Applied Statistics and Probability for Engineers

16-8 ATTRIBUTE CONTROL CHARTS 625

16-8 ATTRIBUTE CONTROL CHARTS

16-8.1 PChart (Control Chart for Proportions)

Often it is desirable to classify a product as either defective or nondefective on the basis of com-

parison with a standard. This classification is usually done to achieve economy and simplicity in

the inspection operation. For example, the diameter of a ball bearing may be checked by deter-

mining whether it will pass through a gauge consisting of circular holes cut in a template. This

kind of measurement would be much simpler than directly measuring the diameter with a device

such as a micrometer. Control charts for attributes are used in these situations. Attribute control

charts often require a considerably larger sample size than do their variable measurements coun-

terparts. In this section, we will discuss the fraction-defectivecontrol chart,or Pchart.

Sometimes the Pchart is called the control chart for fraction nonconforming.

Suppose Dis the number of defective units in a random sample of size n. We assume that

Dis a binomial random variable with unknown parameter p. The fraction defective

of each sample is plotted on the chart. Furthermore, the variance of the statistic is

Therefore, a Pchart for fraction defective could be constructed using pas the center line and

control limits at

(16-22)

However, the true process fraction defective is almost always unknown and must be estimated

using the data from preliminary samples.

Suppose that mpreliminary samples each of size nare available, and let Dibe the number

of defectives in the ith sample. The is the sample fraction defective in the ith

sample. The average fraction defective is

(16-23)

Now may be used as an estimator of P pin the center line and control limit calculations.

P

1

ma

m

i 1

Pˆi

1

mna

m

i 1

Di

PˆiDi n

UCLp 3

B

p 11 p 2

n^ LCLp^3 B

p 11 p 2

n

^2 Pˆ

p 11 p 2

n

Pˆ

Pˆ

D

n

PChart

The center line and upper and lower control limits for the Pchart are

(16-24)

where is the observed value of the average fraction defective.p

UCLp 3

B

p 11 p 2

n^ CLp^ LCLp^3 B

p 11 p 2

n

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