QUALITY CONTROL,

ASSURANCE & RELIABILITY

ET / PE ZC 434
BITS Pilani Lecture 12
Pilani Campus
 RECAP – LECTURE 11

 INTRODUCTION TO CONTROL CHARTS

 Issues in construction of control chart are:
– Selection of control limits,
– Minimizing error in inferences, Operating characteristic
curve (OC curve)
– Sampling size, Average Run Length (ARL)
– Interpretation of control charts

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 OC CURVE

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 OC CURVE AND ARL

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 ARL CURVE

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 TODAY’S OBJECTIVES

 CONTROL CHARTS FOR VARIABLES

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 CONTROL CHART FOR VARIABLES
 Variables are those quality characteristics that are measurable on a
numerical scale, e.g. length, viscosity

 It is necessary to control the mean value of a quality characteristic as

well as its variability

 Mean gives an indication of the central tendency of a process

 Variability provides an idea of the process dispersion

 Control charts aid in detecting such changes in process parameters

 Variables provide more information than attributes

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 CHANGE IN PROCESS MEAN

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 CHANGE IN DISPERSION

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 FACTORS TO BE CONSIDERED

 COSTS : FIXED + VARIABLE

 SELECTION OF CHARACTERISTICS

 SELECTION OF RATIONAL SAMPLES

 SAMPLE SIZE

 FREQUENCY OF SAMPLING

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 X BAR AND R CHART

Why two separate charts for mean and range?

 X bar chart monitors the mean between sample values
 R chart monitors the variation within sample

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 Example

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 Example – contd..

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 EXAMPLE – Contd….

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 X BAR AND R CHART

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 Selection of Control Limits
Let ‘θ’ is the mean diameter of parts produced by the process
and θ* would be sample mean diameter of parts chosen from
the process
– E(θ*) is the mean or expected value and σ(θ*) be the standard
deviation of the estimator θ*
Then the centre line and control limits are given by
• CL = E(θ*)
• UCL = E(θ*) + k σ(θ*)
• LCL = E(θ*) - k σ(θ*)

– Where k = no. of standard deviations of the sample statistic that the

control limits are placed from the centre line
– Generally value of k is chosen as 3

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 X BAR CHART

FOR
INFORMATION

For a normally distributed population, the distribution of the statistic’s

relative range (W) = R / σ and it is dependent on sample size ‘n’.
The mean of “W” is represented by “d2”

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 RANGE STATISTICS AND d2
CONSTANT

FOR
INFORMATION

d2: A constant value that converts the average range (R-bar) to an estimate of the
population standard deviation (σ). It's essentially the expected value of the sample range.

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 R CHART

FOR
INFORMATION

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 RANGE STATISTICS – d3 CONSTANT
FOR
INFORMATION
d3: A constant that is the standard deviation of
the sample range.

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 X BAR AND R CHART

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 WHICH SHOULD BE ANALYZED FIRST – R CHART
R- CHART
OR X BAR CHART

X BAR- CHART

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 EXAMPLE – CONTD..

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 X BAR AND R CHART

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 FACTOR VALUES

REFER APPENDIX A7 IN TEXTBOOK T1

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 EXAMPLE – Contd…

In this case, D3 = 0 for n = 5

and hence LCL = 0
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 EXAMPLE – Contd…

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 FACTOR VALUES

REFER APPENDIX A7 IN TEXTBOOK T1

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 EXAMPLE – CONTD..

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 SAMPLE PROBLEM

Consider a process by which coils are manufactured. Samples of

size 5 are randomly selected from the process, and the
resistance values (in ohms) of the coils are measured. The data
values are given in Table 7.2 (next slide) as well as sample mean
and range R.
Draw a control chart.

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 31
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 X BAR AND R CHART - EXAMPLE

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 PROBLEM - CONTD

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 X BAR AND R CHART - EXAMPLE

ORIGINAL R AND XBAR VALUES (FOR 25 SAMPLES)

REVISED R AND X BAR VALUES AFTER REMOVING 3 SAMPLES

REVISED R = 87 – (8+4+3) = 72 REVISED X BAR = 521 – (20.4+18.6+23) = 459

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 FACTOR VALUES

REFER APPENDIX A7 IN TEXTBOOK T1

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 36
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 STEPS FOR DEVELOPMENT OF X
BAR AND R CONTROL CHARTS
1. MEASURE THE REQUIRED QUALITY CHARACTERISTICS USING
PRESELECTED SAMPLING SCHEME AND SAMPLE SIZE
2. FOR EACH SAMPLE, CALCULATE SAMPLE MEAN AND SAMPLE
RANGE
3. CALCULATE THE CENTER LINE AND CONTROL LIMITS FOR
EACH CHART
4. PLOT THE VALUES ON THE RANGE CHART FIRST, THEN X BAR
CHART. R CHART TO BE ANALYZED FIRST SINCE IT REFLECTS
PROCESS VARIABILITY. OUT OF CONTROL POINTS TO BE
IDENTIFIED AND INVESTIGATED.

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 STEPS FOR DEVELOPMENT OF X
BAR AND R CONTROL CHARTS

5. REMOVE THE OUT OF CONTROL POINTS; CALCULATE THE

REVISED CENTER LINE AND CONTROL LIMITS
6. IMPLEMENT THE CONTROL CHARTS

NOTE :
IN R CHART, IF LCL IS > ZERO, AND IF THERE ARE POINTS
BETWEEN LCL AND ZERO, THEN THOUGH THESE POINTS ARE
OUT OF CONTROL, THESE ARE DESIRABLE BECAUSE THEY
INDICATE LOW VARIABILITY

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 STANDARDIZED CONTROL CHARTS

• WHAT HAPPENS TO A CONTROL CHART WHEN SAMPLE SIZE VARIES?

• When the sample size varies, it results in fluctuating control

limits and hence we need standardized control chart
• Standardized values represent a deviation from the mean in
units of standard deviation
• They are dimensionless and have a mean of zero
• Control limits for this are ±3 and are constant
• Easier to interpret the shift in the process

• WHAT IS TO BE DONE WHEN A TARGET IS GIVEN AS THE MEAN?

• Use a standardized control chart

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 Standardized control chart

A plot of the Zi values on a control chart, with the centerline at 0,

the upper control limit at 3, and the lower control limit at - 3 ,
represents a standardized control chart for the mean.

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 Standardized control chart

– To standardize a Range chart, the range Ri for sample ‘i’ is first divided by
the estimate of the process standard deviation Ri
ri 
ˆ

– The values of ri are then standardized by subtracting its mean d2 and

dividing by its standard deviation d3

ri  d 2
– Standardized value of Range is given by ki 
d3

– Value of ki are plotted with a centre line at ‘0’ and UCL = ‘3’ and LCL = ‘-3’

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 STANDARD R CHART

For R - chart FOR

R INFORMATION
CLR  d 2 . 0 (ˆ  )
d2
UCLR  R  3 . R  d 2 . 0  3 . d 3 0  D2 . 0
LCLR  R  3 . R  d 2 . 0  3. d 3 0  D1. 0

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 STANDARD X BAR CHART

For X-bar chart:

Let X o and  o represent the target values of the process

mean and standard deviation respectively and the centre
line and limits are given by
CL X  X 0 FOR
INFORMATION
0
UCL X  X 0  3.  X 0  A . 0
n
0
LCL X  X 0  3.  X 0  A . 0
n

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 STANDARD X-BAR AND R CHART

For X-bar chart:

CLX  X 0
UCLX  X 0  A . 0
LCLX  X 0  A . 0

For R - chart

CLR  d 2 . 0
UCLR  D2 . 0
LCLR  D1. 0

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 EXAMPLE – STANDARDIZED X BAR
AND R CHART

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 FACTOR VALUES

REFER APPENDIX A7 IN TEXTBOOK T1

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 EXAMPLE – STANDARD X BAR AND
R CHART
OBSERVATIONS:
THERE ARE A FEW POINTS OUT OF CONTROL IN STD. X BAR AND R CHART (REFER
EARLIER SLIDES)
REVISED R AND X BAR VALUES AFTER REMOVING 3 SAMPLES
3.273
. = 1.41

TARGET AVE. RESISTANCE = 21.0 Ω; TARGET S.D. = 1.0 Ω

COMPARE TARGET VALUES WITH REVISED CALCULATED VALUES. YOUR VIEWS?

TARGET S.D. NOT REALISTIC FOR THE PROCESS; TO BE REVIEWED. IF RETAINED, THEN
COMMON CAUSES NEED TO BE EXAMINED TO BRING ABOUT MAJOR CHANGES IN THE
PROCESS TO MEET THE TARGET.

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 RECAP

X BAR AND R CHART – WITHOUT AND WITH GIVEN

STANDARDS

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Control chart for Mean and Standard
deviation (X-bar and s chart)
 CAN X-BAR AND R CHART BE USED FOR LARGE SAMPLE SIZES?

 R-chart is easy to construct and use, but a standard deviation

chart (s-chart) is preferable for large sample sizes (> 10)
 Reason is that range accounts only for the maximum and
minimum sample values and is less effective for large samples,
where as sample standard deviation serves as a better measure
of process variability
 Population distribution of a quality characteristic is assumed to
be normal with a population standard deviation denoted by σ

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 X BAR AND s CHART
No Given Standards
X-bar chart
Centre line and Control limits are given by :

s chart
Centre line and Control limits are given by :

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 X BAR AND s CHART
No Given Standards
X bar – formula derivation (in X-bar and s chart)

Centre line is similar to that of X-bar FOR

in X-bar and R chart INFORMATION

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 X BAR AND s CHART
No Given Standards
s formula derivation (in X-bar and s chart)
FOR
INFORMATION

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 X BAR AND s CHART
With given standard

X bar chart
CLX  X 0 (SAME AS THAT OF X BAR IN XBAR AND R
UCLX  X 0  A 0 STANDARDIZED CHART)

LCLX  X 0  A 0

s chart
CLs  c4 0
UCLs  B6 0
LCLs  B5 0

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 X BAR AND s CHART
With given standard
X bar – formula derivation (in X-bar and s chart)

Let X o and  o represent the target values of the process

mean and standard deviation respectively and the centre
line and limits are given by
CL X  X 0 FOR
INFORMATION
0
UCL X  X 0  3.  X 0  A . 0
n
0
LCL X  X 0  3.  X 0  A . 0
n

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 X BAR AND s CHART
With given standard
s formula derivation (in X-bar and s chart)
With given standard, s chart centerline FOR
INFORMATION

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 EXAMPLE
The thickness of the magnetic coating on audio tapes is an important
characteristic. Random samples of size 4 are selected, and the thickness is measured
using an optical instrument. Table 7-3 shows the mean X and standard deviation s for
20 samples. The specifications are 38 ±4.5 micrometers. If a coating thickness is less
than the specifications call for, that tape can be used for a different purpose by
running it through another coating operation

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 X BAR AND s CHART - EXAMPLE

(a) Find the trial control limits for an X- and an s-chart.

(b) Assuming special causes for the out-of-control points, determine the
revised control limits.

(c) Assuming the thickness of the coating to be normally distributed, what

proportion of the product will not meet specifications?

(d) Comment on the ability of the process to produce items that meet
specifications.

(e) If the process average shifts to 37.8 micrometre, what proportion of the
product will be acceptable?

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 X BAR AND s CHART
No Given Standards
X-bar chart
Centre line and Control limits are given by :

s chart
Centre line and Control limits are given by :

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 EXAMPLE
The thickness of the magnetic coating on audio tapes is an important
characteristic. Random samples of size 4 are selected, and the thickness is measured
using an optical instrument. Table 7-3 shows the mean X and standard deviation s for
20 samples. The specifications are 38 ±4.5 micrometers. If a coating thickness is less
than the specifications call for, that tape can be used for a different purpose by
running it through another coating operation

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 SAMPLE PROBLEM
(a) Find the trial control limits for an X- and an s-chart. (Sum s = 95.8)

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 SAMPLE PROBLEM

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 EXAMPLE - CONTD

(Sum X=743.5)

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 SAMPLE PROBLEM

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 EXAMPLE - CONTD

(b) Assuming special causes for the out-of-control points, determine the revised control
limits.

REVISED CONTROL LIMITS WILL BE THE SAME AS TRIAL CONTROL LIMITS

– SINCE NO OUT OF CONTROL POINTS IDENTIFIED

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 EXAMPLE - CONTD

(c) Assuming the thickness of the coating to be normally distributed, what proportion of
the product will not meet specifications?

The specifications are 38 ±4.5 micrometers

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 SAMPLE PROBLEM

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 RECAP

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 CUMULATIVE STANDARD NORMAL
DISTRIBUTION

REFER APPENDIX A-3 OF TEXTBOOK T1 FOR VALUES

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 EXAMPLE - CONTD

(c) Assuming the thickness of the coating to be normally distributed, what proportion of
the product will not meet specifications?

The specifications are 38 ±4.5 micrometers

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 EXAMPLE - CONTD

(d) Comment on the ability of the process to produce items that meet
specifications.

39.28% OF PRODUCT NOT MEETING SPECIFICATIONS – TOO HIGH.

PROCESS APPEARS TO BE IN CONTROL; YET, HIGH NO. OF NON-CONFORMING

ITEMS.

HENCE, REVIEW FOR COMMON CAUSES.

EXISTING PROCESS NOT CAPABLE OF MEETING STATED SPECIFICATIONS.

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 EXAMPLE - CONTD
(e) If the process average shifts to 37.8 micrometre, what proportion of the
product will be acceptable?

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Control chart for individual units –
Moving Range (MR) Chart
Chart for sample size is 1, i.e. for individual units
Reasons for sample size to be ‘1’
– The rate of production is low
– Testing process may be destructive and the cost of the item is very
high
– If every manufactured unit is inspected
The value of the quality characteristic is expressed as ‘X’
Variability of the process is estimated from the ‘Moving Range’,
that are found from successive observations
– Moving Range of 2 observations is simply the difference between
them
– Moving Range are co-related, because they use common rather than
individual values and hence the pattern of MR chart must be
interpreted carefully

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Example – MR AND X CHART

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Example – MR AND X CHART

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 MR AND X CHART –
NO GIVEN STANDARD

MR CHART PROCESS STD. DEVIATION

MR
UCLMR  MR  3.  D4 MR
d2
MR
LCLMR  MR  3.  D3 MR
d2

X CHART

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 X BAR AND R CHART / MR and X CHART
No Given Standards
X CHART

MR CHART

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 SAMPLE PROBLEM
Table shows the Brinell hardness numbers of 20 individual steel fasteners and the
moving ranges. The testing process dents the parts so that they cannot be used for
their intended purpose. Construct the X-chart and MR-chart based on two
successive observations.
Specification limits are 32 ± 7.

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 EXAMPLE – MR AND X CHART
MR CHART

X CHART

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 EXAMPLE - MR AND X CHART

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 FACTOR VALUES

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 MR AND X CHART -
WITH GIVEN STANDARD

WITH GIVEN STANDARD

PROCESS STD. DEVIATION

ASSUMING n=2

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 OTHER CONTROL CHARTS

 Z-MR CHART
 CUMSUM CHART FOR INFO ONLY
 V-MASK METHOD
 MOVING AVERAGE METHOD
 EXPONENTIALLY WEIGHTED MOVING AVERAGE METHOD
(EWMA)

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 Control chart patterns and
corrective action
A ‘non-random identifiable’ pattern in the plot of a chart might
provide reason to look for special cause in a process
There are about 15 typical patterns identified by Western
Electric company and 9 of them have been discussed here
Natural Patterns
– No identifiable arrangement of the plotted point exists
– No point fall outside the control limits
– Majority of the points are near the centre line and few
points close to control limits
– Demonstrates the presence of stable system of common
causes

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 Control chart patterns – contd..

NATURAL PATTERNS

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 Control chart patterns – Contd.

• Sudden shifts in the level

– Can occur because of changes intentional or otherwise
in process settings (temperature, depth of cut etc.)
– New operators, new equipment, new vendors, new
methods are the reasons for sudden shift

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 Control chart patterns– Contd..

• Gradual shifts in the level

– Occurs when a process parameter changes gradually over a period of
time
– X-bar chart might exhibit such a shift because of incoming quality of
raw materials that would have changed with time
– Change in style of supervision, maintenance program etc.
– R-chart might exhibit such a shift because of new operator, decrease in
worker skill due to fatigue etc.

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 Control chart patterns – Contd..

• Trending pattern
– Differs from gradual shift in level, that trends do not stabilize or settle
down
– Represents changes that steadily increase or decrease
– For X-bar chart, can be due to tool wear, deterioration of equipment,
build up of debris on jigs and fixtures, change in temperature etc.
– For R-chart, it may be due to improvement in operator skill due to on
job training, decrease due to fatigue etc.

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 Control chart patterns – Contd..
• Cyclic patterns
– Characterized by repetitive periodic behaviour in the system
– Cycles of low and high points will appear on the control chart
– X-bar chart may exhibit because of rotation of operators, periodic
change of temperature or humidity, seasonal variation in incoming
components
– R-chart may exhibit this pattern because of operator fatigue and
getting energized in subsequent breaks, a difference between shifts,
periodic maintenance of equipments etc.
– If samples are taken so infrequently, only the high and low points will
be represented

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 Control chart patterns – Contd..

Wild patterns
– Can be classified as Bunches and Freaks
– Cluster of several observation that are different from
other points and special causes are associated with these
points
– Freaks
• are caused by external disturbances that influence one or more
samples
• They are points that are too small or large with respect to
control limits and fall outside the control limits and hence easy
to identify
• Care should be taken that no measurement or recording error
is associated with that freak point
• Some special causes may be sudden, very short-lived power
failures, use of new tool for a brief test period etc.

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 Control chart patterns – Contd..

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 Control chart patterns and corrective action –
Contd..
• Wild patterns – Contd.
– Bunches
• Cluster of several observation that are different from other points
• Possible causes may be use of new vendor, use of a different
machine, use of new operator etc., for a short time period.

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Control chart patterns and corrective action –
Contd..

Mixture patterns
– Effect of two or more populations in the sample
– Characterized by points that fall near the control limits, with
absence of points near the centre line
– Might be due to material from two different vendors, different
production method, two or more machine being represented

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Control chart patterns and corrective action –
Contd..
• Stratification patterns
– Is also due to presence of two or more population distribution
– Output is combined or mixed and samples are selected from it
– Majority of the points fall close to centre line, with very few points
near the control limits
– Can be misinterpreted as indicating unusually good control

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 CONTROL CHART PATTERNS AND
CORRECTIVE ACTION – contd…
Interaction patterns
– Occurs when the level of one variable affects the behaviour
of other variables associated with the quality characteristic
– Interaction pattern can be detected by changing the scheme
for rational sampling
– Example, low pressure and low temperature may produce a
desirable effect on output characteristic
– Effective sampling method would involve controlling the
temperature at several high values and then determining the
effect of pressure on output characteristic for each
temperature value
– If the R-chart shows the sample range to be small, then
information regarding the interaction could be used to
establish desirable process parameter settings

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 Control chart patterns – Contd..
• Interaction patterns

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 RECAP

 CONTROL CHARTS FOR VARIABLES :

 X BAR AND R CHART – NO GIVEN STANDARD

 X BAR AND R CHART – WITH GIVEN STANDARD
 X BAR AND s CHART – NO GIVEN STANDARD
 X BAR AND s CHART – WITH GIVEN STANDARD
 CONTROL CHART PATTERNS

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 SITUATED LEARNING ASSIGNMENT
Submission deadline -9th May, 2025 (e-Learn submission only)

Questions to be addressed in the report:

Q1. Prepare a problem statement for a problem related to your area of work.
Q2. Explain the background of the problem and identify the relevant area(s) of quality improvement
which can be focused upon.
Q3. Identify the special and common causes relevant to the process associated with your problem area.
Identify the causes which you feel contribute more to your problem. How do you propose to
resolve it. Give proper reasoning.
Q4. Which quality tool(s) (from the 7 basic QC tools) can be used to identify the major causes for your
problem. Justify your selection.
Q5. Using the dataset related to your problem, take a sample of adequate size, draw a histogram and
calculate the mean and standard deviation, identify the type of skewness and kurtosis of your
sample data. What are your observations?
Q.6 Identify which type of probability distribution would be appropriate for the dataset associated with
your process under study. Justify your selection. (You may take guidance of some of the methods
discussed in Chapters 4 & 5 of the Textbook)

Report must be verified and signed by your organization mentor.

There will be no extension of given deadline – any submission after the deadline will not be evaluated.

20th Apr’25 Quality Control, Assurance & Reliability BITS Pilani 102
 THANK YOU

20th Apr’25 Quality Control, Assurance & Reliability BITS Pilani 103