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, the D follows a binomial distribution with parameters n and p
|The binomial distribution model for number of defectives in a sample||
|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
|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: