Statistical Quality Control

 W. Edward Deming advocated the implementation of a statistical quality management
approach.
 His philosophy behind this approach is ‘reduce variation’- fundamental to
 the principle of continuous improvement and
 the achievement of consistency, reliability, and uniformity.
 It helps in trustworthiness, competitive position, and success.
 Statistical Quality Control
 Statistics: data sufficient enough to obtain a reliable result.
 Quality: relative term and can be defined as totality of features and characteristics of a
product or service that bear on its ability to satisfy stated or implied need (ISO).
 Control: The operational techniques and activities (a system for measuring and checking)
used to fulfil the requirements for quality. It
 incorporates a feedback mechanism system to explore the causes of poor quality
or unsatisfactory performance and
 takes corrective actions.
 also suggests when to inspect, how often to inspect, and how much to inspect.
Basic Concept

 A quality control system using statistical techniques to control quality by
performing
 inspection,
 testing and
 analysis
to conclude whether the product is as stated or designed quality standard.
 Relying on the probability theory, SQC
 evaluates batch quality and
 controls the quality of processes or products
 It makes the inspection more reliable and less costly.
 The basis of the measurement is the performance indicator, either individual,
group or departmental calculated over time (hourly, daily, or weekly).
 These performance measures are plotted on a chart.
 Pattern obtained from plotting these measures are basis of taking
appropriate actions so that
 The process variation in minimized and
 Major problems are prevented in future.
 The timing and type of, and responsibility for, these actions depends on
whether the causes of variation is controlled or uncontrolled
Basic Concept ….Cont’d

 In repetitive manufacture of a product, even with refined machinery,
skilled operator, and selected material, variations are inevitable in the
quality of units produced due to interactions of various causes.
 Variation may be due to
 Common or random causes of variation (as a result of normal
variation in material, method, and so on that causes natural variation
in product or process quality) resulting in stable pattern of variation.
 Special causes (changes in men, machine, materials or tools, jigs and
fixture and so on) resulting in a shift from the stable pattern of
variation.
 SQC assists in timely identification and elimination of the problem with
an object of reducing variations in process or product.
The application of statistical method of collecting and analyzing
inspection and other data for setting the economic standards of
product quality and maintaining adherence to the standards so that
variation in product quality can be controlled
Basic Concept ….Cont’d

Statistical quality control (SQC) is the term used to describe the
set of statistical tools used by quality professionals
SQC encompasses three broad categories of;
 Descriptive statistics
 used to describe quality characteristics and relationships.
 the mean, standard deviation, and range.
 Acceptance sampling used to randomly inspect a batch of products to
determine acceptance or rejection of entire lot based on the results.
 Does not help to identify and catch the in-process problems
 Statistical process control (SPC)
 Involves inspecting the output from a process
 Quality characteristics are measured and charted
 Helpful in identifying in-process variations
Three SQC Categories

Variability: Sources of Variation
 Variation exists in all processes.
 Variation can be categorized as either;
 Common or Random causes of variation, or
Random causes that cannot be identified
Unavoidable: inherent in the process
Normal variation in process variables such as material,
environment, method and so on.
 Can be reduced almost to zero only through improvements in
the process variables.
 Assignable causes of variation
Causes can be identified and eliminated
e.g. poor employee training, worn tool, machine needing repair
 Can be controlled by operator but it needs attention of
management.

Traditional Statistical Tools
Descriptive Statistics include
Measure of accuracy (centering)
 Measure of central tendency indicating the central
position of the series.
 A measure of the central value is necessary to estimate
the accuracy or centering of a process.
 The Mean- simply the average of a set of data
 Sum of all the measurements/data divided by the
number of observations.
 The Median- simply the value of middle item if the
data are arranged in ascending or descending order.
 Applies directly if the number in the series is odd.
 It lies between two middle numbers if the number
of the series is even.
 The Mode- value that repeat itself maximum number
of times in the series.
Shape of Distribution of Observed Data
 A measure of distribution of data
 Normal or bell shaped
 Skewed
n
x
x
n
1i
i

1
K
j
j
X
K






Distribution of Data
 Also a measure of quality
characteristics.
 Symmetric distribution - same
number of data are observed above
and below the mean.
This is what we see only when
normal variation is present in the
data
 Skewed distribution – a
disproportionate number of data are
observed either above or below the
mean.
Mean and median fall at different
points in the distribution
Centre of gravity is shifted to
oneside or other.

Traditional Statistical Tools …cont’d
Measure of Precision or Spread
 Reveals the extent to which numerical data
tend to spread about the mean value.
 The Range- the simplest possible measure
of dispersion.
 Difference between largest and smallest
observations in a set of data.
o Depends on sample size and it tends
to increase as sample size increases.
o Remains the same despite changes
in values lying between two extreme
values.
 Standard Deviation- a measure deviation
of the values from the mean.
 Small values >> data are closely
clustered around the mean
 Large values >> data are spread out
around the mean.
 
1n
Xx
σ
n
1i
2
i





Statistical Process Control
Process Control
 Refers to procedures or techniques adopted to evaluate, maintain and
improve the quality standard in various stages of manufacture.
 A process is considered satisfactory as long as it produces items within
designed specification.
Process should be continuously monitored to ensure that the
process behaves as it is expected.
 Salient features of process control
Controling the process at the right level and variability.
Detecting the deviation as quickly as possible so as to take
immediate corrective actions.
Ultimate aim is not only to detect trouble, but also to find out the
cause.
Developing an efficient information system in order to establish an
efficient system of process control.

Statistical Process Control (SPC)
 Statistical evaluation of the output of a process during production.
 Goal is to make the process stable over time and then keep it stable unless the
planned changes are made.
 Statistical description of stability requires that ‘pattern of variation’ remains
stable over time, not that there be no variation in the variable measured.
 In statistical process control language:
 A process that is in control has only common or random cause variation -
an inherent variability of the system.
 When the normal functioning of the prosess is disturbed by some
unpredictable events, special cause variation is added to common cause
variation.
Applying SPC to service
 Nature of defect is different in services
 Service defect is a failure to meet customer requirements
 One way to deal with service quality is to devise quantifiable measurement
of service elements
 Number of complaints received per month,
 Number of telephone rings before call is answered

 Hospitals
 timeliness and quickness of care, staff responses to requests, accuracy of lab
tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and
checkouts
 Grocery Stores
 waiting time to check out, frequency of out-of-stock items, quality of food
items, cleanliness, customer complaints, checkout register errors
 Airlines
 flight delays, lost luggage and luggage handling, waiting time at ticket counters
and check-in, agent and flight attendant courtesy, accurate flight information,
passenger cabin cleanliness and maintenance
 Fast-Food Restaurants
 waiting time for service, customer complaints, cleanliness, food quality, order
accuracy, employee courtesy
 Catalogue-Order Companies
 order accuracy, operator knowledge and courtesy, packaging, delivery time,
phone order waiting time
 Insurance Companies
 billing accuracy, timeliness of claims processing, agent availability and response
time

Statistical Process Control: Control Chart
Control Chart
 A graphical display of data over time (data are displayed in time sequence in
which they occurred/measured) used to differentiate common cause variation
from special cause variation.
 Control charts combine numerical and graphical description of data with the use
of sampling distribution
 normal distribution is basis for control chart.
 Goal of using this chart is to achieve and mainatain process stability
 A state in which a process has displayed a certain degree of consistency
 Consistency is characterized by a stream of data falling within the
control limits.
Basic Components of a Control Chart
 A control chart always has
 a central line usually mathematical average of
all the samples plotted;
 upper control and lower control limits defining
the constraints of common variations or range
of acceptable variation;
 Performance data plotted over time.
 Lines are determined from historical data.

Control Chart …Cont’d
When to use a control chart?
 Controlling ongoing processes by finding and correcting problems as they occur.
 Predicting the expected range of outcomes from a process.
 Determining whether a process is stable (in statistical control).
 Analyzing patterns of process variation from special causes (non-routine events)
or common causes (built into the process).
 Determining whether the quality improvement project should aim to prevent
specific problems or to make fundamental changes to the process.
Control Chart Basic Procedure
 Choose the appropriate control chart for the data.
 Determine the appropriate time period for collecting and plotting data.
 Collect data, construct the chart and analyze the data.
 Look for “out-of-control signals” on the control chart.
When one is identified, mark it on the chart and investigate the cause.
Document how you investigated, what you learned, the cause and how it was
corrected.
 Continue to plot data as they are generated. As each new data point is plotted,
check for new out-of-control signals

 Interpretation of control chart
 Points between control limits are due to
random chance variation
 One or more data points above an UCL or
below a LCL mark statistically significant
changes in the process
 A process is in control if
 No sample points outside limits
 Most points near process average
 About equal number of points above and below centerline
 Points appear randomly distributed
 A process is assumed to be out of control if
 Rule 1: A single point plots outside the control limits;
 Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on
the same side of the center line;
 Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same
side of the center line;
 Rule 4: Nine or more consecutive points fall to one side of the center line;
 Rule 5: There is a run of six or more consecutive points steadily increasing or
decreasing
Time period
Measured
characteristics

Setting Control Limits
 Type I error
 Concluding a process is not in control when it actually is.
 Type II error
 Concluding a process is in control when it is not.
In control Out of control
In control No Error
Type I error
(producers risk)
Out of control
Type II Error
(consumers risk)
No error
Mean
LCL UCL
/2 /2
Probability
of Type I error
Mean
LCL UCL
/2 /2
Probability
of Type I error

General model for a control chart
 UCL = μ + kσ
 CL = μ
 LCL = μ – kσ
where
μ is the mean of the variable
σ is the standard deviation of the variable
UCL=upper control limit; LCL = lower control limit;
CL = center line.
k is the distance of the control limits from the center line,
expressed in terms of standard deviation units.
 When k is set to 3, we speak of 3-sigma control charts.
 Historically, k = 3 has become an accepted standard in
industry.

Suggested Number of Data Points
 More data points means more delay
 Fewer data points means less precision, wider limits
 A tradeoff needs to be made between more delay and less
precision
 Generally 25 data points judged sufficient
Use smaller time periods to have more data points
Fewer cases may be used as approximation
Sample Size
 Attribute charts require larger sample sizes
50 to 100 parts in a sample
 Variable charts require smaller sample sizes
2 to 10 parts in a sample

Types of the control charts
 Variables control charts
Variable data are measured on a continuous scale.
 For example: time, weight, distance or temperature can be
measured in fractions or decimals.
Applied to data following continuous distribution
 Attributes control charts
Attribute data are counted and cannot have fractions or
decimals.
 Attribute data arise when you are determining only the
presence or absence of something:
 success or failure,
 accept or reject,
 correct or not correct.
 For example, a report can have four errors or five errors, but it
cannot have four and a half errors.
Applied to data following discrete distribution

 Variable control charts
 R chart (range chart)
 X-bar (mean chart)
 S chart (sigma chart)
 Individual or run chart
i-chart
Moving range chart
Median chart
EWMA (exponentially weighted moving average chart)
 General formulae for a control chart
 UCL or UAL = μ + kσx k = 3 ; Accepted Standard
 UWL = μ + 2/3 kσx
 CL = μ
 LWL = μ – 2/3 kσx
 LCL or LAL = μ – kσx
 

m
i
i
X
X
m

X
n


m: # of sample mean
n: # of observations in each
sample

 Mean control charts
 Used to detect the variations in mean of
a process.
 X-bar chart
 Range control charts
 Used to detect the changes in dispersion
or variability of a process
 R chart
 Use X-bar and R charts together
 Sample size : 2 ~ 10
 Use X-bar and S charts together
 Sample size : > 10
 Use i-chart and Moving range chart
together
 Sample size : 1 or one-at-a-time data
 System can show acceptable central
tendencies but unacceptable variability
or
 System can show acceptable variability
but unacceptable central tendencies
Interpret the R-chart first:
 If R-chart is in control -> interpret the X-bar
chart ->
(i) if in control: the process is in control;
(ii) if out of control: the process average is out
of control
 If R-chart is out of control: the process
variation is out of control
-> Investigate the cause; no need to interpret
the X-bar chart

 Constructing a X-bar Chart:
A quality control inspector at the Cocoa Fizz soft drink company has taken three samples
with four observations each of the volume of bottles filled.
If the standard deviation of the bottling operation is 0.2 ounces, use the below data to
develop control charts with limits of 3 standard deviations for the 16 oz. bottling operation.
 Centerline and 3-sigma
control limit formulas
Time 1 Time 2 Time 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.1 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample
means (X-
bar)
15.875 15.975 15.9
Sample
ranges (R)
0.2 0.3 0.2
3X X
UCL X  
3X X
LCL X  
X
CL X
m
i
i
X
X
m


X
n


Where,
m: # of sample mean
n: # of observations in each sample

Centerline (x-double bar):
Control limits for±3σ limits:
Control Chart
 Plot the sample mean in the
sequence from which it was
generated and interpret the
pattern in the control chart.
 
 
15.875 15.975 15.9
x 15.92
3
 
     
 
 
     
 
x x
x x
.2
UCL x zσ 15.92 3 16.22
4
.2
LCL x zσ 15.92 3 15.62
4

Second Method for X-bar Chart using Range and A2 factor
 Use this method when standard deviation for the process distribution is
unknown.
 Control limits solution:
 Center line and 3-sigma
control Fomulas:
1
2
k
i
i
x
n
R
R
k
R R
or
d d n

 


 

;
;&
2
2
2
2
3
3
x
x
x
CL X
R
UCL X X A R
d n
R
LCL X X A R
d n

   
   

OBSERVATIONS(SLIP-RINGDIAMETER,CM)
SAMPLEk 1 2 3 4 5 x R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
Calculate limits for X-bar chart using sample range

R-Chart:
 Always look at the Range chart first.
 The control limits on the X-bar chart are
derived from the average range, so if the
Range chart is out of control, then the
control limits on the X-bar chart are
meaningless.
 Look for out of control signal.
 If there are any, then the special causes
must be eliminated.
 There should be more than five distinct
values plotted, and no one value should
appear more than 25% of the time.
 If there are values repeated too often,
then you have inadequate resolution
of your measurements, which will
adversely affect your control limit calculations.
 Once the effect of the out of control points
from the Range chart is removed, look at
the X-bar Chart.
Standard Deviation of Range and Standard
Deviation of the process is related as:
Centerline and 3-sigma Control Limit
Formulas:
Where
3
3
2
 R
d
d R
d
 
3 3
4
2 2
3 3
3
2 2
3 1 3
3 1 3
R
R
R
CL R
d d
UCL R R R D R
d d
d d
LCL R R R D R
d d

    
    
( )
( )
d
D
d
  3
4
2
1 3 max( , )
d
D
d
  3
3
2
0 1 3

1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
Calculate limits for R-chart

 S-Chart
 The sample standard deviations are
plotted in order to control the
process variability.
 For sample size (n>12),
 With larger samples, the
resulting mean range does not
give a good estimate of
standard deviation
 the S-chart is more efficient
than R-chart.
 For situations where sample size
exceeds 12, the X-bar chart and the
S-chart should be used to check the
process stability.
Centerline and 3-sigma Control Limit
Formulas:
Where
s
s
s
CL S
c
UCL S S B S
c
c
LCL S S B S
c


  

  
2
4
4
4
2
4
3
4
1
3
1
3
max( , )
c
B
c
c
B
c

 

 
2
4
4
4
2
4
3
4
1
1 3
1
0 1 3
( )
&
kn
j ji
ji
j
Sx x
S S
n k


 

 2
11
1

 Changing Sample Size on the X-bar and R Charts
 In some situations, it may be of interest to know the effect of changing
the sample size on the X-bar and R charts. Needed information:
 = average range for the old sample size
 = average range for the new sample size
 nold = old sample size
 nnew = new sample size
d2(old) = factor d2 for the old sample size
d2(new) = factor d2 for the new sample size
 Centerline and 3-sigma Control Limit Formulas:
oldR
newR
( )
( )
( )
( )
old
old
x chart
d new
UCL x A R
d old
d new
LCL x A R
d old

 
   
 
 
   
 
2
2
2
2
2
2
( )
( )
( )
( )
( )
max ,
( )
old
new old
old
R chart
d new
UCL D R
d old
d new
CL R R
d old
d new
LCL D R
d old

 
  
 
 
   
 
   
   
   
2
4
2
2
2
2
3
2
0

Control Chart: Interpreting the Patterns
Patterns
 A nonrandom identifiable arrangement of plotted points on the chart.
 Provides sufficient reasons to look for special causes.
 Causes that affect the process intermittently and
 can be due to periodic and persistent disturbances
 Natural pattern
 No identifiable arrangement of the plotted points exists
 No point falls outside the control limit;
 Majority of the points are near the centerline; and
 Few points are close to the control limits
 These patterns are indicative of a process that is in control.
 One point outside the control limits
 Also known as freaks and are caused by external disturbance
 Not difficult to identify the special causes for freaks. However, make sure
that no measurement or calculation error is associated with it,
 Sudden, very short lived power failure,
 Use of new tool for a brief test period or a broken tool,
 incomplete operation, failure of components

Interpreting the Patterns …cont’d
 Sudden shift in process mean
 A sudden change or jump in process mean or average service level.
 Afterward, the process becomes stable.
 This sudden change can occur due to changes- intentional or otherwise in
 Process settings e.g. temperature, pressure or depth of cut
 Number of tellers at the Bank,
 New operator, new equipment, new measurement instruments, new vendor
or new method of processing.
 Gradual shift in the process mean
 Such shift occurs when the process parameters change gradually over a period
of time.
 Afterward, the process stabilizes
 X-bar chart might exhibit such shift due to change in incoming quality of raw
materials or components over time, maintenance program or style of
supervision.
 R-chart might exhibit such shift due to a new operator, decrease in worker skill
due to fatigue or monotoy, or improvement in incoming quality of raw
materials.

 Trending pattern
 Trend represents changes that steadily increases or decreases.
 Trends do not stabilize or settle down
 X-bar chart may exhibit a trend because of tool wear, dirt or chip buildup, aging
of equipment.
 R-chart may exhibit trend because of gradual improvement of skill resulting
from on-the-job-training or a decrease in operator skill due to fatigue.
 Cyclic pattern
 A repetitive periodic behavior in the system.
 A high and low points will appear on the control chart
 X-bar chart may exhibit a cyclic behavior because of a rotation of operator,
periodic changes in temperature and humidity, seasonal variation of incoming
components, periodicity in mechanical or chemical properties of the material
 R-chart might exhibit cyclic pattern because of operator fatigue and subsequent
energization following breaks, a difference between shifts, or periodic
maintenance of equipment.
 Graph will not show cyclic pattern, if the samples are taken too infrequently

Zones for Pattern Test
UCL
LCL
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Process
average
3 sigma = x + A2R
=
3 sigma = x - A2R
=
2 sigma = x + (A2R)
= 2
3
2 sigma = x - (A2R)
= 2
3
1 sigma = x + (A2R)
= 1
3
1 sigma = x - (A2R)
= 1
3
x
=
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
UCL
LCL
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Process
average
3 sigma = x + A2R
=
3 sigma = x - A2R
=
2 sigma = x + (A2R)
= 2
3
2 sigma = x - (A2R)
= 2
3
1 sigma = x + (A2R)
= 1
3
1 sigma = x - (A2R)
= 1
3
x
=
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13

Performing a Pattern Test
OBSERVATIONS (SLIP- RING DIAMETER, CM)
SAMPLE k 1 2 3 4 5 x R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15

xx-- barbar
ChartChart
ExampleExample
(cont.)(cont.)
UCL = 5.08
LCL = 4.94
Mean
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
5.10 –
5.08 –
5.06 –
5.04 –
5.02 –
5.00 –
4.98 –
4.96 –
4.94 –
4.92 –
x = 5.01=
UCL = 5.08
LCL = 4.94
Mean
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
5.10 –
5.08 –
5.06 –
5.04 –
5.02 –
5.00 –
4.98 –
4.96 –
4.94 –
4.92 –
x = 5.01=x = 5.01=

 A process is assumed to be out of control if
 Rule 1: A single point plots outside the control
limits;
 Rule 2: Two out of three consecutive points fall
outside the two sigma warning limits on the same
side of the center line;
 Rule 3: Four out of five consecutive points fall
beyond the 1 sigma limit on the same side of the
center line;
 Rule 4: Nine or more consecutive points fall to
one side of the center line;
 Rule 5: There is a run of six or more consecutive
points steadily increasing or decreasing

11 4.984.98 BB —— BB
22 5.005.00 BB UU CC
33 4.954.95 BB DD AA
44 4.964.96 BB DD AA
55 4.994.99 BB UU CC
66 5.015.01 —— UU CC
77 5.025.02 AA UU CC
88 5.055.05 AA UU BB
99 5.085.08 AA UU AA
1010 5.035.03 AA DD BB
SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE
11 4.984.98 BB —— BB
22 5.005.00 BB UU CC
33 4.954.95 BB DD AA
44 4.964.96 BB DD AA
55 4.994.99 BB UU CC
66 5.015.01 —— UU CC
77 5.025.02 AA UU CC
88 5.055.05 AA UU BB
99 5.085.08 AA UU AA
1010 5.035.03 AA DD BB
SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE
11 4.984.98 BB —— BB
22 5.005.00 BB UU CC
33 4.954.95 BB DD AA
44 4.964.96 BB DD AA
55 4.994.99 BB UU CC
66 5.015.01 —— UU CC
77 5.025.02 AA UU CC
88 5.055.05 AA UU BB
99 5.085.08 AA UU AA
1010 5.035.03 AA DD BB
SAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONESAMPLESAMPLE xx ABOVE/BELOWABOVE/BELOW UP/DOWNUP/DOWN ZONEZONE

Control Chart for Attributes
Attributes are discrete events: yes/no or pass/fail
Construction and interpretation are same as that of variable control charts.
Attributes control charts
 p chart
 Uses proportion nonconforming (defective) items in a sample.
 Based on a binomial distribution.
 Can be used for varying sample size.
 np chart
 Uses number of nonconforming items in a sample.
 Should not be used when sample size varies.
 c chart
 Uses total number of nonconformities or defects in samples of constant size.
 Occurence of nonconformities follows poisson distribution.
 u chart
 when the sample size varies, the number of nonconformities per unit is used as a basis for
this control chart.

Control Chart: p chart
 Proportion nonconforming or defectives for each sample are plotted on the p-chart
 The chart is examined to determine whether the process is in control.
 Means to calculate center line and control limits
 No standard or target value of proportion nonconforming is specified
 It must be estimated from sample infromation and
 For each sample, proportion of nonconforming items are determined as
 The average of these individual sample proportion of nonconforming items is used as the
center line (CLp):
As true value of p is not known,
p-bar is used as an estimate
x
p
n

m m
i
i i
p
p x
CL p
m nm
  
 
( )
( )
p
p
p p
UCL p
n
p p
LCL p
n

 

 
1
3
1
3

20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans
NUMBER OFNUMBER OF PROPORTIONPROPORTION
SAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE
11 66 .06.06
22 00 .00.00
33 44 .04.04
:: :: ::
:: :: ::
2020 1818 .18.18
200200
20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans
NUMBER OFNUMBER OF PROPORTIONPROPORTION
SAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE
11 66 .06.06
22 00 .00.00
33 44 .04.04
:: :: ::
:: :: ::
2020 1818 .18.18
200200
UCL = p + z = 0.10 + 3
p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = 0.010
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
= 200 / 20(100) = 0.10
total defectives
total sample observations
p =
UCL = p + z = 0.10 + 3
p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
UCL = p + z = 0.10 + 3
p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = 0.010
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
LCL = 0.010
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
= 200 / 20(100) = 0.10
total defectives
p = = 200 / 20(100) = 0.10
total defectives
p = = 200 / 20(100) = 0.10
total defectives
p =
 If the target or standard value is specified
 Center line is selected as that target value i.e.
CLp= po where, po represent a standard value
 Control limits are also based on the target velue.
 If the lower control limit for p is turned out to be negative, LCL is
simply counted as zero.
 Lowest possible value for proportion of nonconformng item is zero
Control Chart: p chart …Cont’d

Control Chart: p chart …Cont’d
 Variable sample size
 Changes in sample size casues the control limits to change, although the center
line remained fixed.
 Control limits can be constructed:
 For individual samples
 If no standard value is given and sample mean proportion nonconforming
is p-bar, control limit for sample i with size ni are
 Using average sample size
Where

 

 
( )
( )
i
i
p p
UCL p
n
p p
LCL p
n
1
3
1
3
( )
( )
p p
UCL p
n
p p
LCL p
n

 

 
1
3
1
3
m
i
i
n
n
m


1

Control Chart: c chart
 No standard given
 Average number of nonconformities per sample unit is found from the sample
observation and is denoted by c-bar.
 The center line and control limits are:
 If lower control limit is found to be less than zero, it is converted to zero.
 Standard given
 if the specified target for the number of nonconformities per sample unit be co..
The center line and control limits are then calculated from:
c
c
c
CL c
UCL c c
LCL c c

 
 
3
3
c o
c o o
o o o
CL c
UCL c c
LCL c c

 
 
3
3

Control Chart: c chart …Cont’d
Number of defects in 15 sample roomsNumber of defects in 15 sample rooms
1 121 12
2 82 8
3 163 16
: :: :
: :: :
15 1515 15
190190
SAMPLESAMPLE
cc = = 12.67= = 12.67
190190
1515
UCLUCL == cc ++ zzcc
= 12.67 + 3 12.67= 12.67 + 3 12.67
= 23.35= 23.35
LCLLCL == cc ++ zzcc
= 12.67= 12.67 -- 3 12.673 12.67
= 1.99= 1.99
NUMBER
OF
DEFECTS
Number of defects in 15 sample roomsNumber of defects in 15 sample rooms
1 121 12
2 82 8
3 163 16
: :: :
: :: :
15 1515 15
190190
SAMPLESAMPLE
cc = = 12.67= = 12.67
190190
1515
cc = = 12.67= = 12.67
190190
1515
cc = = 12.67= = 12.67
190190
1515
190190
1515
= 12.67 + 3 12.67= 12.67 + 3 12.67
= 23.35= 23.35
= 12.67 + 3 12.67= 12.67 + 3 12.67
= 23.35= 23.35
LCLLCL == cc ++ zzcc
= 12.67= 12.67 -- 3 12.673 12.67
= 1.99= 1.99
NUMBER
OF
DEFECTS
33
66
99
1212
1515
1818
2121
2424
NumberofdefectsNumberofdefects
Sample numberSample number
22 44 66 88 1010 1212 1414 1616
UCL = 23.35
LCL = 1.99
c = 12.67
33
66
99
1212
1515
1818
2121
2424
22 44 66 88 1010 1212 1414 1616
UCL = 23.35
LCL = 1.99
c = 12.67
33
66
99
1212
1515
1818
2121
2424
22 44 66 88 1010 1212 1414 161622 44 66 88 1010 1212 1414 1616
UCL = 23.35
LCL = 1.99
c = 12.67

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