control chart in process capability

1. productionprocess.
Wi
t
h t
h
e hel
pofa control chart,one can fi
n
d out t
h
e natural cbility
ofa
productiontroubles.
Control chart makes possible t
h
e diagnosis
and correction
of th
e ma
n
y
statically
or not
.
The control chart
is used
to check whether t
h
e process
is d
in
a process.
V It isa t
o
olto distinguish between chanceand assignablecauses
of s
time and
variations
of this data.
A con trol chart
is a graph that displays data taken ove
G12.9.1.What
is it
?
12.9. CONTROL CHART

2. ControlCharts &Process Copability
/ The general form of the control chart is illustrated in Fig.12.8.
Quality
characteristic
value
The control chart is used to evaluate process stability and to decide when to
adjust the process.
+36
4 10
Sample values
12 14
Sample number, S
G12.9.2. Types of Control Charts
16
The two basic types of control charts used are:
Fig. 12.8. General form of control chart
1. Control charts for variables, and
2. Controlcharts for attributes.
18
Out-of-control
20
G12.9.3. Analysis of Patterns in Control Charts
UCL
12.25
Centre line
LCL
Control charts for variables require a measurement of the quality
characteristicof interest.
Control charts for attributes require a determination of whether a part is
defective or how many defects are there in the sample.
The main objective of using control charts is to determine when aprocess is
Out-of-controlso thatnecessaryactionsmaybe taken.
V Criteria other than aplotted point falling outside the control limits are also
used todetermine whether aprocess is out-of-control.
Plot patterns often indicate whether the process is in control or not.
Analyzing plot patterns is more difficult than plotting the control charts.

3. 12.28
12.10. CONTROL CHARTS FOR VARIABLES
The quality characteristics which can be measured and expressed in specific
units of measurements are called variables.
Control charts based upon measurements of quality characteristics are called
as control charts for variables.
Types of variable control charts: The most commonly used variable control
charts are:
() X- or average-charts,
() R- or range-charts, and
(üi) s- or standard deviation-charts.
V The X-chart is used to monitor the centering of the process to control its
accuracy.
The R-chart monitors the dispersion or precision of the process.
V The s-chart shows the variation of the process.
12.10.1. Construction of X- and R-Charts
Total QualityManagement
Step 1:Select the characteristics for applying a control chart.
Step2: Select the appropriate type of control chart.
Step 3: Collect the data.
Step 4: Choose the rational sub-group i.e., sample.
Step 5:Calculate the average (X) and range (R) for each sample.
For example, if asub-group contains 5 items whose dimensions (say diameter
length or weight or etc.) are x, x2, X3, X and x5, then
Sub-group average, X
5
and subgroup range, R
Maximumvalue - Minimum value
t selection of sample size: Subgroups (i.e., sample size) must be large enougn
to
detect
points or patterns indicating alack of control when alack of controlexists. The
largerthe
sample size,the better the
chances of
detecting the pattern.

4. 12.29
Control
Charts
&
Process
Capability
5tep
6:
Calculate
the
average
of
the
ayerages
(}
)
and
average
of
range
(R).
Let
N
=
Number
of
sub-groups
Then,
average
of
averages
(or
grand
average)
is
given
by
2X
N
Similarly
average
of
range
(R)
is
given
by
2R
N
R
Step
7:
Calculate
the
control
limits
for
X
and
R-charts.
(a)
Control
limits
of
X-chart
Control
limit
or
centre
line,
CLy
:
X
Upper
control
limit,
UCLy
=
X
+
A
R
Lower
control
limit,
LCL
=
X
-
A,
R
and
Where
A
=
Factor
or
constant
for
X-chart,
taken
from
the
Table
A.2.
(see
Apperndix).
(b)
Control
limits
for
R-chart
R
Control
limit
or
centre
line,
CLp
=
D4
R
Upper
control
limit,
UCLR
:
Lower
control
limit,
LCRR
and
where
Da
and
D4
are
statistical
factors,
taken
from
Table
A,2.
(see
Appendix)
Step
8:
Plot
CL,
UCL
and
LCL
on
the
chart.
Step.9:
Plot
individual
X
and
R
values
on
the
chart.
Step
10:
Check
whether
the
process
is
in
control
or
not.
Step
11:
Revise
the
control
limits
if
the
points
are
out-of-control
by
removing
the
out-of-co
points.
=

5. 3
6.34
6.40
6.34
6.36
2
6.46
6.37
6.36
6.41
1
6.35
6.40
6.32
6.37
Sample
Number
X1
X3
Measurements
now
under
statistical
control.
revised
mean
and
range
charts
and
plot
t
h
e
values.
(vi)
State
whether
t
h
e
process
is
which
are
rectified
now.
Revise
t
h
e
central
line
and
control
limits.
()
Draw
t
h
e
control.
(iv)
If
not,
assume
that
t
h
e
deviation
occurred
due
to
assignable
causes
range
charts
and
plot
t
h
e
values.
(iii)
State
whether
t
h
e
process
is
under
statistical
()
Determine
t
h
e
trial
central
line
and
control
limits.
ii)
Draw
t
h
e
mean
and
[A.U,
Nov/Dec
2O03]
process.
molding
process.
Mean
and
range
charts
are
required
to
be
established
for
this
Example
12.9
plastic
component
in
grams.
This
component
is
manufactured
using
a
plastic
injection
Following
table
contains
t
h
e
data
on
t
h
e
weight
ofa

6. and
R=
N
10
XR
1,
1
1
:
0,111
N
EX
64.22
=
6.422
EX=
64.22
10
6.38
6.40
SR
=
1,
1
1
6.
4
5
6.37
6.
4
0
9
6.56
6.
5
5
0.08
6.
4
5
6.48
6.
5
1
8
6.
3
5
0.11
6.
4
1
6.37
6.
3
8
6.37
7
6.38
0.06
6.44
6.28
6.
5
8
6.42
6.
4
1
0.30
6.40
6.
2
9
6.34
6.36
5
6.
3
8
0.12
6.
3
4
6.44
6.
4
0
6.39
0.10
4
6.69
6.
6
4
6.68
6.59
6.
6
5
0.10
3
6.34
6.
4
0
6.34
6.36
6.36
0.06
2
6.46
6.37
6.
3
6
6.41
6.40
0.10
6.35
6.40
6.
3
2
6.37
6.36
0.08
Number
X1
X3
4
Sample
X1
+
X2
+
X3
+
X4
Xmi
n
Measurements
X
R
=
Xmax
ar
e
tabulated
as
follows:
0Solution:
In
order
to
calculate
X
and
R,
t
h
e
results
of
t
h
e
given
observations
To
find:
Centre
line
and
control
limits
for
X-
and
R-charts
Given
Data:
Subgroup
size,
n
=
4;
N=
10
10
6.38
6.40
6.45
6.
3
7
6.56
6.55
6.45
6.48
6.35
6.41
6.37
6.38
7
6.38
6.44
6.28
6.58
6
6.41
6.40
6.29
6.34
5
6.38
6.34
6.44
6.40
4
6.69
6.64
6.68
6.59
Control
Charts
&
Process
Capability
12.31|
=

7. Fi
g
.
12.14.
X-chart
Subgroup
No.
1
2
10
4
5
6
7
625
6.30
6
LCLZ
=
6341
6.40
-
CL
=
6.422
6.45
.
6
UCL
=
6.503
6.55
6.
6
0
6.65
670
-
and
12.15
respectively.
Using
t
h
e
calculated
control
limits,
t
h
e
X-
and
R-charts
are
shown
in
Figs.12.14
(i)
To
draw
t
h
e
and
R-charts:
and
Lower
control
limit,
LCLe
:
Da.
R
:
0
(0.111)
=
0
Ans.
Upper
control
limit,
UCL
=
D4.
R
:0.253
Ans.
=
2.282
(0.111)
Control
limit
or
Central
line,
CL
:
R
=0.111
Ans.
o
For
R-chart:
:
6.341
Ans.
D
and
Lower
control
limit,
Lcl
=
X-
Az
R=
6.422
-0.729
(0.111)
Upper
control
limit,
UCL,
:6.503
Ans
:
X+
Ag
R=
6.422
+0.729
(0.111)
X
Control
limit
or
Central
line,
CL-
=
X=
6.422
Ans.
For
X-chart:
()
To
determine
trail
centre
line
and
control
limits:
For
a
subgroup
size,
n
=
4,
Table
A.
2
gives
t
h
e
following
statistical
factors:
Ag
=
0.729:
Da
=
0:
and
D
=
2.282
12,32
Total
Quality
Management
35
50

8. Appendix
Observations
inSample, n
3
4
5
6
7
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A2. FACTORS FOR COMPUTING CENTRE LINE AND THREE-SIGMA CONTROL LIMITS
25
Factors for Control Limits
A
2.121 1.880
1.732
1.500
1.342
1.134
1.061
1.000
1.225 0.483
0.949
0.905
0.866
0.832
0.802
0.775
0.750
0.728
0.707
0.688
0.671
0.655
0.640
0.626
K-charts
0.612
0.600
For n> 25,
1.023
0.729
0.577
0.419
0.373
0.337
0.308
0.285
0.249
0.235
0.223
0.212
0.203
0.194
0.187
0.180
0.173
0.167
0.162
0.157
0.153
Source: O
ASTM.
A :
A3
2.659
1.954
1.628
0.266 0.886 0.9776
0.850 0.9794
0.9810
.182
1.099
1.032
1.287 0.9515
0.975
0.927
1.427 0.9400 1.0638
0.817
0.789
0.763
0.739
0.718
0.698
0.680
0.663
0.647
0.633
0.619
0.606
B4 = C4
Factors for
Center Line
1/lc4
CA
0.7979
3
0.8862
0.9213
0.9594
0.9650
0.9693
0.9727
0.9754
olollL
0.9845
0.9854
0.9862
0.9869
0.9876
0.9882
0.9887
0.9892
0.9896
1.2533
3
1.1284
1.0854
0.9823 1.0180
0.9835
1.0510
1.0423
1.0363
1.0317
1.0281
1.0252
1.0229
1.0210
1.0194
1.0148
1.0140
1.0133
1.0126
1.0119
1.0114
1.0109
1.0105
3
S-Charts
c4 Vn
Factors for Control Limits
0.428
1.0168 0.448
and
B3
1.0157 0.466
0.030
0.118
0.185
0.239
0.284
0.321
0.354
0.382
0,406
0.482
0.497
0.510
0.523
0.534
0.545
0.555
0.565
B4
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.761
1.716
1.646
1.618
1.594
1.572
1.679 0.313
1.552
1.534
1.518
1.503
1.490
1.477
1.466
1.455
1,445
1.435
Bs
0.029
0.113
0.179
B, = C4
0.232
0.276
0.346
0.374
0.399
0.421
0.440
0.458
0,475
0.490
0.504
0.516
0.528
0.539
0.549
4(n-1)
0.559
3
B6 1/dg dz
2,606 1.128 0.8865 0.853
2.276 1.693 0.5907 0.888
2.088 2.059 0.4857 0.880
1.964 2.326 0.4299 0.864
1.874 2.534 0.3946 0.848
1.806 2.704 9-3698 0.833
1.751
1.707
1.669
1.637
1.610
1.585
1.563
1.526
1.511
1.496
1.483
3.407
1.544 3.472
1.470
1.459
1.448
1.438
1.429.
C4 4n-3 B 1-
Factors for
Center Line
1.420
2.970
3.078
3.173
3.258
3.532
3.588
3.640
0.3152 0.787
0.3069 0.778
3.336 0.2998 0.770
0.2935 0.763
0.2880
3.689
3.735
3.778
3.819
3.858
3.895
3.931
0.3512
3
0.3367
0.3249 0.797
0.2831
0.2787
0.2747
0.2711
0.2677
0.2647
02618
02592
02567
0.820
02544
0.808
c4V2 (n- 1)
0.756
0.750
0.744
0.739
0.734
0.729
0.724
0.720
0.716
0.712
0.708
R-Charts
Factors for Control Limits
DË
0.204
0.388
0.547
0.687
0.811
0.922
1.025
1.118
1.203
1.282
1.356
1.424
1.487
1.549
1.605
1.659
1.710
1.759
1.806
B = 1+
D2
3.686
4.358
4.698
4.918
5.078
5.204
5.306
5.393
5.469
5.535
5.594
S.647
5.696
S.741
S.782
5.820
5856
5.891
5.921
S.951
S.979
6.006
6.031
6.056
3
D3
0
0.076
0.136
0.184
0.223
0.256
0.283
0.307
0.328
0.347
0.363
0.378
0.391
0.403
0.415
0.425
0.434
0.443
0.451
0.459
C4 V 2(n - 1)
D4
3.267
2.574
2.284
2.114
2.004
1.924
1.864
1.816
1.777
1.744
1.717
1.693
1.672
1.653
1.637
1.622
1.608
1.597
1.583
1.575
1.566
A.3
1.557
1.548
1.541
2.847

9. and
Lower
control
limit,
LCLe
3
D3.
R
=0
(0.073)
:
0
Ans.
Upper
:
0.166
Ans.
control
limit,
UCLR
D4.
R=
2.
2
82
(0.783)
Revised
R
=0.073
Ans,
and
Revi
s
ed
:
N-3
2X-
(6.
6
5
+6.
4
2
+
6.
5
1)
10-3
64.22
-
19.58
6.377
Revised
R
=
N-3
ER
-
(0.
1
0
+
0.
3
0
+
0.
1
1)
Wi
t
h
samples
4,
7
and
9
deleted,
we
get
10-3
1,
1
1
-
0.51
=
0.073
removing
t
h
e
out-of-control
points.
i
n
e
a
n
d
control
limits
can
be
revised.
Revised
control
limits
can
be
determined
by
identified
and
they
ar
e
eliminated
through
appropriate
remedial
actions,
t
h
e
centre
Assuming
that
t
h
e
assignable
causes
f
o
r
these
three
samples
(
4
,
7
and
9)
ar
e
(i
)
To
revise
t
h
e
centre
line
and
con::
ol
limits:
limits,
therefore
t
h
e
process
is
statistically
out
of
control.
Ans.
X-chart
(refer
Fig.12.14).
Since
those
three
points
ar
e
outside
t
h
e
upper
control
chart
(refer
Fig.12.15)
and
samples
4
and
9
ar
e
above
t
h
e
upper
control
limits
of
t
h
e
It
may
be
observed
that
sample
7
is
above
t
h
e
upper
control
limits
on
t
h
e
R
(i
)
To
check
t
h
e
process
f
o
r
statistical
control:
Fi
g
.
12.15.
R-chart
Subgroup
No.
2
4
5
6
7
9
10
0.00
LCLR
=0
0.05
0.10
Range
CLR
=0.111
0.15
4
0.20
0.25
UCLR
=
C.
2
53
0.30
Cont
r
o
l
Charts.
12.33|
f
o
r
R-chart:
control
limits
Control
limit
or
Central
line,
CLg
Process
apability

10. 12.34
Revised control limits for X-chart:
() Todraw the revised X- and R-charts:
(x)
ueen
Now using the revised control limits, the X- and R-charts are drawn as shown in
Figs.12.16 and 12.17 respectively.
6.70
6.65
6.60
6.55
6.50
R 6.45
6.40
6.35
6.30
6.25
0
0.30
0.25
0.20
Control limit or Central = X-A, R = 6.377 - 0.729 (0.073)
: 6.324 Ans.
O 0.15
0.10
0.05
0.00
1
1 2
3 4
3
5
Subgroup No.
4 5
6
Fig. 12.16. Revised X-chart
Subgroup No.
7
6
Total Quality Management
8
7
Fig. 12.17. Revised R-chart
9
UCLy =6.430
CLz =6.377
LCLy =
6324
10
UCLR = 0.166
CLR =0.073
10
LCL =0