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Control Charts for Attributes
By the attribute method we mean the measurement of quality through noting the presence or absence of some characteristic in each of the units, and counting how many units do not posses the quality characteristic. The advantage of the attribute method is that a single chart can be set up for several characteristics, whereas a variables chart must be set up for eachof the characteristics with an accompanying chart for controlling variability.
p-Chart for Fraction Nonconforming
This is also known as the proportion chart. The p-chart configuration is intended to evaluate the process in terms of the proportion or fraction of the total units in a sample in which a designated classification event occurs. This designated classification event may be a deviationmore than the specified on a measurement scale, quasi-measurement scale, go or not-go gauge, judgment, etc. It could also be a nonconformity, defect, blemish, presence or absence of some characteristic, etc. The classification may also be based on several characteristics. Instead of proportions, if percents are used, then the p-chart will stand for percent chart.
Let p stands for the fractionnonconforming of the process and [pic] be the sample fraction nonconforming computed as the ratio of the number of nonconforming units d to the sample size n. That is, [pic]= d/n. Let d follows a binomial distribution with parameters n and p ie
[pic]
It is further known that the mean and variance of [pic] are p and p(1-p)/n respectively. If the true value of p is known, the control limits become[pic]
with the central line being at p. Here p could be a standard value p'.
Suppose that the true fraction nonconforming is unknown. As usual, it is assumed that the total number of units tested from the process is subdivided into m rational subgroups consisting of n1, n2, n3...ni...nm units respectively and a value of the proportion defective is computed for each subgroup. For convenience, oneassumes that the subgroup sizes are all equal. If di is the number of defectives found in the ith subgroup, then the estimate of p is [pic]i = di/n. The average of various [pic]i  values is [pic].  The control limits are set at
[pic]
For example, consider the following data on the number of defectives obtained for 50 subgroups of 100 resistors drawn from a process.

Table 3.12 Nonconformingresistors in various subgroups

i |di |[pic]i |i |di |[pic]i |i |di |[pic]i |i |di |[pic]i |i |di |[pic]i | |1 |0 |0.00 |11 |0 |0.00 |21 |1 |0.01 |31 |1 |0.01 |41 |1 |0.01 | |2 |0 |0.00 |12 |2 |0.02 |22 |2 |0.02 |32 |3 |0.03 |42 |1 |0.01 | |3 |2 |0.02 |13 |1 |0.01 |23 |0 |0.00 |33 |0 |0.00 |43 |3 |0.03 | |4 |0 |0.00 |14 |1 |0.01 |24 |1 |0.01 |34 |1 |0.01 |44 |2 |0.02 | |5 |1 |0.01 |15 |1 |0.01 |25|1 |0.01 |35 |2 |0.02 |45 |1 |0.01 | |6 |0 |0.00 |16 |0 |0.00 |26 |0 |0.00 |36 |2 |0.02 |46 |1 |0.01 | |7 |2 |0.02 |17 |0 |0.00 |27 |0 |0.00 |37 |0 |0.00 |47 |0 |0.00 | |8 |2 |0.02 |18 |0 |0.00 |28 |1 |0.01 |38 |2 |0.02 |48 |0 |0.00 | |9 |1 |0.01 |19 |2 |0.02 |29 |0 |0.00 |39 |2 |0.02 |49 |0 |0.00 | |10 |1 |0.01 |20 |3 |0.03 |30 |0 |0.00 |40 |1 |0.01 |50 |2 |0.02 | |The table given below also givesthe [pic]i values. The value of [pic] is 0.01. The control limits are found as
[pic].
Figure 3.44 provides the MINITAB p-chart output for the above data.
[pic]
Figure 3.44 MINITAB p-chart Output
If the computed value for LCL is negative, it is set at zero. This means that there is no 'control' exercised to detect any quality improvement. If the subgroup sizes are unequal, then p is estimatedas
[pic]
and the (varying) control limits are given by
[pic]
Alternatively, an 'average' sample size [pic] could be used. One can also plot the standardised value of pi against the control limits ±3.
Choice of Subgroup Size
Let p1 be the process average fraction defective (considered as acceptable) and suppose that we wish to detect a kσ level shift in a p-chart. That is, the process...

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