Calculation of moving range

The difference between data point, ${\displaystyle x_{i}}$, and its predecessor, ${\displaystyle x_{i-1}}$, is calculated as ${\displaystyle {MR}_{i}={\big |}x_{i}-x_{i-1}{\big |}}$. For ${\displaystyle m}$ individual values, there are ${\displaystyle m-1}$ ranges.

Next, the arithmetic mean of these values is calculated as

${\displaystyle {\overline {MR}}={\frac {\sum _{i=2}^{m}{MR_{i}}}{m-1}}}$

If the data are normally distributed with standard deviation ${\displaystyle \sigma }$ then the expected value of ${\displaystyle {\overline {MR}}}$ is ${\displaystyle d_{2}\sigma =2\sigma /{\sqrt {\pi }}}$, the mean absolute difference of the normal distribution.

Calculation of moving range control limit

The upper control limit for the range (or upper range limit) is calculated by multiplying the average of the moving range by 3.267:

${\displaystyle UCL_{r}=3.267{\overline {MR}}}$.

The value 3.267 is taken from the sample size-specific D₄ anti-biasing constant for n=2, as given in most textbooks on statistical process control (see, for example, Montgomery^[2]: 725).

Calculation of individuals control limits

First, the average of the individual values is calculated:

${\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{m}{x_{i}}}{m}}}$.

Next, the upper control limit (UCL) and lower control limit (LCL) for the individual values (or upper and lower natural process limits) are calculated by adding or subtracting 2.66 times the average moving range to the process average:

${\displaystyle UCL={\overline {x}}+2.66{\overline {MR}}}$.

${\displaystyle LCL={\overline {x}}-2.66{\overline {MR}}}$

The value 2.66 is obtained by dividing 3 by the sample size-specific d₂ anti-biasing constant for n=2, as given in most textbooks on statistical process control (see, for example, Montgomery^[2]: 725).

Creation of graphs

Once the averages and limits are calculated, all of the individuals data are plotted serially, in the order in which they were recorded. To this plot is added a line at the average value, x and lines at the UCL and LCL values.

On a separate graph, the calculated ranges MR_i are plotted. A line is added for the average value, MR and second line is plotted for the range upper control limit (UCL_r).

Analysis

The resulting plots are analyzed as for other control charts, using the rules that are deemed appropriate for the process and the desired level of control. At the least, any points above either upper control limits or below the lower control limit are marked and considered a signal of changes in the underlying process that are worth further investigation.