Variable Data
• Measured on some continuous segment of
the real number line.
• Assume precision of measurement
instrument is at whatever level necessary.
• May be measuring process output
– Length, diameter, resistance, life
• May be measuring process parameters
– Pressure, temperature, tension
G. Baker, Statistics Department
University of South Carolina: Slide 1
Short Term Variation
• We gather data in subgroups. Each
subgroup estimates the location and
variation of the measured parameter or
characteristic at that moment in time.
Diameter
Time
G. Baker, Statistics Department
University of South Carolina: Slide 2
1
Consistent Short-term Variation and
Consistent Location
Characteristic or Parameter
Value
Process Output Characteristic or Parameter
Time
G. Baker, Statistics Department
University of South Carolina: Slide 3
Consistent Short-term Variation and
Changing Location
Characteristic or Parameter
Value
Process Output Characteristic or Parameter
Time
G. Baker, Statistics Department
University of South Carolina: Slide 4
2
Inconsistent Short-term Variation and
Consistent Location
Characteristic or Parameter
Value
Process Output Characteristic or Parameter
Time
G. Baker, Statistics Department
University of South Carolina: Slide 5
Inconsistent Short-term Variation and
Inconsistent Location
Characteristic or Parameter
Value
Process Output Characteristic or Parameter
18
16
14
12
10
8
6
Time
G. Baker, Statistics Department
University of South Carolina: Slide 6
3
Assignable Cause Variation
• Assignable (special) cause sources of
variation are those that act to create
inconsistent short-term variation or act to
create inconsistencies in the process
average.
• We will use a system of two charts (X-bar
and R) to distinguish assignable cause
variation from common cause variation.
G. Baker, Statistics Department
University of South Carolina: Slide 7
Consider the Following Subgroups of
Process Measurements:
Subgroup
X1
X2
X3
X4
R
X
1
36.13
32.85
34.05
38.04
5.19
35.2675
2
38.68
34.95
32.36
33.68
6.32
34.9175
3
34.34
35.69
35.06
29.72
5.97
33.7025
.
.
.
.
.
.
.
.
.
.
.
.
.
.
20
34.08
28.97
34.76
35.53
6.56
33.3350
93.30
686.9200
Sums
G. Baker, Statistics Department
University of South Carolina: Slide 8
4
X-bar Charts
Control Chart Plotting x-bar
x-bar values
µˆ x + 3σˆ x
µ̂ x
µˆ x − 3σˆ x
Subgroups
G. Baker, Statistics Department
University of South Carolina: Slide 9
X-bar Chart using R
n
• Plot x =
∑ xi
i =1
• Centerline:
n
k
X=
• Control Limits:
∑x
i =1
k
,where k = number of subgroups
X ± A2 R
,where A2 is based on subgroup size
G. Baker, Statistics Department
University of South Carolina: Slide 10
5
R Chart
Control Chart Plotting R
µˆ R + 3σˆ R
R values
µ̂ R
µˆ R − 3σˆ R
Subgroups
G. Baker, Statistics Department
University of South Carolina: Slide 11
R Chart
• Plot the range of each subgroup.
k
• Centerline: R =
• Control Limits:
∑ Ri
i =1
k
,where k = number of subgroups
LCLR = D3 R
UCLR = D4 R
,where D3 and D4 depend on subgroup size
G. Baker, Statistics Department
University of South Carolina: Slide 12
6
R Chart
R Chart for X1 - X4
UCL=10.64
Sample Range
10
5
R=4.665
0
LCL=0
0
10
20
Sam ple Num ber
G. Baker, Statistics Department
University of South Carolina: Slide 13
X-bar Chart
X-bar Chart for X1 - X4
38
UCL=37.65
Sample Mean
37
36
35
Mean=34.35
34
33
32
LCL=31.05
31
30
0
10
20
Sam ple Num ber
G. Baker, Statistics Department
University of South Carolina: Slide 14
7
The Whole Picture
Sample Mean
Xbar/R Chart for X1-X4
38
37
36
35
34
33
32
31
30
UCL=37.74
Mean=34.35
LCL=30.95
Sample Range
Subgroup
0
10
20
UCL=10.64
10
5
R=4.665
0
LCL=0
G. Baker, Statistics Department
University of South Carolina: Slide 15
Using s Instead of R
Subgroup
X1
X2
X3
X4
s
X
1
36.13
32.85
34.05
38.04
2.29
35.2675
2
38.68
34.95
32.36
33.68
2.72
34.9175
3
34.34
35.69
35.06
29.72
2.71
33.7025
.
.
.
.
.
.
.
.
.
.
.
.
.
.
20
34.08
28.97
34.76
35.53
2.97
33.3350
Sums
41.3950 686.9200
G. Baker, Statistics Department
University of South Carolina: Slide 16
8
s Chart
• Plot the sample standard deviation,s, of
n
each subgroup.
2
s = ∑ ( x − x ) /( n − 1)
i
i =1
k
• Centerline: s
•
=
∑s
i
i =1
,where k = number of subgroups
k
Control Limits: LCL = B s
UCL = B s
s
3
s
4
,where B3 and B4 depend on subgroup size
G. Baker, Statistics Department
University of South Carolina: Slide 17
X-bar Chart using s
n
• Plot x =
∑ xi
i =1
• Centerline:
n
k
X=
• Control Limits:
∑x
i =1
k
,where k = number of subgroups
X ± As
3
,where A3 is based on subgroup size
G. Baker, Statistics Department
University of South Carolina: Slide 18
9
R vs s
R
C h a rt fo r X 1 - X 4
U C L = 1 0 .6 4
Sample Range
1 0
5
R = 4 .6 6 5
0
L C L = 0
0
1 0
2 0
S a m p le N u m b e r
S
C h a rt fo r X 1 - X 4
5
U C L = 4 .6 9 0
Sample StDev
4
3
S = 2 .0 7 0
2
1
L C L = 0
0
0
1 0
2 0
S a m p le N u m b e r
G. Baker, Statistics Department
University of South Carolina: Slide 19
X-bar using R vs X-bar using s
X -b a r C h a rt fo r X 1 - X 4
3 8
U C L = 3 7 .6 5
Sample Mean
3 7
3 6
3 5
M e a n = 3 4 .3 5
3 4
3 3
3 2
L C L = 3 1 .0 5
3 1
3 0
0
1 0
2 0
S a m p le N u m b e r
X -b a r C h a rt fo r X 1 - X 4
38
U C L = 3 7 .7 2
Sample Mean
37
36
35
M e a n = 3 4 .3 5
34
33
32
31
L C L = 3 0 .9 8
30
0
10
20
S a m p le N u m b e r
G. Baker, Statistics Department
University of South Carolina: Slide 20
10
Ball Bearing Case Study
A small plant has a single line producing ball
bearings. The process output characteristic
of interest is maximum diameter, measured
to the nearest .01mm.
Starting at 8:00 AM, a sample of 5 ball
bearings is taken and measured. This
continues every half hour until 30 samples
have been taken.
The subgroup size of each sample is 5.
G. Baker, Statistics Department
University of South Carolina: Slide 21
Ball Bearing Data
Sample
x1
x2
8:00
8.25
8:30
10.09 9.68
x3
x4
x5
9.71
10.06
9.65 10.39
9.38
9.76 11.27
.
.
.
.
.
.
.
.
.
.
.
.
22:30
9.55
9.82 10.47 10.72 10.09
G. Baker, Statistics Department
University of South Carolina: Slide 22
11
Ball Bearing Data
Ball Bearing Maximum Diameter (.01mm)
Sample Mean
11
UCL=10.75
10
Mean=9.949
LCL=9.145
9
Subgroup
0
Sample Range
Sample
1
3
10
20
30
12:30
17:30
22:30
1
1
1
UCL=2.950
2
R=1.395
1
0
LCL=0
G. Baker, Statistics Department
University of South Carolina: Slide 23
Ball Bearing Data
Ball Bearing Maximum Diameter (.01mm)
Sample Mean
11
1
UCL=10.58
10
Mean=9.932
LCL=9.279
9
Subgroup
Sample Range
Sample
0
10
20
13:30
19:00
UCL=2.393
2
R=1.132
1
0
LCL=0
After eliminating points at 8:00,12:00,16:00,20:00
G. Baker, Statistics Department
University of South Carolina: Slide 24
12
Ball Bearing Data
Sample Mean
Ball Bearing Maximum Diameter (.01mm)
UCL=10.56
10.5
10.0
Mean=9.898
9.5
LCL=9.236
9.0
Subgroup
0
Sample Range
Sample
5
10
15
20
25
10:30
13:30
16:30
19:00
22:30
UCL=2.424
2
R=1.146
1
LCL=0
0
After eliminating point at 19:30
G. Baker, Statistics Department
University of South Carolina: Slide 25
Chart vs Population Estimate
Estimated Chart and Population Distributions
2
f(x)
1.5
n(9.90,0.22)
1
0.5
n(9.90,0.49)
0
8
8.5
9
9.5
10
10.5
11
11.5
12
Diameter (.01mm)
Note: 0.49/sqrt(5) = 0.22
G. Baker, Statistics Department
University of South Carolina: Slide 26
13
Operations Charts
Ball Bearing Diameter - x-bar Chart
Diameter (.01mm)
11.00
UCL = 10.56
10.50
Centerline = 9.90
10.00
9.50
LCL = 9.24
9.00
Sample Time
Ball Bearing Diameter - R Chart
3.00
UCL = 2.42
Range (.01mm)
2.50
2.00
Centerline = 1.15
1.50
1.00
0.50
LCL = 0.00
0.00
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 27
Operations Charts with Zones
Ball Bearing Diameter - x-bar
10.80
Diameter (.01mm)
10.60
10.40
10.20
10.00
9.80
9.60
9.40
9.20
10.56
10.34
10.12
10.12
9.90
9.90
9.68
9.68
9.46
9.46
9.24
9.24
ZONE A
ZONE A
ZONE B
ZONE B
ZONE C
ZONE C
ZONE C
ZONE C
ZONE B
ZONE B
ZONE
ZONE
AA
9.00
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 28
14
Zone Rules for Control Chart
Interpretation
Two out of Three Points in
Zone A (x-bar only)
x-bar or R value
x-bar or R value
Extreme Points (x-bar & R)
Sample Time
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 29
Zone Rules for Control Chart
Interpretation
Runs Above and Below
Centerline (x-bar & R)
x-bar or R value
x-bar or R value
Four out of Five Points in
Zone B or Beyond (x-bar)
1
Sample Time
6
11
16
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 30
15
Zone Rules for Control Chart
Interpretation
x-bar or R value
Successive
Points
14Six
Successive
Points
Increasing or Decreasing
Oscillate Up and Down
(x-bar & R)
(x-bar & R)
x-bar or R value
Six Successive Points
Increasing or Decreasing
(x-bar & R)
1
6
11
16
1
6
Sample Time
11
16
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 31
Zone Rules for Control Chart
Interpretation
x-bar or R value
Fifteen Successive Points
in Zone C Only (x-bar)
x-bar or R value
Eight Successive Points
that Avoid Zone C (x-bar)
1
3
5
7
Sample Time
9
11
1
6
11
16
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 32
16
Pick Out All the Zone Violations
x-bar or R value
x-bar Chart
1
6
11
16
21
26
Sample Time
G. Baker, Statistics Department
University of South Carolina: Slide 33
17