control chart for attributes

Control Charts for Attributes

Ronald Sukwadi, Ph.D.

Department of Industrial Engineering

TKI 329 Quality Control

Introduction

Data that can be classified into one of several

categories or classifications is known as attribute

data.

Classifications such as conforming and

nonconforming are commonly used in quality

control.

Another example of attributes data is the count of

defects.

Ronald Sukwadi TKI 329 Quality Control

Control Charts for Fraction

Nonconforming

Fraction nonconforming is the ratio of the

number of nonconforming items in a

population to the total number of items in that

population.

Control charts for fraction nonconforming are

based on the binomial distribution.



Nonconforming

Recall: A quality characteristic follows a binomial distribution if:

1. All trials are independent.

2. Each outcome is either a “success” or “failure”.

3. The probability of success on any trial is given as p. The probability of a failure is 1-p.

4. The probability of a success is constant.



Nonconforming

The binomial distribution with parameters n

0 and 0 < p < 1, is given by

The mean and variance of the binomial

distribution are

xnx )p1(px

n)x(p

)p1(npnp 2


Design of Fraction Nonconforming

Chart

Three parameters must be specified

1. The sample size

2. The frequency of sampling

3. The width of the control limits

Common to base chart on 100% inspection of

all process output over a period of time

Rational subgroups may also play role in

determining sampling frequency


Sample size

If p is very small, we should choose n sufficiently

large to find at least one nonconforming unit

Otherwise the presence of only one non-conforming

in the sample would indicate out-of-control

condition (example)

To avoid this, choose n such that the probability of

finding at least one nonconforming per sample is at

least γ (example)


Example

p = 0.01 and n = 8

If there is one nonconforming in the sample, then

p =1/8=0.125

and we conclude that the process is out of control

1155.08

)01.01(01.0301.0

)1(3

n

pppUCL


Sample size

The sample size can be determined so that the

probability of finding at least one nonconforming

unit per sample is at least γ

Example p = 0.01 and γ = 0.95

Find n such that P(D ≥ 1) ≥ 0.95

Using Poisson approximation of the binomial with

λ =np

From cumulative Poisson table λ must exceed 3.00

np ≥ 3 n ≥ 300


Sample size

The sample size can be determined so that a shift of some

specified amount, can be detected with a stated level of

probability (50% chance of detection).

UCL = pout

If is the magnitude of a process shift, then n must satisfy:

Therefore,

n

)p1(pL

)p1(pL

n

2


Positive Lower Control Limit

The sample size n, can be chosen so that the

lower control limit would be nonzero:

and

0n

)p1(pLpLCL

2Lp

)p1(n


Interpretation of Points on the Control

Chart for Fraction Nonconforming


Variable Sample SizeVariable-Width Control Limits


Variable Sample SizeControl Limits Based on an Average Sample Size


Variable Sample SizeThe Standardized Control Chart


The Opening Characteristic Function

and Average Run Length Calculations


Control Charts for Nonconformities

(Defects)Procedures with Constant Sample Size


There are many instances where an item will contain nonconformities but the item itself is not classified as nonconforming.

It is often important to construct control charts for the total number of nonconformities or the average number of nonconformities for a given “area of opportunity”. The inspection unit must be the same for each unit.


Control Charts for Nonconformities

(Defects)Procedures with Variable Sample Size


control chart for attributes

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