Control charts are crucial tools for any Six Sigma efforts. The concept of SPC, Statistical Process Control is centrifugal to the success of any Six Sigma projects and if you don’t use Control Charts as part of SPC, what have you used then? So easy and informative are these charts that these figure in the wish-list of charts of most businesses. The biggest benefit is --- If plotted correctly, these charts help you differentiate between random and assignable causes of variations, i.e. common and special causes of variations respectively.

A basic fundamental --- Control charts operate on the specifications of control limits, which are often given by the data of the process. Walter Shewhart had said that these control limits should be 3 times standard deviation from the center line in order to reduce the probability of error happening in detecting the special causes of variation.

For all practical applications though, especially when you use Statistical Software Applications like Minitab, you would note a concept called control chart constants. Let us in this read try to understand how does one calculate control chart constants for various sub-group variables charts and yes, by now we already know that X bar – R, X bar – S and I-MR are the sub-group charts for our use.

Control limits for X bar – R chart

Let us assume a sub-group size of 4, a grand average of 3.5 and a grand range average of 0.3. Let us use these values and find out the control limits. For this, I need a control chart constant table, which most Belts in Six Sigma niche possess.

Formulas first

For Range Charts –

LCL = D3 * R bar

UCL = D4 * R bar

For Average Charts –

LCL = X dbar – (A2 * R bar)

UCL = X dbar + (A2 * R bar)

Corresponding the sub-group size of 4 with the control chart constants table, the values are

D3 = 0

D4 = 2.28

A2 = 0.729

Substituting them with the values given to us,

For Range Charts

Centre line = 0.3

LCL = 0

UCL = 2.28 * 0.3 = 0.684

Thus the control limits for the range chart are {0, 0.684}

For Average Charts

Centre line = 3.5

LCL = 3.5 – (0.73 * 0.3) = 3.28

UCL = 3.5 + (0.73 * 0.3) = 3.72

Thus, the control limits for the Average chart are {3.28, 3.72}

Control limits for X bar – S chart

Let us assume a sub-group size of 12, a grand average of 3.5 and a sample standard deviation average of 0.3. Let us use these values and find out the control limits. For this, I need a control chart constant table, which most Belts in Six Sigma niche possess.

Formulas first

For Range Charts –

LCL = B3 * s bar

UCL = B4 * s bar

For Average Charts –

LCL = X dbar – (A3 * s bar)

UCL = X dbar + (A3 * s bar)

For a sub-group size of 12, looking into the Control Charts Constants for the Standard Deviations section,

B3 = 0.35

B4 = 1.65

A3 = 0.886

Substituting them into the formulas

For Sigma Chart

LCL = 0.35 * 0.3 = 0.11

UCL = 1.65 * 0.3 = 0.50

Thus the control limits for the sigma chart are {0.11, 0.50}

For Average Chart

Centre line = 3.5

LCL = 3.5 – (0.89 * 0.3) = 3.23

UCL = 3.5 + (0.89 * 0.3) = 3.77

Thus the control limits for the sigma chart are {3.23, 3.77}

Thus, the control limits for the Average chart are {3.28, 3.72}

Control limits for I-MR Chart

IMR Charts are slightly different from other variables charts as the concept of sub-groups doesn’t really apply in here, as the sub-group size is 1.

Formulas for control limits

For Moving Range Charts

LCL = 0

UCL = 3.27 * R bar = 3.27 * 0.3 = 0.98

For Individuals Charts

LCL = X bar – (E2 * R bar)

UCL = X bar + (E2 * R bar)

LCL = 3.5 – (2.67 * 0.3) = 2.699

UCL = 3.5 + (2.67 * 0.3)= 4.30

Thus, the control limits for the Individuals charts are {2.7, 4.3}.

Once you know the control charts constants formulas, calculating the control limits is not as tough as you thought it would be. Once you have these control limits and individual values, plotting a control chart in Excel or any other statistical software is not tough either.

Summary

Knowing how to calculate Control limits is not tough. Yes – Knowing which chart to use when is really important. The ground rule is --- Use IMR for sub-group size 1, X bar – R for sub-group sizes 2-9 and X bar – S for sub-group sizes greater than 10. Apart from these basic conditions, there is the basic assumption of normality you need to consider for IMR Charts.

As easy as it gets…. Eshna