UCL =  m_w   + ks_w
                                   Center Line =  m_w
                                      LCL =  m_w   - ks_w

where k is the distance of the control limits from the center line, expressed in terms of standard deviation units. When k is set to 3, we speak of 3-sigma control charts. 

Historically, k = 3 has become an accepted standard in industry. 

The centerline is the process mean, which in general is unknown. We replace it with a target or the average of all the data. The quantity that we plot is the sample average, xbar. The chart is called the xbar chart. 

We also have to deal with the fact the s is in general unknown. Here we replace s_w with a given standard value, or we estimate it by a function of the average S. This is obtained by averaging the individual standard deviations that we calculated from each of m preliminary (or present) samples, each of  size n.  This function will be discussed shortly. 

It is equally important to examine the standard deviations in ascertaining whether the process is in control.  There is, unfortunately, a slight problem involved when we work with the usual estimate of s.  The following discussion will bring this out: 

However, s, the sample standard deviation is not an unbiased estimator of s.   If the underlying distribution is normal  than s actually estimates c₄ s, where  c₄   is a constant that depends on the sample size n. This constant is tabulated in most text books on Statistical Quality Control and may be calculated using

To compute this we  need a non-integer factorial, which is defined for n/2 as follows:

Mean and standard deviation of the estimates
 
 
 
 

Control limits vs. specifications
 
 
 
 
 

How many samples are needed?
 
 
 
 
 
 
 
 
 

When do we recalculate control limits?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

General rules for detecting out of control or non-random situaltions
 
 
 
 
 
 
 
 
 
 
 
 
 
 

With this definition the reader should have no problem verifying that the c₄ factor for  n = 10 is .9727 

So  the mean or expected value of the sample standard deviation is c₄s.

The standard deviation of the sample standard deviation is 

What are the differences between control limits and specification limits ? 

Control Limits are used to determine if the process is producing consistent output. 

Specification  Limits are used to determine if the product will function in the intended fashion. 

How many data points are needed to set up a control chart?

Shewhart gave the following rule of thumb:

 "It has also been observed that a person would seldom 
 if ever be justified in concluding that a state of 
 statistical control of a given repetitive operation 
 or production process has been reached until he had 
 obtained, under presumably the same essential 
 conditions, a sequence of not less than twenty five 
 samples of size four that are in control."

When do we recalculate control limits? 

Since a control chart “compares” the current performance of the response to the past performance of the response, changing the control limits frequently would negate any usefulness. 

So, only change your control limits if you have a valid, compelling reason for doing so.  Some examples of reasons: 

* When you have at least 30 more data points to add to the chart and there have been no known changes to the process 

 - you get a better estimate of the variability 

* If a major process change occurs and effects the way your process runs. 

* If a known, preventable act changes the way the tool or process would behave (power goes out, consumable is corrupted or bad quality, etc.) 

WECO stands for Western Electric Company  Rules 
 

       Any Point Above +3 Sigma 
 -----------------------------------------------------    +3 s LIMIT 
       2 Out of the Last 3 Points Above +2 Sigma 
 -----------------------------------------------------    +2 s LIMIT 
       4 Out of the Last 5 Points Above +1 Sigma 
 -----------------------------------------------------    +1 s LIMIT 
       8 Consecutive Points on This Side of Control Line 
===================================   CENTER LINE 
       8 Consecutive Points on This Side of Control Line 
 -----------------------------------------------------    -1 s LIMIT 
       4 Out of the Last 5 Points Below - 1 Sigma 
------------------------------------------------------   -2 s LIMIT 
       2 Out of the Last 3 Points Below -2 Sigma 
 -----------------------------------------------------    -3 s LIMIT 
       Any Point Below -3 Sigma 

Trend Rules:   6 in a row trending up or down. 
                       14 in a row alternating up and down 

The WECO rules are based on probability.  We know that, for a normal distribution, the probability of encountering a point outside ± 3s is 0.3%.  This is a rare event.  Therefore, if we observe a point outside the control limits, we conclude the process has shifted and that it is unstable. Similarly, we can identify other events which are equally rare and use them as flags for instability.  The probability of observing two points out of three in a row between 2s and 3s and the probability of observing four points out of five in a row between 1s and 2s are also about 0.3%. 

Note: While the WECO rules increase a Shewhart charts sensitivity to trends or drifts in the mean, there is a severe downside to adding the WECO rules to an ordinary Shewhart control chart that the user should be aware of. When following the standard Shewhart "out of control" rule (i.e. signal if and only if you see a point beyond the plus or minus 3 sigma control limits) you will have "false alarms" every 371 points, on the average (see the description of Average Run Length or ARL on the next page).  Adding the WECO rules increases the frequency of false alarms to about once in every 91.75 points, on the average (see Champ and Woodall, 1987). The user has to decide whether this price is worth paying (some users add the WECO rules, but take them "less seriously" in terms of the effort put into troubleshooting activities when out of control signals occur.) 
 

With this background, the next page will describe how to construct Shewhart variables control charts.