6.
Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts 6.3.3. What are Attributes Control Charts?


\(p\) is the fraction defective in a lot or population 
The proportion or fraction nonconforming (defective) in a
population is defined as the ratio of the number of nonconforming
items in the population to the total number of items in that
population. The item under consideration may have one or more
quality characteristics that are inspected simultaneously. If at
least one of the characteristics does not conform to standard, the
item is classified as nonconforming.
The fraction or proportion can be expressed as a decimal, or, when multiplied by 100, as a percent. The underlying statistical principles for a control chart for proportion nonconforming are based on the binomial distribution. Let us suppose that the production process operates in a stable manner, such that the probability that a given unit will not conform to specifications is \(p\). Furthermore, we assume that successive units produced are independent. Under these conditions, each unit that is produced is a realization of a Bernoulli random variable with parameter \(p\). If a random sample of \(n\) units of product is selected and if \(D\) is the number of units that are nonconforming, then \(D\) follows a binomial distribution with parameters \(n\) and \(p\) so that 

The binomial distribution model for number of defectives in a sample 
$$ P(D=x) = \left( \begin{array}{c} n \\ x \end{array} \right)
p^x (1p)^{nx} \, , \,\,\,\,\,\, x = 0, \, 1, \, \ldots, \, n \, , $$


\(p\) control charts for lot proportion defective  If the true fraction conforming \(p\) is known (or a standard value is given), then the center line and control limits of the fraction nonconforming control chart is $$ \begin{eqnarray} UCL & = & p + 3\sqrt{\frac{p(1p)}{n}} \\ \mbox{Center Line} & = & p \\ LCL & = & p  3\sqrt{\frac{p(1p)}{n}} \, . \end{eqnarray} $$ When the process fraction (proportion) \(p\) is not known, it must be estimated from the available data. This is accomplished by selecting \(m\) preliminary samples, each of size \(n\). If there are \(D_i\) defectives in sample \(i\), the fraction nonconforming in sample \(i\) is $$ \hat{p}_i = \frac{D_i}{n} \, , \,\,\,\,\, i = 1, \, 2, \, \ldots, \, m \, , $$ and the average of these individuals sample fractions is $$ \bar{p} = \frac{\sum_{i=1}^m D_i}{mn} = \frac{\sum_{i=1}^m \hat{p}_i}{m} \, . $$ The \(\bar{p}\) is used instead of \(p\) in the control chart setup.  
Example of a \(p\)chart 
A numerical example will now be given to illustrate the above
mentioned principles. The location of chips on a wafer is measured
on 30 wafers.
On each wafer 50 chips are measured and a defective is defined whenever a misregistration, in terms of horizontal and/or vertical distances from the center, is recorded. The results are


Sample proportions control chart 
The corresponding control chart is given below:
