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274 Part IV: Quality AssurancePart IV.B.4time, or both, we do not want to take the measurements. We may prefer to use amore economical method such as a go/no-go gauge. Therefore, it is important thatwe study control charts that are appropriate for quality characteristics that can’t bemeasured numerically. Such control charts are called control charts for attributes. Inthis section, we study various control charts for attributes for detecting large processshifts, which usually occur in phase I implementation of SPC.When a quality characteristic can not be measured numerically we classifythe product as defective or nondefective. In SPC it has become more common touse the terminology conforming or nonconforming instead of nondefective or defective.Thus, in this section, we shall continue use of the terminology conforming ornonconforming. A quality characteristic that classifies any product as conformingor nonconforming is called an attribute.For instance, quality characteristics such as the determination that a soft drinkcan is not leaking, a stud has regular edges, a rod fits into a slot, a 100-watt lightbulb meets the desired standard, or a steel rivet meets the manufacturer’s qualityspecifications are some examples of attributes. Note that the data collectedon a quality characteristic that is an attribute is simply count data. Moreover, thesample sizes when using control charts for attributes are normally much larger(usually in the hundreds) than the sample of size four or five that we usually usein control charts for variables.In general, variable control charts are more informative and they are veryeffective in detecting a defect before even it occurs whereas attribute charts areused only after the defects have occurred. There are cases, however, when variablecontrol charts show some limitations. For example, consider a product that isnonconforming due to any one of 10 quality characteristics that do not conform tospecifications. Clearly, in this case we can not control all 10 quality characteristicsby using one variable control chart, since one variable control chart can controlonly one quality characteristic at a time. Thus, in this case, to control all 10 qualitycharacteristics we would have to use 10 different variable control charts. On theother hand, however, one attribute control chart can study all the quality characteristicsbecause a nonconforming unit is nonconforming irrespective of the numberof quality characteristics that do not conform to specifications. Thus, we canconclude that both variable and attribute control charts have their pros and cons.In some cases the quality characteristic is such that instead of classifying aunit as conforming or nonconforming, we record the number of nonconformitiesper manufactured unit. For example, the number of holes in a roll of paper, thenumber of irregularities per unit area of a spool of cloth, the number of blemisheson a painted surface, the number of loose ends in a circuit board, the number ofnonconformities per unit length of a cable, the number of nonconformities of alltypes in an assembled unit, and so on. In such cases, we use control charts thatfall under the category of control charts for attributes, and such control charts areused to reduce the number of nonconformities per unit length, area, or volume ofa single manufactured unit, or to reduce the number of nonconformities per manufacturedor assembled unit.The control charts for attributes are quite similar to the control charts for variables,that is, the center line and the control limits are set in the same manner asin the case of control charts for variables. However, it is important to note that thepurposes of using control charts for variables and control charts for attributes are

Chapter 19: B. Statistical Process Control 275Table 19.4 Control charts for attributes.Control chartp chartnp chartc chartu chartQuality characteristic under investigationPercent or fraction of nonconforming units in a subgroup or a sample, wheresample size can be variableNumber of nonconforming units in a sampleNumber of nonconformities in a sample or in one or more inspection unitsNumber of nonconformities per unit, where sample size can be variablequite distinct. As noted earlier, the purpose of using the control charts for variablesin any process is to reduce the variability due to special or assignable causes,whereas the control charts for attributes are used to reduce the number of nonconformingunits or to reduce the number of nonconformities per manufactured orassembled unit, or simply the number of nonconformities per unit length, area,or volume of a single manufactured unit.In this section, we are going to study four types of control charts for attributes:the p chart, np chart, c chart, and u chart. In Table 19.4, we give a very brief descriptionof these charts, which can help to determine the appropriate type of controlchart that should be used for the quality characteristic under investigation.The p Chart: Control Chart for Fraction of Nonconforming UnitsThe most frequently used attribute control chart is the p chart. This is used wheneverwe are interested in finding the fraction or percent of units that do not conformto the specifications in a situation where the observed quality characteristicis an attribute or a variable measured by a go/no-go gauge. A p chart can be usedto study one or more quality characteristics simultaneously. Since each inspectedunit is classified as conforming or nonconforming and it is assumed that the conformityor nonconformity of each unit is defined independently, which is trueonly if the process is stable, the probability of occurrence of a nonconforming unitat any given time is the same. Then the basic rules of the p chart are governedby the binomial probability distribution with parameters n and p, where n is thesample size and p is the fraction of nonconforming units produced by the processunder investigation.The binomial probability distribution function of a random variable X withparameters n and p is defined byPart IV.B.4nP X xx p xp n x( = )= ( 1)x = 0, 1,..., n . (19.29)The mean and the standard deviation of the random variable X are given bynp andnp( 1 p)

Chapter 19: B. Statistical Process Control 275

Table 19.4 Control charts for attributes.

Control chart

p chart

np chart

c chart

u chart

Quality characteristic under investigation

Percent or fraction of nonconforming units in a subgroup or a sample, where

sample size can be variable

Number of nonconforming units in a sample

Number of nonconformities in a sample or in one or more inspection units

Number of nonconformities per unit, where sample size can be variable

quite distinct. As noted earlier, the purpose of using the control charts for variables

in any process is to reduce the variability due to special or assignable causes,

whereas the control charts for attributes are used to reduce the number of nonconforming

units or to reduce the number of nonconformities per manufactured or

assembled unit, or simply the number of nonconformities per unit length, area,

or volume of a single manufactured unit.

In this section, we are going to study four types of control charts for attributes:

the p chart, np chart, c chart, and u chart. In Table 19.4, we give a very brief description

of these charts, which can help to determine the appropriate type of control

chart that should be used for the quality characteristic under investigation.

The p Chart: Control Chart for Fraction of Nonconforming Units

The most frequently used attribute control chart is the p chart. This is used whenever

we are interested in finding the fraction or percent of units that do not conform

to the specifications in a situation where the observed quality characteristic

is an attribute or a variable measured by a go/no-go gauge. A p chart can be used

to study one or more quality characteristics simultaneously. Since each inspected

unit is classified as conforming or nonconforming and it is assumed that the conformity

or nonconformity of each unit is defined independently, which is true

only if the process is stable, the probability of occurrence of a nonconforming unit

at any given time is the same. Then the basic rules of the p chart are governed

by the binomial probability distribution with parameters n and p, where n is the

sample size and p is the fraction of nonconforming units produced by the process

under investigation.

The binomial probability distribution function of a random variable X with

parameters n and p is defined by

Part IV.B.4

n

P X x

x p x

p n x

( = )=

( 1

)

x = 0, 1,..., n . (19.29)

The mean and the standard deviation of the random variable X are given by

np and

np( 1 p)

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