Applied Statistics and Probability for Engineers

16-8 ATTRIBUTE CONTROL CHARTS 627

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0.1

0.2

0.3

0.4

0.6

LCL

p

UCL

Sample number

Sample fraction defective,

p

0.5

Figure 16-16 Pchart

for a ceramic substrate.

The center line and upper and lower control limits on the Uchart are

(16-27)

where is the average number of defects per unit.u

UCLu 3

B

u

n^ CLu^ LCLu^3 B

u

n

UChart

16-8.2 UChart (Control Chart for Defects per Unit)

It is sometimes necessary to monitor the number of defects in a unit of product rather than

the fraction defective. Suppose that in the production of cloth it is necessary to control the

number of defects per yard or that in assembling an aircraft wing the number of missing riv-

ets must be controlled. In these situations we may use the control chart for defects per unit,

or the Uchart.Many defects-per-unit situations can be modeled by the Poisson distribution.

If each sample consists of nunits and there are Ctotal defects in the sample,

is the average number of defects per unit. A Uchart may be constructed for such data.

If the number of defects in a unit is a Poisson random variable with parameter , the mean and

variance of this distribution are both. Each point on the chart is U, the average number of defects

per unit from a sample of nunits. Therefore, the mean of Uis and the variance of Uis n.

(16-25)

If there are mpreliminary samples, and the number of defects per unit in these samples are U 1 ,

U 2 , ..., Um, the estimator of the average number of defects per unit is

(16-26)

The parameters of the Uchart are defined as follows.

U

1

ma

m

i 1

Ui

LCL 3

B



n

UCL 3

B



n

U

C

n

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