Download Module 4: Shewhart Control Charts

Statistical Quality Control in Textiles
Module 4:
Shewhart Control Charts
Dr. Dipayan Das
Assistant Professor
Dept. of Textile Technology
Indian Institute of Technology Delhi
Phone: +91-11-26591402
E-mail: [email protected]
Introduction
Why Control Charts? [1]
Input
Yes
Manufacturing
Process
No
Scrap or
Rework
No
Yes
Customer
Is output
acceptable?
Is process
under
control?
Correction
Is input
acceptable?
Yes
Output
The control charts are required to know whether the
manufacturing process in under control or out of
control.
Basis of Control Charts
The basis of control charts is to checking whether the variation in the
magnitude of a given characteristic of a manufactured product is
arising due to random variation or assignable variation.
Random variation: Natural variation or allowable variation, small
magnitude
Assignable variation: Non-random
variation, relatively high magnitude.
variation
or
preventable
If the variation is arising due to random variation, the process is said
to be under control. But, if the variation is arising due to assignable
variation then the process is said to be out of control.
Types of Control Charts
Control Charts
Other Control
Charts*
Shewhart
Control Charts
Variables
x
Chart
R
Chart
s
Chart
Attributes
p
Chart
np
Chart
c
Chart
* This will be
discussed later on.
Basics of Shewhart Control Charts
Major Parts of Shewhart Control Chart
Lower Control Limit (LCL):
This indicates the lower limit of
tolerance.
If m is the underlying statistic
so that E  m   m & Var  m  2m
CL = m
UCL = m  3m
LCL = m  3m
Out of control
Quality Scale
Central
Line
(CL):
This
indicates the desired standard
or the level of the process.
Upper Control Limit (UCL):
This indicates the upper limit of
tolerance.
UCL
3
CL
3
LCL
Out of control
1 2 3 4 5 6 7 8 9 10
Sample Number
Why 3?
Let us assume that the probability distribution of the sample statistic
m is (or tends to be) normal with mean m and standard deviation m .
Then
P  m  3m  m  m  3m   0.9973
This means the probability that a random value of m falls in-between
the 3- limits is 0.9973, which is very high. On the other hand, the
probability that a random value of m falls outside of the 3- limits is
0.0027, which is very low. When the values of m fall in-between the 3 limits, the variations are attributed due to chance variation, then the
process is considered to be statistically controlled. But, when one or
many values of m fall out of the 3- limits, the variations are
attributed due to assignable variation, then the process is said to be
not under statistical control.
Analysis of Control Chart: Process Out of Control
The following one or more incidents indicate the process is said to
be out of control (presence of assignable variation).
A point falls outside any
of the control limits.
UCL
Eight consecutive points fall
within 3 limits.
UCL
3
CL
CL
LCL
Standard Control Action
LCL
3
Analysis of Control Chart: Process Out of Control…
Two out of three consecutive
points fall beyond 2 limits.
UCL
UCL
CL
LCL
Four out of five consecutive
points fall beyond 1 limits.
2
2
CL
1
1
LCL
Note: Sometimes the 2 limits are called as warning limits. Then, the 3 
limits are called as action limits.
Analysis of Control Chart: Process Out of Control…
Presence of upward or
downward trend
Presence of cyclic trend
UCL
UCL
CL
CL
LCL
LCL
Analysis of Control Chart: Process Under Control
When all of the following incidents do not occur, the process is said
to be under control (absence of assignable variation).
1) A point falls outside any of the control limits.
2) Eight consecutive points fall within 3 limits.
3) Two out of three consecutive points fall beyond 2 limits.
4) Four out of five consecutive points fall beyond 1 limits.
5) Presence of upward or downward trend
6) Presence of cyclic trend
Shewhart Control Charts for Variables
The Mean Chart (x-Chart)
Let xij , j  1, 2, , n be the measurements on ith sample (i=1,2,…,k).
The mean xi , range Ri , and standard deviation si for ith sample are
given by
xi 
j n
1
xij

n j 1
j n
j n
Ri  max  xij   min  xij 
j 1
j 1
si 
j n

j 1
x
ij
 xi 
2
n
Then the mean x of sample means, the mean R of sample ranges,
and the mean s of sample standard deviations are given by
1 ik
x   xi
k i 1
1 ik
R   Ri
k i 1
1 i k
s   si
k i 1
The Mean Chart (x-Chart)…
Let us now decide the control limits for xi .
When the mean  and standard deviation  of the population from
which samples are taken are given.
CL = E  xi   
 3 
     A
 n
UCL= E  xi  +3 Var  xi     
A
 3 
     A
 n
LCL = E  xi   3 Var  xi     
A
The Mean Chart (x-Chart)…
When the mean  and standard deviation  are not known.
CL= x
  3 

 x  
 s  x  A1s 
  c2 n 



 A1
UCL= 

  3 

x

R

x

A
R
2 
  d n 

  2

 A2


  3 

 x  
 s  x  A1s 
  c2 n 



 A1
LCL= 

  3 

x

R

x

A
R
2 
  d n 

  2

 A2


The Range Chart (R-Chart)
Let xij , j  1, 2, , n be the measurements on ith sample (i=1,2,…,k).
The range Ri for ith sample is given by
j n
j n
j 1
j 1
Ri  max  xij   min  xij 
Then the mean R of sample ranges is given by
1 ik
R   Ri
k i 1
The Range Chart (R-Chart)…
Let us now decide the control limits for Ri .
When the standard deviation  of the population from which
samples are taken is known.
CL = E  Ri   d2
UCL= E  Ri  +3 Var  Ri   d 2  3d3   d 2  3d3    D2
 D2
LCL = E  Ri   3 Var  Ri   d 2  3d3   d 2  3d3    D1
 D1
The Range Chart (R-Chart)…
When the standard deviation  of the population is not known.
CL = E  Ri   R
 3d3 
 3d3 
UCL= E  Ri  +3 Var  Ri   R  
 R  1 
 R  D4 R
d2 
 d2 

 D4
 3d3 
 3d3 
LCL = E  Ri   3 Var  Ri   R  
 R  1 
 R  D3 R
d2 
 d2 

 D3
The Standard Deviation Chart (s-Chart)
Let xij , j  1, 2, , n be the measurements on ith sample (i=1,2,…,k).
The standard deviation si for ith sample is given by
si 
j n

j 1
x
ij
 xi 
2
n
Then the mean s of sample standard deviations is given by
1 i k
s   si
k i 1
The Standard Deviation Chart (s-Chart)…
Let us now decide the control limits for si .
When the standard deviation  of the population from which
samples are taken is known.
CL = E  si   c2
UCL= E  si  +3 Var  si   c2  3c3   c2  3c3    B2
 B2
LCL = E  si   3 Var  si   c2  3c3   c2  3c3    B1
 B1
The Standard Deviation Chart (s-Chart)…
When the standard deviation  of the population is not known.
CL =
E  si   s
UCL=
 3c3 
c3
E  si  +3 Var  si   s  3 s  1 
 s  B4 s
c2
c2 

 B4
LCL =
 3c3 
c3
E  si   3 Var  si   s  3 s  1 
 s  B3 s
c2
c2 

 B3
Table
Sample
size
Mean Chart
Standard deviation chart
Range chart
n
A
A1
A2
c2
B1
B2
B3
B4
d2
D1
D2
D3
D4
2
2.121
3.760
1.886
0.5642
0
1.843
0
3.297
1.128
0
3.686
0
3.267
3
1.232
2.394
1.023
0.7236
0
1.858
0
2.568
1.693
0
4.358
0
2.575
4
1.500
1.880
0.729
0.7979
0
1.808
0
2.266
2.059
0
4.698
0
2.282
5
1.342
1.596
0.577
0.8407
0
1.756
0
2.089
2.326
0
4.918
0
2.115
6
1.225
1.410
0.483
0.8686
0.026
1.711
0.030
1.970
2.534
0
5.078
0
2.004
7
1.134
1.277
0.419
0.8882
0.105
1.672
0.118
1.882
2.704
0.205
5.203
0.076
1.924
8
1.061
1.175
0.373
0.9027
0.167
1.638
0.185
1.815
2.847
0.387
5.307
0.136
1.864
9
1.000
1.094
0.337
0.9139
0.219
1.609
0.239
1.761
2.970
0.546
5.394
0.184
1.816
10
0.949
1.028
0.308
0.9227
0.262
1.584
0.284
1.679
3.078
0.687
5.469
0.223
1.777
11
0.905
0.973
0.285
0.9300
0.299
1.561
0.321
1.646
3.173
0.812
5.534
0.256
1.744
12
0.866
0.925
0.266
0.9359
0.331
1.541
0.354
1.618
3.258
0.924
5.592
0.284
1.716
13
0.832
0.884
0.249
0.9410
0.359
1.523
0.382
1.594
3.336
1.026
5.646
0.308
1.692
14
0.802
0.548
0.235
0.9453
0.384
1.507
0.406
1.572
3.407
1.121
5.693
0.329
1.671
15
0.775
0.816
0.223
0.9499
0.406
1.492
0.428
1.552
3.472
1.207
5.779
0.348
1.652
16
0.759
0.788
0.212
0.9523
0.427
1.478
0.448
1.534
3.532
1.285
5.817
0.364
1.636
17
0.728
0.762
0.203
0.9951
0.445
1.465
0.466
1.518
3.588
1.359
5.854
0.379
1.621
18
0.707
0.738
0.194
0.9576
0.461
1.454
0.482
1.503
3.640
1.426
5.888
0.392
1.668
19
0.688
0.717
0.187
0.9599
0.477
1.443
0.497
1.499
3.689
1.490
5.922
0.404
1.596
20
0.671
0.697
0.180
0.9619
0.491
1.433
0.510
1.477
3.735
1.548
5.950
0.414
1.586
25
0.600
0.610
0.153
0.9696
0.548
1.392
0.565
1.435
3.931
1.804
6.058
0.459
1.541
Illustration
Sample
No.  i 
Yarn strength
xicNtex 1  Ri cNtex 1  si cNtex 1 
xij cNtex 1 







1
14.11 13.09
12.52
13.40
13.94 13.40 12.72
11.09
13.28 12.34
12.99
3.02
0.83
2
14.99 17.97
15.76
13.56
13.31 14.03 16.01
17.71
15.67 16.69
15.57
4.66
1.54
3
15.08 14.41
11.87
13.62
14.84 15.44 13.78
13.84
14.99 13.99
14.19
3.57
0.97
4
13.14 12.35
14.08
13.40
13.45 13.44 12.90
14.08
14.71 13.11
13.47
2.36
0.64
5
13.21 13.69
13.25
14.05
15.58 14.82 14.31
14.92
10.57 15.16
13.96
5.01
1.36
6
15.79 15.58
14.67
13.62
15.90 14.43 14.53
13.81
14.92 12.23
14.55
3.67
1.07
7
13.78 13.90
15.10
15.26
13.17 13.67 14.99
13.39
14.84 14.15
14.23
2.09
0.72
8
15.65 16.38
15.10
14.67
16.53 15.42 15.44
17.09
15.68 15.44
15.74
2.42
0.69
9
15.47 15.36
14.38
14.08 14.08 14.84 14.08
14.62
15.05 13.89 14.59
1.58
0.54
10
14.41 15.21
14.04
13.44
14.94
14.84 16.19
14.85
2.75
0.81
14.41
3.11
0.92
15.85 14.18 15.44
Average
xcNtex 1   14.41


RcNtex 1   3.11


scNtex 1   0.92



Illustration (x-chart)
CL= x  14.11 cN  tex 1
xicNtex 1 
UCL=
UCL

 x  A2 R 
 14.11   0.308  3.11 cN  tex 
1
 15.37 cN  tex 1

CL
LCL
LCL=
 x  A2 R 
 14.11   0.308  3.11 cN  tex 1 
 13.45 cN  tex 1
0
i  
The process average is out of control.
Illustration (R-chart)
CL=
R  3.11cN  tex
1
UCL= D4 R 
Ri cNtex 1 


UCL
 1.777  3.11cN  tex 1 
 5.50 cN  tex 1
LCL= D3 R 
 0.223  3.11cN  tex 1 
 0.69cN  tex 1
CL
LCL
i
The process variability is in control.
Illustration (s-chart)
CL=
si cNtex 1 
s  0.92cN  tex 1

UCL= B4 s 
1
 1.716  0.92cN  tex 

UCL
 1.58 cN  tex 1
CL
LCL= B3 s 
1
 0.284  0.92 cN  tex 
 0.26 cN  tex 1
LCL
i
The process variability is in control.
Illustration (Overall Conclusion)
Although the process variability is in control, the process cannot
be regarded to be in statistical control since the process average is
out of control.
Shewhart Control Charts for Attributes
Control Chart for Fraction Defective (p-Chart)
The fraction defective is defined as the ratio of the number of
defectives in a population to the total number of items in the
population.
Suppose the production process is operating in a stable manner, such
that the probability that any item produced will not conform to
specifications is p and that successive items produced are
independent. Then each item produced is a realization of a Bernouli
random variable with parameter p. If a random sample of n items of
product is selected, and if D is the number of items of product that
are defectives, then D has a binomial distribution with parameter n
n x
and p; that is
The mean and
P  D  x  nCx p x 1  p  , x  0,1, , n.
variance of the random variable D are np and np 1  p  , respectively.
Control Chart for Fraction Defective (p-Chart)…
The sample fraction defective is defined as the ratio of the number of
p  D n .
defective items in the sample of size n; that is
The
distribution of the random variable p  can be obtained from the
binomial distribution. The mean and variance of p  are p and
p 1  p 
, respectively.
n
When the mean fraction of defectives p of the population from which
samples are taken is known.
CL=
p
p 1  p 
UCL= p  3
n
p 1  p 
LCL= p  3
n
Control Chart for Fraction Defective (p-Chart)…
When the mean fraction of defectives p of the population is not
known.
Let us select m samples, each of size n. If there are Di defective items in
ith sample, then the fraction defectives in the ith sample is
pi  Di n , i  1, 2,
defectives is
, m. The average of these individual sample fraction
i m
p 
CL= p 
i m
 D  p
i 1
mn
UCL= p  3
i

i 1
i
m
p 1  p 
n
LCL= p  3
p 1  p 
n
Control Chart for Number of Defectives (np-Chart)
It is also possible to base a control chart on the number of defectives
rather than the fraction defectives.
When the mean number of defectives np of the population from
which samples are taken is known.
CL= np
UCL= np  3 np 1  p 
LCL= np  3 np 1  p 
When the mean number of defectives np of the population is not
known.
CL= np
UCL= np  3 np 1  p
LCL= np  3 np 1  p
Illustration [2]
The following refers to the number of defective knitwears in samples of size 180.
Sample
No.
No. of
defectives
Sample
No.
No. of
defectives
Sample
No.
No. of
defectives
1
05
11
36
21
24
2
08
12
24
22
17
3
10
13
19
23
12
4
12
14
13
24
08
5
12
15
05
25
17
6
29
16
02
26
19
7
25
17
11
27
04
8
13
18
08
28
09
9
09
19
15
29
05
10
20
20
20
30
12
Illustration…
Here, n=180 and p  423 30 180  0.0783.
CL=
np  14.09
UCL= np  3 np 1  p  24.90
npi
UCL
CL
LCL= np  3 np 1  p  3.28
LCL
i
The process is out of control.
Control Chart for Defects (c-Chart)
Consider the occurrence of defects in an inspection of product(s).
Suppose that defects occur in this inspection according to Poisson
distribution; that is
e c c x
P  x 
, x  0,1, 2,
x!
,
Where x is the number of defects and c is known as mean and/or
variance of the Poisson distribution.
When the mean number of defects c in the population from which
samples are taken is known.
CL= c
UCL= c  3 c
LCL= c  3 c
Note: If this calculation yields a negative value of LCL then set LCL=0.
Control Chart for Defects (c-Chart)…
When the mean number of defects c in the population is not known.
Let us select n samples. If there are ci defects in ith sample, then the
average of these defects in samples of size n is
i n
c 
CL= c
i n
 c  c
i 1
n
i

i 1
i
n
UCL= c  3 c
LCL= c  3 c
Note: If this calculation yields a negative value of LCL then set LCL=0.
Illustration [2]
The following dataset refers to the number of holes (defects) in knitwears.
Sample
No.
No. of
holes
Sample
No.
No. of
holes
Sample
No.
No. of
holes
1
4
11
3
21
2
2
6
12
7
22
1
3
3
13
9
23
7
4
8
14
6
24
6
5
12
15
10
25
5
6
9
16
11
26
9
7
7
17
7
27
11
8
2
18
8
28
8
9
11
19
9
29
3
10
8
20
3
30
2
Illustration…
Consider ci denote the number of holes in ith sample.
CL= c  6.57
UCL
ci
UCL= c  3 c  14.26
CL
LCL= c  3 c  1.12 0
LCL
i
The process is in control.
Frequently Asked Questions & Answers
Frequently Asked Questions & Answers
Q1: Is it not possible that a process turns to be out of control because of presence of
random variation?
A1: Yes, it is possible, but the probability of such occurrence is very low, that is,
0.0027.
Q2: Is it so that a process can be found to be out-of control even if there is no point
falling out of 3-sigma limits?
A2: Yes, it is possible, the presence of a run, trend, etc. can do so.
Q3: If process mean is in control, but the process variability is not in control, can the
process be said to be under control?
A3: No.
Q4: Is Shewhart control chart able to detect a small shift in process mean?
A4: No.
Frequently Asked Questions & Answers
Q5: Name the probability distribution that the process defectives can be regarded to
follow?
A5: Binomial distribution.
Q6: Name the probability distribution that the process defects can be regarded to
follow?
A6: Poission distribution.
References
1.
Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan
Chand & Sons, New Delhi, 2007.
2.
Leaf, G. A. V., Practical Statistics for the Textile Industry: Part II, The Textile
Institute, UK, 1984.
Sources of Further Reading
1.
Leaf, G. A. V., Practical Statistics for the Textile Industry: Part I, The Textile
Institute, UK, 1984.
2.
Leaf, G. A. V., Practical Statistics for the Textile Industry: Part II, The Textile
Institute, UK, 1984.
3.
Gupta, S. C. and Kapoor, V. K., Fundamentals of Mathematical Statistics,
Sultan Chand & Sons, New Delhi, 2002.
4.
Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan
Chand & Sons, New Delhi, 2007.
5.
Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley &
Sons, Inc., Singapore, 2001.
6.
Grant, E. L. and Leavenworth, R. S., Statistical Quality Control, Tata McGraw
Hill Education Private Limited, New Delhi, 2000.
7.
Montgomery, D. C. and Runger, G. C., Applied Statistics and Probability for
Engineers, John Wiley & Sons, Inc., New Delhi, 2003.