control charts for attributes 1 introduction data that can be classified into one of several...

Control Charts for Attributes 1

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.


Type of Attributes Control Chart

• Control Chart for Fraction Nonconforming — p chart

• Control Chart for Nonconforming — np chart

• Control Chart for Nonconformities — c chart

• Control Chart for Nonconformities per unit — u chart


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.


The characteristic of binomial distribution

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.


The Binomial distribution

• 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


Development of the Fraction Nonconforming Control Chart1

Assume • n = number of units of product selected

at random. • D = number of nonconforming units from

the sample• p= probability of selecting a

nonconforming unit from the sample.• Then: xnx )p1(p

x

n)xD(P



• The sample fraction nonconforming is given as

where is a random variable with mean and variance

n

Dp̂

p̂

n

)p1(pp 2



Standard Given• If a standard value of p is given, then the

control limits for the fraction nonconforming are

UCL pp p

nCL p

LCL pp p

n

31

31

( )

( )



No Standard Given• If no standard value of p is given, then the

control limits for the fraction nonconforming are

wheren

pppLCL

pCL n

pppUCL

)1(3

)1(3

m

p̂

mn

Dp

m

1ii

m

1ii



Trial Control Limits• Control limits that are based on a preliminary

set of data can often be referred to as trial control limits.

• The quality characteristic is plotted against the trial limits, if any points plot out of control, assignable causes should be investigated and points removed.

• With removal of the points, the limits are then recalculated.


Example1

A process that produces bearing housings is investigated. Ten samples of size 100 are selected.

• Is this process operating in statistical control?

Sample # 1 2 3 4 5 6 7 8 9 10

#Nonconf 5 2 3 8 4 1 2 6 3 4


Example2

n = 100, m = 10

Sample # 1 2 3 4 5 6 7 8 9 10 # Nonconf. 5 2 3 8 4 1 2 6 3 4 Fraction Nonconf.

0.05 0.02 0.03 0.08 0.04 0.01 0.02 0.06 0.03 0.04

038.0m

p̂p

m

1ii


Example3

Control Limits are:

002.0100

)038.01(038.03038.0LCL

038.0CL

095.0100

)038.01(038.03038.0UCL


Example4

Sample Number

109876543210

0.10

0.05

0.00

Prop

ortio

n

P Chart for C1

P=0.03800

3.0SL=0.09536

-3.0SL=0.000


Design of the Fraction Nonconforming Control Chart

• Sample size

• Frequency of sampling

• Width of the control limits


Sample size of fraction nonconforming control chart

• If p is very small, we should choose n sufficiently large so that we have a high probability of finding at least one nonconforming unit in the sample.

• Otherwise, we might find that the control limits are such that the presence of only one nonconforming unit in the sample would indicate an out-of control condition.


Sample size of fraction nonconforming control chart1

• Choose the sample size n so that the probability of finding at least one nonconforming unit per sample is at least γ.

De

P(D 1) P D 0 1D!


Example1

• If P=0.01, n=8, then UCL=0.1155

• If there is one nonconforming unit in the sample, then p=1/8=0.1250, and we can conclude that the process is out of control.


Example2

Suppose we want the probability of at least one nonconforming unit in the sample to be at least 0.95.

0e

P(D 1) 0.95 P D 0 1 0.950!3

np n 300



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, Duncan, 1986). If is the magnitude of a process shift, then n must satisfy:

Therefore,

n

)p1(pL

)p1(pL

n2


Example

• Suppose that p=0.01, and we want the probability of detecting a shift to p=0.05 to be 0.50.

0.01(1 0.01)0.05 0.01 3

nn 56



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


Example

• If p=0.05 and three-sigma limits are used, the sample size must be

2(1 0.05)n 3 171

0.05


Width of the control limits

• Three-sigma control limits are usually employed on the control chart for fraction nonconforming.

• Narrower control limits would make the control chart more sensitive to small shifts in p but at the expense of more frequent “false alarms.”


Interpretation of p Chart

• Care must be exercised in interpreting points that plot below the lower control limit.– They often do not indicate a real improvement in

process quality.– They are frequently caused by errors in the

inspection process or improperly calibrated test and inspection equipment.

– Inspectors deliberately passed nonconforming units or reported fictitious data.


The np control chart

• The actual number of nonconforming can also be charted. Let n = sample size, p = proportion of nonconforming. The control limits are:

(if a standard, p, is not given, use )p

)p1(np3npLCL

npCL

)p1(np3npUCL


Variable Sample Size1

• In some applications of the control chart for the fraction nonconforming, the sample is a 100% inspection of the process output over some period of time.

• Since different numbers of units could be produced in each period, the control chart would then have a variable sample size.


Three Approaches for Control Charts with Variable Sample Size

1.Variable Width Control Limits

2.Control Limits Based on Average Sample Size

3.Standardized Control Chart


P Charts with Variable Sample Size — Example

Week Total Requests Second Visit Required

1 200 6

2 250 8

3 250 9

4 200 3

5 150 2


Variable Width Control Limits

• Determine control limits for each individual sample that are based on the specific sample size.

• The upper and lower control limits are

in

)p1(p3p


Control Limits Based on an Average Sample Size• Control charts based on the average sample

size results in an approximate set of control limits.

• The average sample size is given by

• The upper and lower control limits arem

nn

m

1ii

n

)p1(p3p


平均管制界限與個別不良率之分析 1

• 點落在管制界限內，其樣本大小，小於平均樣本大小時– 不須計算個別管制界限，該點在管制狀態內。

• 點落在管制界限內，其樣本大小，大於平均樣本大小時– 計算個別管制界限，再對該點樣本進行判斷。


平均管制界限與個別不良率之分析 2

• 點落在管制界限外，其樣本大小，大於平均樣本大小時– 不須計算個別管制界限，該點在管制狀態外。

• 點落在管制界限外，其樣本大小，小於平均樣本大小時– 計算個別管制界限，再對該點樣本進行判斷。


The Standardized Control Chart

• The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties:

– Centerline at 0– UCL = 3 LCL = -3– The points plotted are given by:

i

ii

n)p1(p

pp̂z


Control Charts for Nonconformities (Defects)1

• 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)2

Poisson Distribution• The number of nonconformities in a given area can

be modeled by the Poisson distribution. Let c be the parameter for a Poisson distribution, then the mean and variance of the Poisson distribution are equal to the value c.

• The probability of obtaining x nonconformities on a single inspection unit, when the average number of nonconformities is some constant, c, is found using:

!x

ce)x(p

xc


c-chart

• Standard Given:

• No Standard Given:

c3cLCL

cCL

c3cUCL

c3cLCL

cCL

c3cUCL


Choice of Sample Size: The u Chart

• If we find c total nonconformities in a sample of n inspection units, then the average number of nonconformities per inspection unit is u = c/n.

• The control limits for the average number of nonconformities is

n

u3uLCL

uCLn

u3uUCL


Three Approaches for Control Charts with Variable Sample Size

1. Variable Width Control Limits

2. Control Limits Based on Average Sample Size

3. Standardized Control Chart


Variable Width Control Limits

• Determine control limits for each individual sample that are based on the specific sample size.

• The upper and lower control limits are

in

u3u


Control Limits Based on an Average Sample Size• Control charts based on the average sample

size results in an approximate set of control limits.

• The average sample size is given by

• The upper and lower control limits arem

nn

m

1ii

n

u3u


The Standardized Control Chart

• The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties:

– Centerline at 0– UCL = 3 LCL = -3– The points plotted are given by:

i

ii

nu

uuz


Demerit Systems1

• When many different types of nonconformities or defects can occur, we may need some system for classifying nonconformities or defects according to severity; or to weigh various types of defects in some reasonable manner.


Demerit Schemes2


Demerit Schemes3

• Let ciA, ciB, ciC, and ciD represent the number of units in each of the four classes.

• The following weights are fairly popular in practice:

– Class A-100, Class B - 50, Class C – 10, Class D - 1

di = 100ciA + 50ciB + 10ciC + ciD

di - the number of demerits in an inspection unit


Demerit Systems—Control Chart Development1

• Number of demerits per unit:

where n = number of inspection units

D =

n

Du i

n

1iid


Demerit Systems—Control Chart Development2

where and

u

u

ˆ3uLCL

uCL

ˆ3uUCL

DCBA uu10u50u100u

2/1

DC2

B2

A2

u n

uu10u50u100ˆ


Dealing with Low-Defect Levels1

• When defect levels or count rates in a process become very low, say under 1000 occurrences per million, then there are long periods of time between the occurrence of a nonconforming unit.

• Zero defects occur• Control charts (u and c) with statistic

consistently plotting at zero are uninformative.



• Chart the time between successive occurrences of the counts – or time between events control charts.

• If defects or counts occur according to a Poisson distribution, then the time between counts occur according to an exponential distribution.



• Exponential distribution is skewed.• Corresponding control chart very asymmetric.• One possible solution is to transform the

exponential random variable to a Weibull random variable using x = y1/3.6 (where y is an exponential random variable) – this Weibull distribution is well-approximated by a normal.

• Construct a control chart on x assuming that x follows a normal distribution.

• See Example 6-6, page 304.



• Data transformation: defect (Poisson distribution) → time between event (Exponential distribution) → x=y1/3.6=y0.2777

→ X bar – MR control chart


Choice Between Attributes and Variables Control Charts• Each has its own advantages and disadvantages• Attributes data is easy to collect and several

characteristics may be collected per unit.• Variables data can be more informative since specific

information about the process mean and variance is obtained directly.

• Variables control charts provide an indication of impending trouble (corrective action may be taken before any defectives are produced).

• Attributes control charts will not react unless the process has already changed (more nonconforming items may be produced).


Guidelines for Implementing Control Charts

1. Determine which process characteristics to control.

2. Determine where the charts should be implemented in the process.

3. Choose the proper type of control chart.4. Take action to improve processes as the result

of SPC/control chart analysis.5. Select data-collection systems and computer

software.

control charts for attributes 1 introduction data that can be classified into one of several...

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