Process Variation
Expected Variation
Variation inherent within
the process.
It is expected on an
ongoing basis.
It affects every unit.
Common cause variation
Chance cause variation
Chronic problems
Unexpected Variation
Arises due to special
circumstances.
It occurs sporadically.
It affects only certain
units.
Special cause variation
Assignable cause
variation
Sporadic problems
G. Baker, Statistics Department
University of South Carolina
Shewhart Control Chart
Control Chart
Measurement
Center line: µ̂
Center line:
µ̂
Sample
G. Baker, Statistics Department
University of South Carolina
1
Shewhart Control Chart
Control Chart
Measurement
Upper control limit: µˆ + 3σˆ
Center line:
µ̂
Lower control limit: µˆ − 3σˆ
Sample
G. Baker, Statistics Department
University of South Carolina
Shewhart Control Chart
Control Chart
Measurement
Assignable
Cause
Upper control limit: µˆ + 3σˆ
Common
Cause
Center line:
µ̂
Lower control limit: µˆ − 3σˆ
Assignable
Cause
Sample
G. Baker, Statistics Department
University of South Carolina
2
In-Control and Out-of-Control
• A process is said to be “in control” when it
demonstrates only common cause variation.
The process is predictable.
• A process is said to be “out of control”
when it demonstrates assignable, as well as
common cause, variation. The process is
not predictable.
G. Baker, Statistics Department
University of South Carolina
Variation over Time
In Control
Out of Control
Common Cause Variation
Measurements
Measurements
Common & Assignable Cause Variation
Time
Time
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University of South Carolina
3
Types of Data
• Variable Data: quality characteristic that one
measures.
– Weight of a unit
– Fill of a bottle
– Resistance of an electrical component
• Attribute data: quality characteristic defined in
terms of “presence or absence of”.
– Defects per square yard of material
– Color inconsistencies on a painted surface
– Go/no-go gage on a variable characteristic
G. Baker, Statistics Department
University of South Carolina
Some Terms used when Dealing with
Attribute Data
• Product Control Point of View: a defect is
counted only if it influences a units function.
• Process Control Point of View: a defect is
counted if the mechanism creating the defect is the
same regardless of the defect’s effect on unit
function.
• It is essential from a process control standpoint to
observe and note each occurrence of each type of
defect.
G. Baker, Statistics Department
University of South Carolina
4
Binomial Review
•
You have a binomial experiment if
(1) The experiment consists of a repetition of n identical
and independent trials.
(2) Each trial results in two possible outcomes, success
and failure.
(3) The probability of success, denoted by p, remains the
same from trial to trial.
•
You can approximate a binomial with a normal
if np and n(1-p) > 5. (i.e., for large sample sizes
the shape of a binomial distribution will
resemble that of a normal distribution.)
G. Baker, Statistics Department
University of South Carolina
Relationship between the Normal and
Binomial Distributions
• Consider b(x;15,0.4). Bars are calculated
from binomial. Curve is normal
approximation to binomial.
b(x;n=15, p=.40)
0.25
f(x)
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
x
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University of South Carolina
5
Binomial Review
n
for x = 1, 2, … ,n
f ( x ) =   p (1 − p )
 x
µ = np
σ = np(1 − p )
( n− x )
x
2
Let n = subgroup size; k = number of subgroups; x = number of
defects in a subgroup
k
pˆ =
∑x
i =1
k
∑n
i =1
=p
G. Baker, Statistics Department
University of South Carolina
Welcome to Baker’s Better Beads, Inc.
Supplier of beads to the rich and
famous!
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University of South Carolina
6
Motto
We work hard and make
no defects!
G. Baker, Statistics Department
University of South Carolina
Work Instructions
• Take paddle and dip it into container.
• Shake gently so that 50 beads remain.
• Count defective beads and record on
“Defectives Record Sheet”.
• Return beads to container. Spills will not
be tolerated!
G. Baker, Statistics Department
University of South Carolina
7
Bead Production
• In our demonstration
– What does the bowl of beads represent?
– What does the paddle represent?
– What type of variation should we expect to find
in our analysis?
G. Baker, Statistics Department
University of South Carolina
np Control Chart
Control Chart
Measurement
Upper control limit:µ
ˆx
+ 3σˆ = np + 3 np (1 − p )
∑x
Center line: µ̂ = np
= np
ˆ =n
∑n
x
i =1
k
x
ˆx
Lower control limit:µ
k
i =1
+ 3σˆ x = np − 3 np (1 − p )
Sample
G. Baker, Statistics Department
University of South Carolina
8
np Control Chart - Example
Assume 25 subgroups of size n = 50 and
25
∑ x = 98
i =1
k
pˆ = p =
∑x
i =1
kn
=
98
= 0.0784
( 25)(50)
µˆ = np = 50(0.0784) = 3.92
x
G. Baker, Statistics Department
University of South Carolina
np Control Chart - Example
µˆ ± 3σˆ = np ± 3 np (1 − p )
x
x
= −1.78, 9.62
G. Baker, Statistics Department
University of South Carolina
9
np Control Chart
np Chart of Defectives
Number of Defectives
12
9.62
10
8
6
3.92
4
2
0.00
0
0
5
10
15
20
25
30
Sample
G. Baker, Statistics Department
University of South Carolina
Conclusions
• Process is “in control” (stable, predictable)
• We expect a sample of 50 units to contain
between 0 and 9.62 defective units,
averaging 3.92.
G. Baker, Statistics Department
University of South Carolina
10
P Control Chart
• Generally we are interested in percent of
successes, not the number of successes in a
sample of size n.
• So instead of plotting estimates of np,
number of successes in a sample of size n,
we plot estimates of p, proportion of
successes in the population.
G. Baker, Statistics Department
University of South Carolina
p Control Chart
Control Chart
Measurement
Upper control limit:
µˆ + 3σˆ = p + 3
p
Center line:
Lower control limit: µ
ˆp
p
µ̂ =
p
k
∑x
i =1
k
∑n
p (1 − p )
n
=p
i =1
+ 3σˆ = p − 3
p
p (1 − p )
n
Sample
G. Baker, Statistics Department
University of South Carolina
11
np Control Chart - Example
Assume 25 subgroups of size n = 50 and
25
∑ x = 98
i =1
k
pˆ = p =
∑=1 x
i
kn
=
98
= 0.0784
( 25)(50)
µˆ = p = 0.0784
p
G. Baker, Statistics Department
University of South Carolina
np Control Chart - Example
µˆ ± 3σˆ = p ± 3
p
p
p (1 − p )
n
= −.0356, .1924
G. Baker, Statistics Department
University of South Carolina
12
p Chart (equal sample sizes)
p Chart of Defectives
0.25
.1924
Percent Defective
0.20
0.15
0.10
0.0784
0.05
0.00
0
5
10
15
20
25
0.0000
30
Sample
G. Baker, Statistics Department
University of South Carolina
np & p Charts (equal sample sizes)
p Chart of Defectives
np Chart of Defectives
0.25
.1924
9.62
10
0.20
Percent Defective
Number of Defectives
12
8
6
3.92
4
0.15
0.10
.0784
0.05
2
0.00
0
0
5
10
15
Sample
20
25
0.0000
0.00
30
0
5
10
15
20
25
30
Sample
G. Baker, Statistics Department
University of South Carolina
13
Minitab
• Load Data in spreadsheet window
• STAT Control Charts np or p
• At a minimum, fill in variable and subgroup
size.
• Other options:
– Test
– Stamp
– Annotation
G. Baker, Statistics Department
University of South Carolina
Application of p Chart
• Usually used by management to answer:
– Is rate stable?
– Is rate location acceptable?
– Is rate variation acceptable?
– Has a system change made a difference?
G. Baker, Statistics Department
University of South Carolina
14
Before and After Process Change
P Chart for Defectiv by Change
Before
After
Proportion
0.2
UCL=0.1310
0.1
P=0.044
LCL=0
0.0
0
10
20
30
40
50
60
Sample Number
G. Baker, Statistics Department
University of South Carolina
Sample sizes are not always equal
• We can still use a p chart – plot pˆ =
k
• Centerline:
µ̂ =
p
∑x
i =1
k
∑n
x
n
=p
i =1
• Control limits will depend on sample size
and will change as sample size changes.
µˆ + 3σˆ = p ± 3
p
p
p(1 − p )
n
G. Baker, Statistics Department
University of South Carolina
15
p Chart with Varying Sample Sizes
p Chart of Defectives
UCL=0.2133
Proportion
0.2
0.1
P=0.09116
LCL=0
0.0
0
5
10
15
20
25
Sample Number
Note, varying sample sizes
G. Baker, Statistics Department
University of South Carolina
16