Statistical Quality Control Lecture Notes.pdf

Control is a system for measuring and checking or inspecting a phenomenon. It suggests when to inspect, how often to inspect and how much to inspect. Control ascertains quality characteristics of an item, compares the same with prescribed quality characteristics of an item, compares the same with prescribed quality standards and separates defective item from non-defective ones.
Statistical Quality Control (SQC) is the term used to describe the set of statistical tools used by quality professionals.
SQC is used to analyze the quality problems and solve them. Statistical quality control refers to the use of statistical methods in the monitoring and maintaining of the quality of products and services.
Variation in manufactured products is inevitable; it is a fact of nature and industrial life. Even when a production process is well designed or carefully maintained, no two products are identical.
The difference between any two products could be very large, moderate, very small or even undetectable depending on the sources of variation.
For example, the weight of a particular model of automobile varies from unit to unit, the weight of packets of milk may differ very slightly from each other, and the length of refills of ball pens, the diameter of cricket balls may also be different and so on.
The existence of variation in products affects quality. So the aim of SQC is to trace the sources of such variation and try to eliminate them as far as possible.

1. Chap – 5:
Statistical
Quality
Control
May 21
2022
For B.Com/M.Com Students
Quantitative
Techniques

2. Chap – 5: Statistical Quality Control


2
Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

╬ Table of Contains
• Causes of variations in quality characteristics
• Quality Control Charts - purpose and logic
• Process under control and out of control, Warning limits
• Types of Charts
o Variable charts (Mean & Range Charts)
o Attribute charts ( p, C charts)
• Acceptance Sampling (Theory Only)
• Numerical/Practical Questions

4.1 Introduction
CCCC Statistics:
Statistics means the good amount of data to obtain reliable results.
The Science of statistics handles this data in order to draw certain
conclusions. Its techniques find extensive applications in quality
control, production planning and control, business charts, linear
programming etc.
C Quality:
Quality refers to the sum of the attributes or prop erties that
describe a product (item). These are generally expressed in terms of
specific product characteristics such as length, width, weight, color
etc.
In simple words quality means degree of perfection.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

C Control:
Control is a system for measuring and checking or i nspecting a
phenomenon. It suggests when to inspect, how often to inspect and
how much to inspect. Control ascertains quality characteristics of
an item, compares the same with prescribed quality characteristics
of an item, compares the same with prescribed quality standards
and separates defective item from non-defective ones.
Statistical Quality Control (SQC) is the term used to describe the set
of statistical tools used by quality professionals.
SQC is used to analyze the quality problems and sol ve them.
Statistical quality control refers to the use of statistical methods in
the monitoring and maintaining of the quality of pr oducts and
services.
4.2 Causes of variations in quality characteristics
Variation in manufactured products is inevitable; it is a fact of
nature and industrial life. Even when a production process is well
designed or carefully maintained, no two products are identical.
The difference between any two products could be ver y large,
moderate, very small or even undetectable depending on the
sources of variation.
For example, the weight of a particular model of automobile varies
from unit to unit, the weight of packets of milk may differ very
slightly from each other, and the length of refills of ball pens, the
diameter of cricket balls may also be different and so on.
The existence of variation in products affects quality. So the aim of
SQC is to trace the sources of such variation and try to eliminate
them as far as possible.
The causes of variation are broadly classified into two categories:

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

1. Chance causes, and
2. Assignable causes.

Chance Causes
Chance causes are also known as random or natural or common
causes. Even in a well designed or carefully maintained production
process, variability exists in the product due to s ome
natural/random causes.
Even if the process is operated under the same conditions, that is,
the quality of raw materials used is same and there is no change in
the machine settings, operators or the environment, there is a
specific pattern of variability in the product.
For 18 Process Control example, the diameter of ball bearings varies
slightly, there is a slight variation in the weight of cricket balls, the
fuel efficiency of a particular model of automobile is varies slightly
and so on. Such variability is due to different common or chance
causes, may which affect the process output in minor ways.
Such causes are known as chance causes of variation. These may
arise due to, inflexibility of aged machines, variability in purchased
material, poor lighting, extent of worker training or other non
obvious reasons.
These may or may not be present at the same time, but when taken
together produce random results. If the quality of the output varies
too much due to chance causes, the process must be redesigned or
modified to eliminate one or more of these causes. Since process
redesigning or modification is the responsibility of the management,
the elimination of common/chance causes of variation is usually
the responsibility of the management and not that of workers.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

It may not be possible to eliminate all chance causes in a process.
However, even if variation due to chance causes is present in the
production process, it is still said to be under statistical control.

Assignable Causes
Another kind of variability may be present occasion ally in the
output of a process. The causes of such type of variability are not
due to the process design, but take place because of changes in raw
material, machine, operator, environment or any other component
of the process.
These causes are called assignable causes and are also known as
special or non-random or unnatural causes. Accident al improper
setting of the machine, a worker falling ill and still continuing to
work change of operators or shift, breakages, misreading of scales,
batch of defective raw material, etc. is examples of assignable
causes.
Since the effect of assignable causes is localized within a process,
these may be eliminated by workers or their immediate supervisor.
The variability due to assignable causes is generally larger than the
variability due to chance causes and it usually rep resents an
unacceptable level of process performance.
A process that is operating in the presence of assignable causes is
said to be an out-of-control process.
4.3 Quality Control Charts - purpose and logic
You will agree that graphical representation is one of the most
sensitive statistical instruments. So we can represent the quality
characteristic of the output product such as weight, length,

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

diameter, defects, etc. graphically to understand, describe or
monitor process variation.
The idea of representing quality characteristics graphically was first
given by Walter A. Shewhart. He invented control cha rts for the
industrial processes to distinguish acceptable (chance) variation
from the assignable variation.
He observed that with the help of control charts, the occurrence of
assignable causes of variation could be detected qu ickly and
corrective action could be taken to eliminate them.
A control chart is a two-dimensional graphical display of a quality
characteristic that has been measured or computed i n terms of
mean or other statistic from samples and plotted ag ainst the
sample number or time at which the sample is taken f rom the
process.
The concept of control chart is based on the theory of sampling and
probability. In a control chart, a sample statistic of a quality
characteristic such as mean, range, proportion of defective units,
etc. is taken along the Y-axis and the sample number or time is
taken along the X-axis. A control chart consists of three horizontal
lines, which are described below:
1.
Centre Line (CL) – The centre line of a control chart
represents the value which can have three different
interpretations depending on the available data. First, it can
be the average value of the quality characteristic or the
average of the plotted points. Second, it can be a standard or
reference value, based on representative prior data or an
aimed (targeted) value based on specifications. Third, it can be
the population parameter if that value is known. The centre
line is usually represented by a solid line.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

2. Upper Control Line – The upper control line represents the
upper value of the variation in the quality characteristic. So
this line is called upper control limit (UCL). Usually, the UCL
is shown by a dotted line.


3. Lower Control Line – The lower control line represents the
lower value of the variation in the quality characteristic. So
this line is called lower control limit (LCL). Usually, the LCL is
shown by a dotted line.

The UCL and LCL also have three interpretations depending on the
available data same as the centre line. These limits are obtained
using the concept of 3σ (three sigma) limits.
4.4 3σ Limits
The quality characteristic can be described by a pr obability
distribution or a frequency distribution.
In most situations, a quality characteristic follows a normal
distribution or can be approximated by a normal distribution.
A normally distributed random variable (X) lies between
µ − 3σ to
µ + 3σ is 0.9973 where μ and σ are the mean and the standard
deviation of the random variable (X) respectively.
Thus,pnrM 2 o1 v e v M T o1q= 0.9973.
So the probability that the random variable X lies outside the limits
M o1 is 1 − 0.9973 = 0.0027which is very small. It means that if we
consider 100 samples, most probably 0.27 of these may fall outside
thepM o1 limits. So if an observation falls outside the 3
σlimits in
100 observations, it is logical to suspect that something might have
gone wrong.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Therefore, the control limits on a control chart are set up by using
3
σ limits. The UCL and LCL of a control chart are called 3σ limits of
the chart. The question is: How do we calculate 3σ limits?
Suppose M is a sample statistic (e.g., mean, range, proportion of
defectives, etc.) that measures some quality charac teristic of
interest.
Further suppose that M
# and 1
# are the mean and standard error
(standard deviation) of the sample statistic M, respectively. Then the
centre line and control limits for controlling the quality
characteristic are given by:
• Central Limit (CL) g c
# ----------------------------------------- (1)
• Upper Control Limit (UCL)lg c
#T o1
# ----------------------- (2)
• Lower Control Limit (LCL) g c
#2 o1
# ----------------------- (3)










The chart in above figure shows the centre line (CL), lower control
limit (LCL) and upper control limit (UCL). The UCL and LCL are set
at the distance M
# o1
# from the centre lineppM
#. Note from the

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

above the area covered between the UCL and LCL is 0 .9973
(99.73%).
So the probability that an observation falls outside these limits is
0.0027.
If the sample points fall between the control lines, the process is
said to be under statistical control. But, if one or more points lie
outside the control limits, control chart alarms (indicates) that the
process is not under statistical control. Some assignable causes are
present in the process. To bring the process under statistical
control, it is necessary to investigate the assignable causes and take
corrective action to eliminate them and then contin ue the
production process.
h
Types of Control Charts
Control charts can be used to measure any character istic of a
product, such as the weight of a cereal box, the nu mber of
chocolates in a box, or the volume of bottled water. The different
characteristics that can be measured by control cha rts can be
divided into two groups: variables and attributes.













Control
Charts
Variable
Charts
Attributes
Charts
e$− %ℎ'() *− %ℎ'() +− %ℎ'() ,+− %ℎ'()
%− %ℎ'()

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

A control chart for variables is used to monitor characteristics
that can be measured and have a continuum of values , such as
height, weight, or volume.

A soft drink bottling operation is an example of a variable measure,
since the amount of liquid in the bottles is measured and can take
on a number of different values.

Other examples are the weight of a bag of sugar, the temperature of
a baking oven, or the diameter of plastic tubing.

A control chart for attributes, on the other hand, is used to
monitor characteristics that have discrete values a nd can be
counted. Often they can be evaluated with a simple y es or no
decision.

Examples include color, taste, or smell.

The monitoring of attributes usually takes less time than that of
variables because a variable needs to be measured (e.g., the bottle
of soft drink contains 15.9 ounces of liquid).

An attribute requires only a single decision, such as yes or no, good
or bad, acceptable or unacceptable.
e.g., the apple is good or rotten,
the meat is good or stale,
the shoes have a defect or do not have a defect,
the light bulb works or it does not work

or counting the number of defects (e.g., the number of broken
cookies in the box, the number of dents in the car, the number of
barnacles on the bottom of a boat).

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

h Decision tree: Which chart is used in particular data?





h
-.− /0123
The x-bar chart is quality control chart used to monitor the mean
and variation of a process based on samples taken in a given time.
The control limits on both chats are used to monitor the mean and
variation of the process going forward. If a point is out of the control
limits, it indicates that the mean or variation of the process is out-
of-control; assignable causes may be suspected at this point.

On the x-bar chart, the y-axis shows the grand mean4e56 and the
control limits while the x-axis shows the sample group.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

• List of useful formulae for making -. - chart:
- 7w8 g y5T o1
- 8w8 g y52 o1
- w8 g y5=
∑:$
;

- v g <
∑4=
>?:56
@
;

- , = AB.BC D'E+FGD

• Steps for making -. - chart:
-
To make a table for finding necessary UCL, LCL and CL.
-
To calculate UCL, LCL and CL.
-
To take the sample data on X – axis and e5 - values on Y – axis.
-
To make a straight line parallel to X – axis, from the pointpe5.
-
To make doted straight lines from the values of UCL and LCL
which are lies on Y – axis.

• Conclusion derived from the H. - chart:

1.
Controlled Variation
-
Controlled variation is characterized by a stable and
consistent pattern of variation over time, and is associated
with common causes. A process operating with controlled
variation has an outcome that is predictable within the
bounds of the control limits.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

2. Uncontrolled Variation
- Uncontrolled variation is characterized by variation that
changes over time and is associated with special causes.
The outcomes of this process are unpredictable; a customer
may be satisfied or unsatisfied given this unpredictability.






Ex – 1: JetMate, a manufacturer of jet engines, intended to closely
monitor the maximum power generated by its engines. JetMate
obtained a set of 40 readings on the maximum power for its
successively produced engines, as given in the following table.
Consider sub-samples consists of every 4 consecutive observations
in the sample (read left to right row wise).Here, there are 10 such
sub-samples. Compute the average for each sub-sample, which can
be used as the sub-sample statistic, and construct an e$ – chart.
And make a conclusion about it.
Reading on maximum power delivered by
,=40 jet engines
125 120 121 123 122 130 124 122 120 122
118 119 123 124 122 124 121 122 138 149
123 128 122 130 120 122 124 134 137 128
122 121 125 120 132 130 128 130 122 124

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Solu: For preparing of e$ - chart, first we have to find the following.
- 7w8 g y5T o1
- 8w8 g y52 o1
- w8 g y5=
∑:$
;

- v g <
∑4=
>?:56
@
;

- , = AB.BC D'E+FGD


4-consecutive sub-
groups
e$ J
K−e5 4J
K−e56
L

Sample sub-groups 1 2 3 4
1 125 120 121 123 122.25 -3.03 9.18
2 122 130 124 122 124.5 -0.78 0.61
3 120 122 118 119 119.75 -5.53 30.58
4 123 124 122 124 123.25 -2.03 4.12
5 121 122 138 149 132.5 7.22 52.13
6 123 128 122 130 125.75 0.47 0.22
7 120 122 124 134 125 -0.28 0.08
8 137 128 122 121 127 1.72 2.96
9 125 120 132 130 126.75 1.47 2.16
10 128 130 122 124 126 0.72 0.52
Total 102.56

Now, e5=
∑:$
;
=
MLNL.ON
MP
= 125.28
v gl<
∑4=
>?:56
@
;
= <
MPL.NS
MP
= 3.20
7w8 g y5h rv g L(Q (R h rr3.20q= 134.88
8w8 g y52 rv g L(Q (R 2 rr3.20q= 115.68
w8 g y5= 125.28

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Sample Mean







Sample sub-groups
As per above mentioned graph, all the points are in between the
LCL and UCL therefore quality of product is under control.
h
U − /0123
R charts are used to monitor the variation of a process based on
samples taken from the process at given times (hours, shifts, days,
weeks, months, etc.). The measurements of the samples at a given
time constitute a subgroup. Typically, an initial series of subgroups
is used to estimate the standard deviation of a pro cess. The
standard deviation is then used to produce control limits for the
range of each subgroup. During this initial phase, the process
should be in control. If points are out-of-control during the initial
(estimation) phase, the assignable cause should be determined and
the subgroup should be removed from estimation. Determining the
process capability (see R & R Study and Capability Analysis
procedures) may also be useful at this phase. Once the control
110
120
130
140
150
0 1 2 3 4 5 6 7 8 9 10
UCL=134.68
CL=125
LCL=115.68

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

limits have been established for the R chart, these limits may be
used to monitor the variation of the process going forward. When a
point is outside these established control limits, it indicates that the
variation of the process is out-of-control. An assignable cause is
suspected whenever the control chart indicates an out-of control
process.

• List of useful formulae for making U - chart:
- 7%8
V= W
X*$
- 8%8
V= W
Y*$
- %8
V= *$=
∑V
;

- * = Z'E+FG (',[G = J
\]=− J
\K;
- , = AB.BC D'E+FGD
- W
Y and W
X are the tabulated values depends on sample size.
Ex -1: The following data structure is to have the data in five
columns, with one subgroup per row. Construct the co ntrol chart
based on the range. Given that according to sample size the value of
W
Y= 0 and W
X= 2.115.
Sub-group Z
M Z
L Z
Y Z
X Z
N
1 2 6 3 8 5
2 8 8 7 7 9
3 6 2 2 4 3
4 5 6 7 6 10
5 8 2 6 5 0

Solu: For preparing of * - chart, first we have to find the following.
- 7%8
V= W
X*$
- 8%8
V= W
Y*$
- %8
V= *$=
∑V
;

- * = Z'E+FG (',[G = J
\]=− J
\K;

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Sub-group Z
M Z
L Z
Y Z
X Z
N
Sample Range
R
1 2 6 3 8 5 8−2=6
2 8 8 7 7 9 9−7=2
3 6 2 2 4 3 6−2=4
4 5 6 7 6 10 10−5=5
5 8 2 6 5 0 8−0=8
Total 25

Now *$=
∑V
;
=
LN
N
= 5
Given that W
Y= 0 and W
X= 2.115.
7%8
V= W
X*$= 2.115 × 5 = 10.575 = 10.58
8%8
V= W
Y*$= 0 × 5 = 0
%8
V= *$= 5
Sample range







Sample sub-groups

As per above mentioned graph, all the points are in between the
LCL and UCL therefore quality of product is under control.
2
4
6
8
10
0 1 2 3 4 5
CL=5
LCL=0
UCL=10.58

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

h _ − /0123
A p-chart is an attributes control chart used with data collected in
subgroups of varying sizes. Because the subgroup size can vary, it
shows a proportion on nonconforming items rather than the actual
count. P-charts show how the process changes over tim e. The
process attribute (or characteristic) is always described in a yes/no,
pass/fail, go/no go form.
For example, use a p-chart to plot the proportion of incomplete
insurance claim forms received weekly. The subgroup would vary,
depending on the total number of claims each week. The p-chart is
used to determine if the process is stable and predictable, as well as
to monitor the effects of process improvement theories.
• List of useful formulae for making _ - chart:
-
7%8
`= +̅ + 3<
`̅rM?`̅q
;$

- 8%8
`= +̅ − 3<
`̅rM?`̅q
;$

- %8
`= +̅ =
∑;`
∑;

- ,$ =
∑;
b
; Where b = ,cEdG( BC FB)D/e)GED and , = D'E+FG DefG.
• How do you create a p – Chart?
1. Determine the subgroup size. The subgroup size must be large
enough for the p chart; otherwise, control limits may not be
accurate when estimated from the data.
2. Calculate each subgroups non conformities rate= np/n
3. Compute +̅ = total number of defectives / total number of
samples =Σnp/Σn
4. Calculate upper control limit (UCL) and low control limit (LCL).
If LCL is negative, then consider it as 0. Since the sample sizes

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

are unequal, the control limits vary from sample interval to
sample interval.
5. Plot the graph with proportion on the y-axis, lots on the x-axis:
Draw centerline, UCL and LCL.


6. Finally, interpret the data to determine whether the process is
in control.

Ex -1: ABC manufacturing produces thousands of tubes every day.
A Quality inspector randomly drawn variable samples for 10 days
and reported the defective tubes for each sample size. Based on the
given following data, prepare the control chart for fraction defective
and determine the process in statistical control?
Lot 1 2 3 4 5 6 7 8 9 10
Sample
size
,
1250 1300 1350 1200 1050 1050 1200 1100 1000 600
No. of
defective
in the
sample
,+
18 15 13 16 8 6 18 14 22 12

Solu: For preparing of + - chart, first we have to find the following.
-
7%8
`= +̅ + 3<
`̅rM?`̅q
;$

- 8%8
`= +̅ − 3<
`̅rM?`̅q
;$

- %8
`= +̅ =
∑;`
∑;

,B.BC FB)D
WGCGg)ehG (')G=+=
,+
,

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

- ,$ =
∑;
b
; Where b = ,cEdG( BC FB)D/e)GED and , = ,cEdG( BC D'E+FGD.


Now ,$ =
∑;
i
=
MLNPjMYPPjMYNPjMLPPjMPNPjMLPPjMMPPjMPPPjSPP
MP
=
MPPNP
MP
= 1005



%8
`= +̅ =
∑;`
∑;

=
MkjMNjMYjMSjkjSjMkjMXjLLjML
MLNPjMYPPjMYNPjMLPPjMPNPjMLPPjMMPPjMPPPjSPP

=
MXL
MPPNP

= 0.014


7%8
`= +̅ + 3<
`̅rM?`̅q
;$
= 0.014 +<
P–PMXrM?P–PMXq
MPPN
= 0.018


8%8
`= +̅ − 3<
`̅rM?`̅q
;$
= 0.014 −<
P–PMXrM?P–PMXq
MPPN
= 0.010

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.


Lot No.
Sample
Size
,
No. of
defective
in the
sample
,+
Defective
Rate
+=,+/,
1 1250 18 0.0144
2 1300 15 0.0115
3 1350 13 0.0096
4 1200 16 0.0133
5 1050 8 0.0076
6 1050 6 0.0057
7 1200 18 0.0150
8 1100 14 0.0127
9 1000 22 0.0220
10 600 12 0.0200











0 1 2 3 4 5 6 7 8 9 10
0.005
0.010
0.015
0.020
0.025
Lot Number
Defective rate p

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

The proportion of defectives on day 9 and 10 are higher than the
upper control limit (UCL). Therefore the process is out of control.
Black belts or statisticians to identify the root cause for the cause
and take appropriate corrective action to bring the process in
control.
h
l − /0123 In statistical quality control, the c-chart is a type of control
chart used to monitor "count"-type data, typically total number of
nonconformities per unit. It is also occasionally used to monitor the
total number of events occurring in a given unit of time.
The c-chart differs from the p-chart in that it accounts for the
possibility of more than one nonconformity per inspection unit, and
that (unlike the p-chart and u-chart) it requires a fixed sample size.
The p-chart models "pass"/"fail"-type inspection only, while the c-
chart (and u-chart) give the ability to distinguish between (for
example) 2 items which fail inspection because of one fault each
and the same two items failing inspection with 5 faults each; in the
former case, the p-chart will show two non-conformant items, while
the c-chart will show 10 faults.
Nonconformities may also be tracked by type or location which can
prove helpful in tracking down assignable causes.
Examples of processes suitable for monitoring with a c-chart
include:
• Monitoring the number of voids per inspection unit in injection
molding or casting processes
• Monitoring the number of discrete components that must be re-
soldered per printed circuit board
• Monitoring the number of product returns per day
The Poisson distribution is the basis for the chart and requires the
following assumptions:

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

• The number of opportunities or potential locations for
nonconformities is very large.
• The probability of nonconformity at any location is small and
constant.
• The inspection procedure is same for each sample and is carried
out consistently from sample to sample.

• List of useful formulae for making l - chart:
-
7%8
m= g̅ + 3√g̅

- 8%8
m= g̅ + 3√g̅

- %8
m= g̅ =
opq]r ;s\tuv pw xuwumqy wps;x p; K;y`umqKp;
zp.pw {]\`ru


Ex – 1: In a certain sampling inspection, the number of defectives
found in 10 samples out of 100 as given below:
Casting 1 2 3 4 5 6 7 8 9 10
No. of defects found on
inspection
16 18 11 18 21 10 20 18 17 21
Do these indicate that the quality characteristic under
inspection in under statistical quality control? Answer the
same using c – chart.
Solu:
For preparing of g - chart, first we have to find the following.
-
7%8
m= g̅ + 3√g̅

- 8%8
m= g̅ + 3√g̅

- %8
m= g̅ =
opq]r ;s\tuv pw xuwumqy wps;x p; K;y`umqKp;
zp.pw {]\`ru

24. Chap – 5: Statistical Quality Control


24
Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Now, %8
m= g̅ =
opq]r ;s\tuv pw xuwumqy wps;x p; K;y`umqKp;
zp.pw {]\`ru

=
MSjMkjMMjMkjLMjMPjLPjMkjMOjLM
MP

=
MOP
MP

%8
m= 17

• 7%8
m= g̅ + 3√g̅

= 17 + 3√17
v 3k y noI.34n3U
= 17 + 12.3693
7%8
m= 29.3693

• 8%8
m= g̅ − 3√g̅

= 17 − 3√17
v 3k e noI.34n3U
= 17 − 12.3693
8%8
m= 4.6307

25. Chap – 5: Statistical Quality Control

25
Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.















0
5
10
15
20
25
30
No. of Defects

1 2
5: Statistical Quality Control
Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.



Casting No’s
8 7 9 4 6 3 5
5: Statistical Quality Control

Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.
10