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X-Bar, R Bar Control Charts and Zones
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Zones and WECO, (Western Electric Company Rules) Using Pre-Control limits Statistical Process Control Information

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The two control charts most often associated with statistical process control, (SPC), are the X-Bar and Range or R charts. X-Bar is just a fancy word for mean, also known as average. Range charts are just that, the range that the sample lay between. You should always set up the Range or R-Bar control charts first.

The charts are compared with control limits. Control limits, are typically calculated based on the number of original samples, or first samples, taken on any given process. (see upper control limit, lower control limit and Pre-Control Limit) The number of samples to start with will vary based on any given process, but should be sufficient to represent the population of the widget being manufactured. Motorola in the 1980ís require their suppliers to supply 75 pieces of data for first article inspection.

Once you have taken the data using a stratified sampling plan, the control limits are established by using a factor scale that will estimate the sigma, or standard deviation. Using this estimate, you can add and subtract 3 times one sigma to get your control limits and zones.

Remember that the bell of a bell shaped curve is +/- 3 sigma. A different factor is used for the range control limits. Note that if the number of observations are less than 7, then the lower control limit on the range charts is always zero, (0). For more information on how to calculate upper control limits and lower control limits, including the factor tables, see our glossary.

Once you have established your control limits, periodically you take more samples, usually 2 to 5 more sample pieces, and calculate the mean and range and plot them on the chart.

For additional information see ZeroRejects and Statistical Process Control, (SPC)

A typical X-Bar chart should show a saw toothed pattern when the data points are connected as seen below. Note that the control limits are the top most and bottom most line on the chart below.

Typical X-Bar Saw Tooth Pattern

Zones and
WECO, (Western Electric Company Rules)

To create the zones, first find the upper and lower control limits and then divide the section of the chart between the UCL, (upper control limit), and the mean and the LCL, (lower control limit), and the mean evenly. You do not calculate the zones individually.

zone A = mean +/- 3 sigma
zone B = mean +/- 2 sigma
zone C = mean +/- 1 sigma

X-Bar Chart with WECO Zones

The established way to trend these charts is by using the WECO, (Western Electric Company), rules


  • A single point falls outside the 3 Sigma limit, i.e. beyond Zone A
  • Two out of three successive points fall in Zone A or beyond, the odd point may be anywhere
  • Four out of Five successive points fall in Zone B or beyond, the odd point may be anywhere
  • Eight successive points fall in Zone C or beyond

For additional information see What are the WECO rules for signaling "Out of Control"?

Another way to envision the zones are shown below.
X-Bar WECO Rules VS Sigma Zones

There are many different rules that one could use. These type of rules are to evaluate trends. Since every process can be different, the rules are sometimes modified as to not stop production for an out of control condition when the design margin is still being adhered to.

Pre-Control Limits

Pre-Control limits are sometimes used instead of the calculated upper and lower control limits for control charts. This is done due to the fact that some processes may typically run skewed, or shifted from the mean or median. There are established formulas to calculate pre-control, but as with all cases, to include the WECO rules, you may have to modify them to fit you particular process.

Using ZeroRejects, you can easily see where your data is in relation to your  specification and design margin.

For addition information...

Control Charts as a Tool in Data Quality Improvement (U.S. Department of Transportation)

* Reference Quality Planning and Analysis - Juran/Gryan

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Last Updated: Monday, 17-Apr-06 03:37:20 PDT