Transcript Slide 1

Individual values VS Averages
Comparison of individual values compared to averages
2
Individual values VS Averages
Calculations of the average for both the individual values
and for the subgroup averages are the same. However the
sample standard deviation is different.
x 

n
where
 x  population standard deviation of subgroup averages
  population standard deviation of individual values
n  subgroup size
if we assume normality, then the population standard
deviation can be estimated from : ˆ 
s
c4
s  standard deviation of samples, c  factor (in Table B) for computing ˆ
if n  20 , c4  4(n  1 ) ( 4n  3 )
Besterfield Quality Control 8th Ed
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Central Limit Theorem
If the population from which samples are taken
is not normal, the distribution of sample
averages will tend toward normality provided
that the sample size, n, is at least 4. This
tendency gets better and better as the sample
size gets larger.
Besterfield Quality Control 8th Ed
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Central Limit Theorem
Illustration of central limit theorem
5
Central Limit Theorem
Dice illustration of central limit theorem
Besterfield Quality Control 8th Ed
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Control Limits & Specifications
Figure 5-21 Relationship of limits, specifications, and distributions
Besterfield Quality Control 8th Ed
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Control Limits & Specifications
 The control limits are established as a function
of the average (ค่า control limits เป็ นค่าที่คานวนจากค่าเฉลี่ยที่
ได้จากการเก็บข้อมูล)
 Specifications are the permissible variation in
the size of the part and are, therefore, for
individual values (ค่า Specifications เป็ นค่าที่กาหนดขึ้นเพื่อ
ใช้เป็ นขอบเขตของการกระจายสาหรับค่าของข้อมูลแต่ละตัว)
 The specifications or tolerance limits are
established by design engineers or customers
to meet a particular function
Besterfield Quality Control 8th Ed
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Process Capability & Tolerance
 The process spread will be referred to as
the process capability and is equal to 6σ
 The difference between specifications is
called the tolerance
 When the tolerance is established by the
design engineer without regard to the
spread of the process, undesirable
situations can result
Besterfield Quality Control 8th Ed
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Process Capability & Tolerance
Three situations are possible:
 Case I: When the process capability is
less than the tolerance 6σ<USL-LSL
 Case II: When the process capability is
equal to the tolerance 6σ=USL-LSL
 Case III: When the process capability is
greater than the tolerance 6σ >USL-LSL
Besterfield Quality Control 8th Ed
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Process Capability & Tolerance
Case I: When the process capability is less
than the tolerance 6σ<USL-LSL
Case I 6σ<USL-LSL
Besterfield Quality Control 8th Ed
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Process Capability & Tolerance
Case II: When the process capability is
less than the tolerance 6σ=USL-LSL
Case I 6σ=USL-LSL
Besterfield Quality Control 8th Ed
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Process Capability & Tolerance
Case III: When the process capability is
less than the tolerance 6σ>USL-LSL
Case I 6σ>USL-LSL
Besterfield Quality Control 8th Ed
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Capability Index
Process capability and specifications or tolerance
are combined to form the capability index, Cp.
USL  LSL
Cp 
6 0
where C p  capability index
USL  LSL  tolerance or specificat ion
6 0  process capability
 0  standard deviation of averages of subgroup
after the averages due to designable causes are discarded
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Capability Index
The capability index does not measure process
performance in terms of the nominal or target
value. This measure is accomplished by Cpk.
Min{(USL  X ), ( X  LSL)
C pk 
3 o
where C p k  capability index
USL - LSL  tolerance or specificat ion
Besterfield Quality Control 8th Ed
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Capability Index
1. The Cp value does not change as the process
center changes
2. Cp = Cpk when the process is centered
3. Cpk is always equal to or less than Cp
4. A Cpk = 1 (and Cpk = 1.33) is a de facto
standard. It indicates that the process is
producing product that conforms to
specifications
5. A Cpk < 1 indicates that the process is producing
product that does not conform to specifications
Besterfield Quality Control 8th Ed
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Capability Index
6. A Cp < 1 indicates that the process is not
capable
7. A Cpk = 0 indicates the average is equal
to one of the specification limits
8. A negative Cpk value indicates that the
average is outside the specifications
Besterfield Quality Control 8th Ed
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Cpk Measures
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1 (and Cp = 1)
Cpk > 1
1-How to estimate Process Capability
0
This following method of calculating the process capability
assumes that the process is stable or in statistical
control:
 Take 25 (g) subgroups of size 4 for a total of 100
measurements
 Calculate the range, R, for each subgroup
R 
Calculate the average range, R = ΣR/g

d
 Calculate the estimate of the population standard
2
deviation using:
R
o 
d2
 Process capability will equal 6σ0
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2-How to estimate Process Capability
The process capability can also be obtained by using the standard
deviation:
 Take 25 (g) subgroups of size 4 for a total of 100
measurements
 Calculate the sample standard deviation, s, for each
subgroup
 Calculate the average sample standard deviation, s = Σs/g
 Calculate the estimate of the population standard
deviation
s
o 
d2
Process capability will equal 6σo
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Capability Index (EX. 5-4)
1.
A new process is started, and the sum of the sample standard
deviations for 25 groups of size 4 is 105. Approximate the process
capability.
2.
Determine the capability index before (σo = 0.038) and after (σo =
0.03) improvement using specification of 6.40 ± 0.15.
What is the Cpk value after improvement for Question 1 when the
process center is 6.40? When the process center is 6.30?
A new process is started, and the sum of the sample standard
deviations for 25 subgroups of size 4 is 750. If the specifications
are 700 ± 80, what is the process capability index? What action
would you recommend?
3.
4.
Additional control charts
1. Standard deviation chart (or s chart)
•
This chart is nearly the same X-bar and R chart. However, for
subgroup sizes ≥ 10, an s chart is more accurate than an R Chart
2. Moving average and Moving range chart
•
This chart is used to combine a number of individual values and plot
them on the chart. This technique is quite common in the chemical
industry, where only one reading (datum) is possible at a time ( Can
chemical engineering students give an example?)
3. Exponential Weighted Moving-Average (EWMA) chart
•
The EWMA chart gives the greatest weight to the most recent data
and less weight to all previous data. It primary advantage is the
ability to detect small shifts in the process average; however, it does
not react as quickly to large shifts as the X-bar chart.
S Control Chart
For subgroup sizes ≥10, an s chart is more
accurate than an R Chart. Trial control limits are
given by:

x
g
i 1
x
g

s
g
i 1
s
,
g
UCLx  x  A3 s
UCLs  B4 s
LCLx  x  A3 s
LCLs  B3 s
where
A 3 , B3 , B4  factors for the control limits found in Table B
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Revised Limits for
S control chart
x0  xnew
xx


s0  snew
ss


d
g  gd
g  gd
d
,
s0
0 
c4
UCLx  x0  A 0
UCLs  B6 0
LCLx  x0  A 0
LCLs  B5 0
where
sd  discarded subgroup averages
c 4 , A, B5 , B6  factors found in Table B
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S chart (Ex. 5-5)
Measurement
Average
S.D.
X-bar
S
Subgroup
No.
Date
Time
X1
X2
X3
X4
1
12/26
8:50
6.35
6.40
6.32
6.37
6.35
0.034
2
11:30
6.46
6.37
6.36
6.41
6.46
0.045
3
1:45
6.34
6.40
6.34
6.36
6.34
0.028
4
3:45
6.69
6.64
6.68
6.59
6.69
0.045
5
4:20
6.38
6.34
6.44
6.40
6.38
0.042
8:35
6.42
6.41
6.43
6.34
6.42
0.041
7
9:00
6.44
6.41
6.41
6.46
6.44
0.024
8
9:40
6.33
6.41
6.38
6.36
6.33
0.034
9
1:30
6.48
6.44
6.47
6.45
6.48
0.018
10
2:50
6.47
6.43
6.36
6.42
6.47
0.045
8:30
6.38
6.41
6.39
6.38
6.38
0.014
12
1:35
6.37
6.37
6.41
6.37
6.37
0.020
13
2:25
6.40
6.38
6.47
6.35
6.40
0.051
14
2:35
6.38
6.39
6.45
6.42
6.38
0.032
15
3:35
6.50
6.42
6.43
6.45
6.50
0.036
6
11
12/27
12/28
Comment
New, temporary operator
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S chart (Ex. 5-5)
Measurement
Average
Range
X-bar
S
Subgroup
No.
Date
Time
X1
X2
X3
X4
16
12/29
8:25
6.33
6.35
6.29
6.39
6.34
0.042
17
9:25
6.41
6.4
6.29
6.34
6.36
0.056
18
11:00
6.38
6.44
6.28
6.58
6.42
0.125
19
2:35
6.35
6.41
6.37
6.38
6.37
0.025
20
3:15
6.56
6.55
6.45
6.48
6.51
0.054
9:35
6.38
6.4
6.45
6.37
6.40
0.036
22
10:20
6.39
6.42
6.35
6.4
6.39
0.029
23
11:35
6.42
6.39
6.39
6.36
6.39
0.024
24
2:00
6.43
6.36
6.35
6.38
6.38
0.036
25
4:25
6.39
6.38
6.43
6.44
6.41
0.029
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12/30
Comment
Damaged oil line
Bad material
Moving average and
Moving range chart
This chart is used to combine a number of individual values and plot
them on the chart. This technique is quite common in the chemical
industry, where only one reading (datum) is possible at a time
Value
Three-period moving sum
X-bar
R
35
-
-
-
26
-
-
-
28
35+26+28
=89
(35+26+28)/3
=29.6
35-26 =
9
32
86
28.6
6
36
96
32
8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Sx-bar =
SR =
Exponential Weighted MovingAverage (EWMA) chart
The EWMA is defined by the equation :
where Vt  X t  (1   )Vt 1
Vt  the EWMA of the most recent plotted point
Vt 1  the EWMA of the previous plotted point
  the weight given to the subgroup average or individual value
X t  the subgroup average or individual value
Exponential Weighted MovingAverage (EWMA) chart
The value of lamda,  , should be between 0.05 and 0.25, with lower valu es
giving a better ability to detect smaller shifts.
Values of 0.08, 0.10, and 0.15 work well
In order to start the sequential calculatio ns, Vt-1 , is equal to X .

UCL  X  A2 R
(2   )

LCL  X  A2 R
(2   )
Exponential Weighted Moving-Average
(EWMA) chart (EX. 5-6)
Exponential Weighted Moving-Average
(EWMA) chart
Quiz 5-1
Quiz 5-2
Control charts for X-bar and R are to be established on a certain dimension part,
measured in millimeters. Data were collected in subgroup sizes of 6 and are given
below. Determine the trial central and control limits. Assume assignable causes and
revise the central line and limits.
Subgroup
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
X-bar
20.35
20.40
20.36
20.65
20.20
20.40
20.43
20.37
20.48
20.42
20.39
20.38
20.4
R
0.34
0.36
0.32
0.36
0.36
0.35
0.31
0.34
0.30
0.37
0.29
0.30
0.33
Subgroup
Number
14
15
16
17
18
19
20
21
22
23
24
25
X-bar
20.41
20.45
20.34
20.36
20.42
20.50
20.31
20.39
20.39
20.40
20.41
20.40
R
0.36
0.34
0.36
0.37
0.73
0.38
0.35
0.38
0.33
0.32
0.34
0.30
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Quiz 5-3
Control charts for X-bar and s are to be established on the Brinell hardness of hardened
tool steel in kilograms per square millimeter. Data for subgroup sizes of 8 are shown
below. Determine the trail central lime and control limits for the X-bar and s charts.
Assume that the out-of-control points have assignable causes. Calculate the revised
limits and central line.
Subgroup
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
X-bar
540
534
545
561
576
523
571
547
584
552
541
545
546
S.D.
26
23
24
27
25
50
29
29
23
24
28
25
26
Subgroup
Number
14
15
16
17
18
19
20
21
22
23
24
25
X-bar
551
522
579
549
508
569
574
563
561
548
556
553
S.D.
24
29
26
28
23
22
28
33
23
25
27
23
34
Quiz 5-4
Use data in Quiz 5-2 to establish EWMA chart, using  = 0.1
Subgroup
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
X-bar
20.35
20.40
20.36
20.65
20.20
20.40
20.43
20.37
20.48
20.42
20.39
20.38
20.4
R
0.34
0.36
0.32
0.36
0.36
0.35
0.31
0.34
0.30
0.37
0.29
0.30
0.33
Subgroup
Number
14
15
16
17
18
19
20
21
22
23
24
25
X-bar
20.41
20.45
20.34
20.36
20.42
20.50
20.31
20.39
20.39
20.40
20.41
20.40
R
0.36
0.34
0.36
0.37
0.73
0.38
0.35
0.38
0.33
0.32
0.34
0.30
35