, . \end{eqnarray} $$  
Example of individuals chart 
The control chart is given below
The process is in control, since none of the plotted points fall outside either the \(UCL\) or \(LCL\). 

Alternative for constructing individuals control chart 
Note: Another way to construct the individuals chart is by
using the standard deviation. Then we can obtain the chart from
$$ \bar{x} \pm 3s/c_4 \, .$$
