6.
Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts


During the 1920's, Dr. Walter A. Shewhart proposed a general model for control charts as follows:  
Shewhart Control Charts for variables 
Let \(w\)
be a sample statistic that measures some continuously
varying quality characteristic of interest (e.g., thickness), and
suppose that the mean of \(w\)
is \(\mu_w\),
with a standard deviation of \(\sigma_w\).
Then the center line, the \(UCL\), and the \(LCL\) are
Center Line = \(\mu_w\) \(LCL = \mu_w  k \sigma_w\) 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, \(\overline{X}\). The chart is called the \(\overline{X}\) chart. We also have to deal with the fact that \(\sigma\) is, in general, unknown. Here we replace \(\sigma_w\) with a given standard value, or we estimate it by a function of the average standard deviation. 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 estimator of \(\sigma\). The following discussion will illustrate this. 

Sample Variance  If \(\sigma^2\) is the unknown variance of a probability distribution, then an unbiased estimator of \(\sigma^2\) is the sample variance, $$ s^2 = \frac{\sum_{i=1}^n \left( x_i  \overline{x} \right)^2}{n1} \, . $$ However, \(s\), the sample standard deviation, is not an unbiased estimator of \(\sigma\). If the underlying distribution is normal, then \(s\) actually estimates \(c_4 \cdot \sigma\), where \(c_4\) 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  
\(c_4\) factor 
$$ c_4 = \sqrt{\frac{2}{n1}} \frac{ \left( \frac{n}{2}1 \right)!} {\left( \frac{n1}{2}1 \right)!} \, . $$
Note that in some sources the formula is given in terms of \(\sigma = c_4 \cdot s\), in which case \(c_4\) will be the reciprocal of the formula given above. 

Fractional Factorials 
To compute this we need a noninteger factorial, which
is defined for \(n/2\)
as follows:
$$ \left( \frac{n}{2} \right) ! =
\left( \frac{n}{2} \right)
\left( \frac{n}{2}  1 \right)
\left( \frac{n}{2}  2 \right)
\cdots
\left( \frac{1}{2} \right) \sqrt{\pi} \, . $$
For example, let \(n=7\). Then \(n/2 = 7/2 = 3.5\) and $$ \left( \frac{7}{2} \right) ! = (3.5)! = (3.5)(2.5)(1.5)(0.5)(1.77246) = 11.632 \, . $$ With this definition the reader should have no problem verifying that the \(c_4\) factor for \(n=10\) is 0.9727. 

Mean and standard deviation of the estimators 
So the mean or expected value of the sample standard deviation
is \(c_4 \cdot \sigma\).
The standard deviation of the sample standard deviation is $$ \sigma_s = \sigma \sqrt{1  c_4^2} \, . $$ 

What are the differences between control limits and specification limits ?  
Control limits vs. specifications 
Control Limits are used to determine if the process is in a state of
statistical control (i.e., 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?  
How many samples are needed? 
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."It is important to note that control chart properties, such as false alarm probabilities, are generally given under the assumption that the parameters, such as \(\mu\) and \(\sigma\), are known. When the control limits are not computed from a large amount of data, the actual properties might be quite different from what is assumed (see, e.g., Quesenberry, 1993). 

When do we recalculate control limits?  
When do we recalculate control limits? 
Since a control chart "compares" the current performance of the
process characteristic to the past performance of this
characteristic, 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:


What are the WECO rules for signaling "Out of Control"?  
General rules for detecting out of control or nonrandom situaltions 
WECO stands for Western Electric Company Rules
Any Point Above +3 Sigma
 
WECO rules based on probabilities  The WECO rules are based on probability. We know that, for a normal distribution, the probability of encountering a point outside \(\pm 3 \sigma\) 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 is unstable. Similarly, we can identify other events that are equally rare and use them as flags for instability. The probability of observing two points out of three in a row between \(2 \sigma\) and \(3 \sigma\) and the probability of observing four points out of five in a row between \(1 \sigma\) and \(2 \sigma\) are also about 0.3 %.  
WECO rules increase false alarms 
Note: While the WECO rules increase a Shewhart chart's
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 understand. 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. 