Quality Control
Introduction
• Statistical Quality Control is a process of
maintaining quality of industrial products under
some standard specifications.
• The term ‘quality’ in statistical quality control
means an attribute of the product that determines
its fitness for use.
• In fact, quality means level of standard which in
turn depends on four M’s and they are- materials,
manpower, machines and management.
• Two apparently identical parts made under
carefully controlled conditions, from the same
batch of raw material, and only seconds apart by
the same machine, can nevertheless be different in
many respects.
• Any manufacturing process, however good, is
characterized by a certain amount of variability,
which is of random nature and which cannot be
completely eliminated.
• In fact, variation in the quality of manufactured
product in the repetitive process in industry is
inherent and inevitable.
• These variations are broadly classified as being due
to the two causes, they are …
– Chance causes
– Assignable causes
• Chance causes of variations results from many
minor causes that behave in a random manner.
The variations due to these causes is beyond the
control of human hand and cannot be prevented/
eliminated under any circumstances.
• The range of such variation is known as ‘natural
tolerance of the process’.
• The second type of variation attributed to any
production process is due to non-random causes,
so called assignable causes.
• These type of variation may occur at any stage of
production process, i.e., right from the arrival of
raw material to final delivery of the manufactured
products.
• Some of the important factors of assignable causes
of variation are– defective raw materials
– new technology or operation
– negligence of operators
– improper handling of machines
– faulty equipments
– unexperienced technical staffs, etc.
• These causes can be identified and eliminated and
are to be discovered in a production process
before the production becomes defective.
• Statistical Quality Control (SQC) is the method of
identifying presence of assignable causes in any
production process.
• The main purpose of SQC is to devise statistical
technique which would help us in separating the
assignable causes from the chance causes, thus
enabling us to take immediate remedial action
whenever assignable causes are present.
• A production process is said to be statistically
under control, if it is governed by the random or
chance causes alone, in the absence of assignable
causes of variations.
• SQC helps in the detection and correction of many
production troubles and substantial improvement
in the product quality.
• It tells us when to leave a process alone and when
to take action to correct troubles, thus preventing
frequent and unwanted adjustments.
• It provides better quality assurance at lower
inspection cost.
• SQC reduce waste of time and material to the
absolute minimum by giving an early warning of
the cost of production and hence may lead to
more profit.
• One of the methods used to detect the presence of
assignable cause of variation in a production
process is ‘Cotrol Chart’.
Control Chart - Introduction
• Control chart is a simple pictorial device for
detecting unnatural pattern of variation in data
resulting from repetitive processes.
• The concept of control chart was first discovered
and developed by Walter A. Shewart, a physicist,
in 1924.
• These charts provide criteria for detecting lack of
statistical control.
• Control charts are based on the theory of
probability and sampling.
• Control charts provide a powerful tool of
discovering and correcting the assignable causes of
variation, thus enabling us to stabilize and control
processes at desired performance and thus bring
the process under statistical control.
• Control charts are simple to construct and easy to
interpret.
• In fact, control charts tell us whether the sample
points fall within 3 limits or not.
• Any sample point going outside the 3 limits is an
indication of lack of control, i.e., presence of some
assignable causes of variation which must be
identified and eliminated.
Basic Theory of Control Chart
• No production process is perfect enough to
produce all the items exactly alike. Some amount
of variation in the produced items is inherent in
any production scheme. This variation is the result
of both chance causes and assignable causes.
• Considering that the measurements are sample
statistics from a normal population X, we have
P(  - 3σ  X   + 3σ) = 0.99
• These two limits  - 3σ and  + 3σ are used in
control charts for detecting whether a process is in
control or not. The limit represented by  - 3σ is
called lower control limit (LCL) and the limit
represented by  + 3σ is called upper control limit
(UCL). So,
– LCL =  - 3σ, and
– UCL =  + 3σ
• Since for normal population X,  = E(X) and σ2 =
V(X) , so it follows that
LCL  E( X )  3 V ( X )
UCL  E( X )  3 V ( X )
Types of Control Charts
• Following two types of control charts are used as
most important statistical tools for data analysis in
quality control– A) Control charts for measurements (variables)
– B) Control charts for attributes
Control Chart for Measurement
• Control charts for measurement includes
(a) Control Charts of process mean and
(b) Control charts for process variability
Control Chart for Process Mean
• Control charts for process mean is also called X
chart.
• It is a statistical tool used to detect the presence of
assignable cause of variation in a production
process by studying the variation in means of
sample observations.
• The process of constructing X chart consists of
drawing ‘k’ number of samples each containing ‘n’
units as follows-
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Obervations of Transaction Time
63
55
56
53
60
63
60
65
57
60
61
65
58
64
60
61
79
68
65
61
55
66
62
63
57
61
58
63
58
51
61
57
65
66
62
68
73
66
61
70
57
63
56
64
66
63
65
59
63
53
69
60
68
67
59
58
70
62
66
80
65
59
60
61
63
69
68
56
61
56
62
59
65
57
69
62
70
60
67
79
61
61
66
57
74
56
55
66
61
72
62
70
61
65
71
62
70
61
55
58
• Then the means of all ‘k’ samples are computed
and these means are plotted against the sample
number as follows:
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Obervations of Transaction Time
63
55
56
53
60
63
60
65
57
60
61
65
58
64
60
61
79
68
65
61
55
66
62
63
57
61
58
63
58
51
61
57
65
66
62
68
73
66
61
70
57
63
56
64
66
63
65
59
63
53
69
60
68
67
59
58
70
62
66
80
65
59
60
61
63
69
68
56
61
56
62
59
65
57
69
62
70
60
67
79
Average
61
57.6
61
61.8
66
61.8
57
60
74
69.4
56
60.4
55
58.8
66
58.6
61
64.4
72
68.4
62
60.4
70
64.6
61
61.2
65
63.4
71
69.8
62
61.4
70
65.2
61
59.8
55
61.6
58
66.8
Average
80
70
60
50
40
Average
30
20
10
0
0
5
10
15
20
25
• The final step is to locate three lines, namely,
central line (CL), lower control line (LCL) and upper
control line (UCL) in the chart.
• We consider following cases for constructing these
control lines
• Case I – When process mean and standard
deviation are known (with values  and )
• In this case CL  
LCL    3
UCL    3

n

n
• Sometime, CL  
LCL    A
UCL    A
where, 𝑨 =
𝟑
𝒏
its value for different levels of n
are given in control chart table.
The control chart table is shown below-
• In above example, if it is known from any source
that process mean (or population mean) is 60 and
process standard deviation is 2.5, then
CL    60

2.5
LCL    3
 60  3 
 56.65
n
5

2.5
UCL    3
 60  3 
 63.35
n
5
• So that complete X-bar chart looks like
UCL
CL
LCL
CL    60

2.5
LCL    3
 60  3 
 56.65
n
5

2.5
UCL    3
 60  3 
 63.35
n
5
• In the control chart, if all the points lie between
LCL and UCL, then it is interpreted that there are
only random causes of variation and there are no
assignable cause of variation in the process, or, the
process is in controlled condition.
• If many points lie beyond LCL and UCL, then it is
interpreted that there are assignable causes of
variation in the process, or, the process is out of
control.
• Case II – When process mean and standard
deviation are unknown
• In this case, it is necessary to estimate them on the
basis of preliminary samples.
• For the estimation of process mean, usually the
grand mean of all ‘k’ samples is calculated as 𝒙
and it is used as the estimate of .
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
60
57
58
79
55
57
58
65
73
57
66
63
68
70
65
63
61
65
70
63
60
64
68
66
61
51
66
66
63
63
53
67
62
59
69
56
57
60
60
61
60
65
62
58
61
62
61
56
65
69
59
66
60
68
62
69
67
65
65
61
61
63
63
57
68
70
64
59
60
58
80
61
56
59
62
79
Grand Average
61
66
57
74
56
55
66
61
72
62
70
61
65
71
62
70
61
55
58
61.8
61.8
60
69.4
60.4
58.8
58.6
64.4
68.4
60.4
64.6
61.2
63.4
69.8
61.4
65.2
59.8
61.6
66.8
62.77
• For the estimation of , there are two methods.
One is to consider sample standard deviation and
another is to consider sample range.
• For the construction of X-bar chart based on
sample standard deviation, first of all the standard
deviation of all ‘k’ samples (each containing ‘n’
units) is computed by formula

1 n
si 
xij  xi

n j 1

2
• Then the average of these standard deviations is
computed by formula
1 k
s   si
k i 1
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Obervations of Transaction Time
63
55
56
53
60
63
60
65
57
60
61
65
58
64
60
61
79
68
65
61
55
66
62
63
57
61
58
63
58
51
61
57
65
66
62
68
73
66
61
70
57
63
56
64
66
63
65
59
63
53
69
60
68
67
59
58
70
62
66
80
65
59
60
61
63
69
68
56
61
56
62
59
65
57
69
62
70
60
67
79
Grand Average
Average Std. Dev.
61
57.6 4.219005
61
61.8 2.167948
66
61.8 3.701351
57
60 2.738613
74
69.4 7.162402
56
60.4 4.722288
55
58.8 3.193744
66
58.6 5.504544
61
64.4 2.880972
72
68.4 4.929503
62
60.4 3.646917
70
64.6 4.037326
61
61.2 5.761944
65
63.4 4.615192
71
69.8 6.723095
62
61.4 2.302173
70
65.2 5.80517
61
59.8 2.387467
55
61.6 5.727128
58
66.8 8.408329
62.77 4.531756
• Next, after plotting sample means in graph the
control lines are determined by using formulas
CL  x
LCL  x  A1 s
UCL  x  A1 s
• Here, A1 is a constant and its value is determined
from control chart table shown below-
• For the construction of X-bar chart based on
sample range, first of all the ranges of all ‘k’
samples are computed and then the average of
these ranges is computed as 𝑹.
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Obervations of Transaction Time
63
55
56
53
60
63
60
65
57
60
61
65
58
64
60
61
79
68
65
61
55
66
62
63
57
61
58
63
58
51
61
57
65
66
62
68
73
66
61
70
57
63
56
64
66
63
65
59
63
53
69
60
68
67
59
58
70
62
66
80
65
59
60
61
63
69
68
56
61
56
62
59
65
57
69
62
70
60
67
79
Grand Average
Average Range
61
57.6
10
61
61.8
5
66
61.8
9
57
60
7
74
69.4
18
56
60.4
11
55
58.8
8
66
58.6
15
61
64.4
7
72
68.4
12
62
60.4
8
70
64.6
11
61
61.2
16
65
63.4
10
71
69.8
18
62
61.4
6
70
65.2
14
61
59.8
6
55
61.6
14
58
66.8
21
62.77
11.3
• Then, after plotting sample means in graph the
control lines are determined by using formulas
CL  x
LCL  x  A2 R
UCL  x  A2 R
• Here, the value of constant A2 is obtained from
control chart table.
• For observation given in above table, X-bar chart
based on R is constructed as follows• Here x  62.77
• From control chart table, for n = 5, A2 = 0.577
• Also, R  11.3
• So,
CL  x  62.77
LCL  x  A2 R  62.77  0.577 11.3  56.25
UCL  x  A2 R  62.77  0.577 11.3  69.29
UCL
CL
LCL
Control Chart for Process Variability
• In controlling a process, it may not be enough to
monitor the process mean, but it is also required
to monitor process variability. Although an
increase in process variability may become more
apparent from increased fluctuations on the X ’s,
a more sensitive test of shifts in process variability
is provided by separate control charts namely (a)
-chart and (b) R-chart.
 - Chart
• Not required
R - Chart
• R-chart measures process variability based on
sample ranges.
• In this chart, it is assumed that the distribution of
range of samples is approximately normal.
• For the construction of R-chart, ‘k’ number of
samples each containing ‘n’ units are taken
randomly.
• Then the range of each sample is obtained.
• Next, these ranges are plotted against the sample
number.
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Obervations of Transaction Time
63
55
56
53
60
63
60
65
57
60
61
65
58
64
60
61
79
68
65
61
55
66
62
63
57
61
58
63
58
51
61
57
65
66
62
68
73
66
61
70
57
63
56
64
66
63
65
59
63
53
69
60
68
67
59
58
70
62
66
80
65
59
60
61
63
69
68
56
61
56
62
59
65
57
69
62
70
60
67
79
Range
61
61
66
57
74
56
55
66
61
72
62
70
61
65
71
62
70
61
55
58
10
5
9
7
18
11
8
15
7
12
8
11
16
10
18
6
14
6
14
21
25
20
15
R
a
n
g
e
10
5
0
0
5
10
15
Sample Number
20
25
• For determination of control lines in the chart we
consider following cases• Case I – When process standard deviation is known
• Let the known value of population standard
deviation be , then control lines are given by
CL  d 2
LCL  D1
UCL  D2
• Where the values of d2, D1 and D2 are given in
control chart tables for different values of ‘n’.
• Case II – When process standard deviation is
unknown
• In this case, the average of sample ranges is
computed as 𝑹.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
63
60
57
58
79
55
57
58
65
73
57
66
63
68
70
65
63
61
65
70
55
63
60
64
68
66
61
51
66
66
63
63
53
67
62
59
69
56
57
60
56
60
61
60
65
62
58
61
62
61
56
65
69
59
66
60
68
62
69
67
53
65
65
61
61
63
63
57
68
70
64
59
60
58
80
61
56
59
62
79
61
61
66
57
74
56
55
66
61
72
62
70
61
65
71
62
70
61
55
58
Average
10
5
9
7
18
11
8
15
7
12
8
11
16
10
18
6
14
6
14
21
11.3
• Finally, the control lines are determined by
following formulas
CL  R
LCL  D3 R
UCL  D4 R
• In above problem, 𝑹 = 11.3. From control chart
table, for n = 5, D3 = 0 and D4 = 2.114, so
CL  R  11.3
LCL  D3 R  0
UCL  D4 R  2.114 11.3  23.9
25
UCL
20
15
R
a
n
g
e
CL
10
5
LCL
0
0
5
10
15
Sample Number
20
25
Control Chart for Attributes
• Although more complete information can usually
be gained from measurements on a finished
product, it is often quicker and cheaper to check
the product against specifications on an
“attribute” basis.
• In spite of wide applications of 𝑿 and R (or )
charts for detecting the assignable causes of
variations in a production process, their use is
restricted because firstly they are charts for
variables only, i.e., for quantity characteristics
which can be measured and expressed in number
and secondly, in some cases they are
impracticable and un-economic to construct.
• In situations like this it would be more
appropriate to construct the control charts for
attributes.
• There are following types of control charts of
attributes1. Control chart for the number of defectives (np
or d-chart)
2. Control charts for fraction defectives (p-chart)
3. Control chart for number of defective per unit
(c-chart)
d-Chart or np-Chart
• In manufacturing process, it is also called ‘control
chart for number of defectives’.
• For the construction of d-chart, first ‘k’ samples
each of size ‘n’ are taken randomly and the
number of defectives in each sample is observed.
• Let ‘d’ be number of defectives in a sample, then
in this type of chart ‘d’ is plotted against the
sample number as follows:
• Data on 30 days for number of late flights out of
240 takeoffs daily are presented below26
19
26
22
24
19
19
20
18
18
17
9
13
10
12
14
14
13
9
10
12
15
14
15
16
18
17
16
18
17
30
26
26
25
24
22
20
20
19
19 19
18 18
18
17
18
17
16
15
15
14 14
13
16
15
Ряд1
14
13
12
10
17
12
10
10
9
9
5
0
0
5
10
15
20
25
30
35
• For determination of control lines we proceed as
follows:
• Here, ‘d’ is number of defectives in a sample of
size ‘n’, so sample fraction defective is
d
p
n
• So that d  p  n
• If the actual or true proportion of defectives in
the population from which samples are drawn is
‘P’, then ‘d’ is a random variable having binomial
distribution with parameters ‘n’ and ‘P’, i.e.,
𝒅~𝑩 𝒏, 𝑷
• So that, 𝑬 𝒅 = 𝒏𝑷 and 𝑽 𝒅 = 𝒏𝑷 𝟏 − 𝑷
• Thus, 3-control limits for d-chart are given by
E(d )  3 V (d )  nP  3 nP(1  P)
• We consider following cases• Case I – When population proportion is known
• Let the known population proportion be P0, then
control lines are given by
CL  E (d )  nP0
LCL  E (d )  3 V (d )  nP0  3 nP0 (1  P0 )
UCL  E (d )  3 V (d )  nP0  3 nP0 (1  P0 )
• Case II – When population proportion is unknown
• In this case since population fraction defective is
not known, so the sample fraction defective (or
sample proportion) of all samples are calculated
by
di
pi 
ni
• Then average of all sample proportions are
calculated by
1 k
p   pi
k i 1
• This value is used in place of population
proportion for defining control lines as-
CL  n p
LCL  n p  3 n p (1  p )
UCL  n p  3 n p (1  p )
• Since p cannot be negative, so, if LCL given by the
above formula comes out to be negative, then it is
taken to be 0.
p-Chart
• It is also called ‘control chart for fraction
defectives’.
• For the construction of p-chart, first ‘k’ samples
each of size ‘n’ are taken randomly and the
number of defectives in each sample is observed.
• Let ‘d’ be the number of defectives in a sample.
• So that sample proportion is given by
d
p
n
• In p-chart, the sample proportion ‘p’ is plotted
against the sample number.
• For example, for data on flight delays, presented
above, sample proportions are calculated and pchart is constructed as shown below-
• Data on 30 days for number of late flights out of
240 takeoffs daily are presented below26
19
26
22
24
19
19
20
18
18
17
9
13
10
12
14
14
13
9
10
12
15
14
15
16
18
17
16
18
17
• Sample proportions are calculated below
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Delays
26
19
26
22
24
19
19
20
18
18
17
9
13
10
12
Proportion
Day
0.11
0.08
0.11
0.09
0.1
0.08
0.08
0.08
0.08
0.08
0.07
0.04
0.05
0.04
0.05
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Delays Proportion
14
14
13
9
10
12
15
14
15
16
18
17
16
18
17
0.06
0.06
0.05
0.04
0.04
0.05
0.06
0.06
0.06
0.07
0.08
0.07
0.07
0.08
0.07
Formula ??
p=d/n=26/240
• p-chart is shown below0,12
0,1
0,08
0,06
Ряд1
0,04
0,02
0
0
5
10
15
20
25
30
35
• For determination of control lines we proceed as
follows• As discussed above in ‘d’ chart 𝒅~𝑩 𝒏, 𝑷
• So that
E (d )  nP
V (d )  nP(1  P)
• Since, sample proportion ‘p’ is given by
d
p
n
• So,
1
d  1
E ( p)  E    E (d )   nP  P
n
n n
• Also,
1
P(1  P)
d  1
V ( p)  V    2 V (d )  2  nP(1  P) 
n
n
n n
• We consider following cases• Case I – When population proportion is known
• Let the known population proportion be P0, then
control lines are given by
CL  E ( p)  P0
P0 (1  P0 )
LCL  E ( p)  3 V ( p)  P0  3
n
P0 (1  P0 )
UCL  E ( p)  3 V ( p)  P0  3
n
• Case II – When population proportion is unknown
• In this case since population fraction defective is
not known, it is replaced by average of sample
fraction defective, or average of sample
proportions given by
k
1
p   pi
k i 1
• Or, equivalently
k
p
k
d n p
i 1
k
i
n
i 1
i

i 1
k
i
i
n
i 1
i
• So, control lines are given by
CL  p
p(1  p)
LCL  p  3
n
p(1  p)
UCL  p  3
n
• For example, in above problem, average of sample
proportions is calculated as follows:
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Delays
26
19
26
22
24
19
19
20
18
18
17
9
13
10
12
Proportion
Day
0.11
0.08
0.11
0.09
0.1
0.08
0.08
0.08
0.08
0.08
0.07
0.04
0.05
0.04
0.05
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Delays Proportion
14 0.06
k
1
14 0.06
p
pi 
13 0.05
k i 1
9 0.04
10 10.04
12 0.050.11  0.08  ......  0.07
30
15 0.06
14 0.06
15 0.06
 0.068
16 0.07
18 0.08
17 0.07
16 0.07
18 0.08
17 0.07



• Alternatively, it can also be calculated by using
formula
k
Day Delays Proportion
Day Delays Proportion
di

1
26 0.11 16
14 0.06
2
19 0.08 17
14 0.06
p  i k1 
3
26 0.11 18
13 0.05
 ni
4
22 0.09 19
9 0.04
5
6
7
8
9
10
11
12
13
14
15
24
19
19
20
18
18
17
9
13
10
12
0.1
0.08
0.08
0.08
0.08
0.08
0.07
0.04
0.05
0.04
0.05
20
21
22
23
24
25
26
27
28
29
30
i 1
10 0.04
12 26
0.05 19  26  ........  17
15 0.06
240

240

.........30times
14 0.06
15 0.06
16 0.07  490  0.068
18 0.08
7200
17 0.07
16 0.07
18 0.08
17 0.07

• Thus control lines are given by
CL  p  0.068
p(1  p)
0.068(1  0.068)
LCL  p  3
 0.068  3
n
240
p(1  p)
0.068(1  0.068)
UCL  p  3
 0.068  3
n
240
CL  0.068
LCL  0.019
UCL  0.12
• Complete p-chart is shown belowUCL
CL
LCL
c-Chart
• It is also called ‘control chart for number of
defects’.
• Before discussing the theory behind c-chart, it is
required to distinguish between defect and
defective.
• An article which does not confirm to one or more
of the specifications is termed as defective while
any instance of article’s lack of confirming to
specification is a defect.
• Thus every defective contains one or more of the
defects.
• Unlike a ‘d’ chart, which is related to the number
of defectives in a sample, c-chart is related to the
number of defects per unit.
• Sample size for c-chart may, thus, be a single unit
like a radio or a group of units.
• While considering the distribution of number of
defects in a manufacturing process, we may need
to note two essential points• The chance for a defect to occur in any position
of the manufactured product is sufficiently
large.
• The chance of a defect occurring in any
particular position is very small.
• In such situation, from statistical theory we know
that the patterns of variations in data can be
represented by Poisson distribution and
consequently the 3 control limits based on
Poisson distribution are used.
• Thus c-chart is based on Poisson distribution.
• If c denotes the number of defects in an item,
then c is Poisson distribution with parameter, say
.
• Here, E(c) =  and V(c) = , so control limits are
given by:
CL  E (c)  
LCL  E (c)  3 V (c)    3 
UCL  E (c)  3 V (c)    3 
• Case I – If population parameter of the underlying
Poisson distribution is known (to be 0, which is
the average number of defects in entire
population). In this case,
CL  0
LCL  0  3 0
UCL  0  3 0
• Case II – If population parameter is not known• In this case grand average of number of defects of
entire samples is calculated by
1 k
c   ci
k i 1
where, ci is the number of defects in the ith
sample. The unknown population parameter is
replaced by it, so the control lines are given by
CL  c
LCL  c  3 c
UCL  c  3 c
• Problem –
A flow solder machine is used to make mechanical
and electrical connections of the leaded
components on a printed circuit board. The
boards are run through the flow solder process
almost continuously, and every hour five boards
are selected and inspected for process control
purpose. The number of defects in each sample of
five boards is given below-
Sample
1
2
3
4
5
6
7
8
9
10
No. of
Defects
6
4
8
10
9
12
16
2
3
10
Sample
11
12
13
14
15
16
17
18
19
20
No. of
Defects
9
15
8
10
8
2
7
1
7
13
Construct c-chart for above data to determine
whether the process is in controlled condition.
c-chart is shown below by plotting number of
defects against sample number-
• To determine whether the process is in controlled
condition, we need to identify CL, LCL and UCL.
• Since the process parameter is not known, so we
use
CL  c
LCL  c  3 c
UCL  c  3 c
• Here,
1
c
 6  4  ......  13  8
20
• So,
CL  c  8
LCL  c  3 c  8  3 8  0, set to 0
UCL  c  3 c  8  3 8  16.484
• Complete c-chart is shown below-
UCL
CL
LCL
u-Chart
• If number of unit in difference samples is not
same, then c-chart is meaningless. In such case,
instead of actual number of defects in the sample,
average number of defects in each sample is
plotted and such a chart is called u-chart.
• It is also called defects per unit charge.
• If ‘c’ is number of defects in the sample of ‘n’ units
then average number of defects per unit is
c
u
n
• If process parameter is not known, it is replaced
by
1 k
u   ui
k i 1
where ui is the average number of defects in the
ith sample.
• So, control lines are given by
CL  u
u
LCL  u  3
n
u
UCL  u  3
n
• Problem –
To draw u-chart for data presented above in
construction of c-chart.
• Working table is shown below-
Sample no.
No. of defects Defects per
unit
Sample no.
No. of defects Defects per
unit
1
2
6
4
1.2
0.8
11
12
9
15
1.8
3.0
3
4
5
8
10
9
1.6
2.0
1.8
13
14
15
8
10
8
1.6
2.0
1.6
6
7
12
16
2.4
3.2
16
17
2
7
0.4
1.4
8
9
10
2
3
10
0.4
0.6
2.0
18
19
20
1
7
13
0.2
1.4
2.6
Total
32
• Here to find defects per unit, number of defects is
divided by sample size, i.e., by 5.
• Here,
1 k
1
u   ui 
x32  1.6
k i 1
20
• So,
CL  u  1.6
u
1.6
LCL  u  3
 1.6  3
 3.3
n
5
u
1.6
UCL  u  3
 1.6  3
 0 so set to 0.
n
5
• U-chart is shown belowUCL
CL
LCL
Control chart for individual measurement
• In all types of control charts so far we discussed,
more than one units are taken as sample. Taking
observations of many units, in some cases, may
be too costly and more time consuming.
• In such cases, only one unit is observed in each
sample.
• Each observation is plotted in graph and to define
control lines, difference between successive
observations are calculated.
• The absolute value of such difference is called
moving range (MR).
• Following expressions are used to define control
lines for such control chart of individual
observations-
CL  x
MR
LCL  x  3
d2
MR
UCL  x  3
d2
where 𝒙 is mean of individual observations and
𝑴𝑹 is average of moving ranges. The value of d2 is
taken to be 1.128 from control chart table for n =
2, since moving range is obtained by considering
two sample units.
• Problem –
Batches of a particular chemical product are
selected from a process and the purity on each is
measured. Data for 15 successive batches are
given below0.77, 0.76, 0.77, 0.72, 0.73, 0.73, 0.85, 0.70, 0.75,
0.74, 0.75, 0.84, 0.79, 0.72, 0.74.
To draw control chart-
Batch
Purity (x)
Moving Range |MR|
1
0.77
(Leave it blank)
2
0.76
0.01
3
0.77
0.01
4
0.72
0.05
5
0.73
0.01
6
0.73
0.00
7
0.85
0.12
8
0.70
0.15
9
0.75
0.05
10
0.74
0.01
11
0.75
0.01
12
0.84
0.09
13
0.79
0.05
14
0.72
0.07
15
0.74
0.02
Total
11.36
0.65
1
x  11.36  0.757
15
1
MR   0.65  0.046
14
CL  x  0.757
MR
0.046
LCL  x  3
 0.057  3 
 0.635
d2
1.128
MR
0.046
UCL  x  3
 0.057  3 
 0.879
d2
1.128
UCL
CL
LCL