Chapter 7: Control Charts For
Attributes
By Drew Kelly
IET 603
Introduction to Statistical Quality Control.
Douglass C. Montgomery
7-1 Introduction
 In this Chapter we deal with what is called Attributes Data.
 Attributes Data deals with classifications of a product that is
either defective or non-defective.
Taken from Google
7-1 Introduction
 The terms nonconformity and nonconforming are used
more in today’s time.
 However many feel that saying defect and defective gives a
more clear distinction between the two.
 Attribute Charts are not as statistically informing as Variable
charts. Simple due to the amount of numerical information
Variable charts have to offer
7-2 The Control Chart for Fraction
Nonconforming
 The fraction nonconforming is the ratio of the number of
nonconforming units in a population to the total number of
units in the population.
 Again nonconforming item simply means that the entire
product is defective. And should not be put out for use.
7-2-1 Development and Operation of
the Control Chart
 The control chart is also called the P- Chart.
 Again this is for the fraction of nonconforming
 Equations for the control chart are:
Taken from Google
 The control limits given here are called trial control limits.
7-2-1 Continued
 There are three parameters that must be need to be specified:
the sample size, the frequency of sampling, and the widths of
the control limits.
 When interpreting points on the control charts, one must
use caution.
 Often it is found that points below the lower control limit do
not represent a real improvement in process quality.
NP Control Chart
 The np chart is used when you want to display the number
of nonconforming instead of the fraction
nonconforming.
 The equations for the np chart are :
 Look at example 7.2 on page 310
Taken from Google
NP Control Chart
Taken From Google.
Questions?
7-2-2 Variable Sample Size
 In some occasions the sample is a inspection of process
output over a period of time. This will cause the control
chart to have a variable sample size.
 There are three methods to construct control charts with
variable sample sizes.
Variable-Width Control Limits
 Determine the control limits for each individual sample that
are based on the specific sample size.
 The equations the same as if you were using it for the p chart
 If you will look at page 311 in your book to see an example
Control limits based on an average
sample size
 Based on average sample size
 Which will give an approximation of the control charts.
 This approach assumes that future sample sizes will not differ
from the ones previously used.
 To find the average sample size you take the total of the
sample size, denoted by ni, and divide by the total number
of observations.
 Then solve like you would for the p chart
 Look at page 312 if you feel the need to.
The Standardized control Chart
 Where the points are plotted in standard deviation units.
 Center line must be at 0
 And control limits are at +3 and -3
7-2-4 Operating-Characteristic Function
 The OC function is of the fraction of nonconforming control
chart; is a graphic display of the probability of incorrectly
accepting the hypothesis of SC against the process fraction
nonconforming.
 This mean it is a type II error (failure to reject the incorrect
hypothesis)
Questions?
7-3 Control charts for Nonconformities
 Review: A nonconforming item is an unit of product that
does not meet one or more of the specifications for that
particular product. ( Defective unit )
 A nonconformity is the specific point where the specification
is not met. ( Defect within a unit )
 Not all products with nonconformities will be
nonconforming.
 Depending on the degree of severity it may still pass
operations
Constant Sample size
 Control charts can be created for either the total number of
nonconformities in a unit or the average.
 The control chart is called a c-chart. And is for the number
of nonconformities, or defects.
Taken from Google
Further Analysis of Nonconformities
 Other useful techniques of further analyzing nonconformities
are cause and effect diagrams.
 Defect or nonconformity data is more useful that defective
data because there are always going to be more
nonconformity data.
 Out-of-control-action plans can be and should be done when
your process is out of control.
Choice of sample size
 The sample size should be chosen according to statistical
considerations, such as picking a size large enough to make
positive lower control limits.
 Economic factors should be considered when determining
the sample size.
U chart
 The best approach for setting up an u chart is to use the
following equations:
Taken From Google
 The u chart is used when the sample size is considered a
variable sample size
 And is the control chart for average number of
nonconformities per unit.
7-3-3 Demerit Systems
 Demerit systems are ways to classify the
seriousness of defect within the unit.
 This method is best used when you have a very
complex product such as a car, computer,
electrical appliances.
Demerit System Classification
 Class A-Very serious. Unit is completely unfit for service.
 Class B- Serious. The unit will most likely suffer a class A
operating failure.
 Class C- Moderately Serious. The unit will possibly fail
service or cause a good deal of trouble
 Class D- Minor. The unit will not fail but has minor defects.
Demerit Class Weights
 Class A-100
 Class B-50
 Class C-10
 Class D-1
 From this you can get the equation:
D=100ciA+50ciB+10ciC+1ciD
Dealing with low Defect levels
 When the defect level drops, c and u charts become
ineffective.
 Time between occurrence control charts are a good way to
deal with this
 They chart the time between the successive occurrences of
the defect.
Taken from
pqsystems.com
Questions?
7-4 Choice Between Attributes and
Variable Control Charts
 Attribute Control Charts
 Easy to understand
 Easy to make
 Avoids the hassle of having to make several Xbar or Rbar charts,
like you would have to with Variable Control Charts.
7-4 Choice Between Attributes and
Variable Control Charts
 Variable Control Charts
 Provide much more useful information about process
performance.
 Produce identification of impending trouble and allow
management to take action to prevent defects from even
happening.
7-5 Guidelines for Implementing
Control Charts
 Every process can benefit from SPC
 There are general guide lines that are used in order to
implement the correct control chart.
 In total there are 5
Guidelines for selecting the appropriate
control chart
1.
2.
3.
4.
5.
Determining which process characteristic to control
Determining where the charts should be implemented
Choosing the proper type of control chart
Taking action to improve processes as the result of control
chart analysis
Selecting data collection systems and computer software.
Review
 Attribute charts are mostly used where there is not much
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information to be given
Nonconforming and nonconformities are not the same thing.
Nonconforming means defective.
Nonconformity mean defect.
Variable charts are more complex to make but provide useful
information
 Any final Questions?
References
 http://www.transtutors.com/homework-help/operations
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management/quality-control/p-charts.aspx
http://www.qimacros.com/control-chart-formulas/uchart-formula/
http://www.qimacros.com/control-chart-formulas/npchart-formula/
http://www.six-sigma-material.com/SPC-Charts.html
http://www.pqsystems.com/qualityadvisor/DataAnalysisTo
ols/t_chart.php
Douglass C. Montgomery, Intro to Statistical Quality
Control.