Download Certified Quality Engineer Body of Knowledge

Chapter 6
Statistical Process
Control (SPC)
1
Descriptive Statistics
1. Measures of Central Tendencies (Location)
•
Mean

x
i
N
•
Median = The middle value
•
Mode - The most frequent number
2. Measures of Dispersion (Spread)
•
Range
R=Maximum-Minimum
•
Standard Deviation
•
Variance

 (x
i
  )2
N
2
2
The Standard Deviation
(x-µ)
1
2
3
x
x
x
x
x
4
5
6
7
µ
 
(x
i
  )2
N
8
River Crossing Problem
River
A
B
C
1
2
3
3
1
1
3
3
1
1
6
1
3
3
3
3
3
6
6
1
6
Average
3
2.5
2
2
2.5
2
1
2.5
1
Range
22
51
51
St Dev
0.7071
1.5092
2.4152
Inferential Statistics
Population (N)
Parameters
Samples (n)
Statistics
1. Central Tendency:
x
i

N
1. Central Tendency:
2. Dispersion:
2. Dispersion:
s   n 1 
  n 
 ( xi
  )2
N
xi

X
n
 X )2
n 1
 ( xi
5
The Normal (Gaussian) Curve
-3
-2
-1

+1
+2
+3
68.26%
95.46%
99.73%
6
Red Bead Experiment
20
18
16
14
Series1
12
Series2
10
Series3
Series4
8
Series5
6
4
2
0
1
2 3
4
5 6
7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
7
Types of Control Charts
Quality Characteristic
Attribute
Variable
n>1
No
Yes
n>6
Yes
X-bar
and s
chart
No
X and
MR
chart
X-bar
and R
chart
Type of Attribute
Defective
Constant
sample
size?
No
p-chart
Defect
Yes
np-chart
Constant
sampling
unit?
No
Yes
c-chart
u-chart
8
Data
Stats
Information
1. Central Tendency
2. Dispersion
3. Shape
Decision
No Action
Action
9
The Shape of the Data Distribution
• “Box-and-Whisker” Plot
mean = median = mode
mode
median
median
mean
mode
mean
Skewed to the right (positively skewed)
Skewed to the left (negatively skewed)
• Pearsonian Coefficient of Skewness
SK 
3( mean  median )

10
Control Charts
Special Cause
(Assignable)
+3σ
Common Cause
(Chance or
Random)
Average
-3σ
Special Cause
(Assignable)
11
Central Limit Theorem
X
Sample (x-bar) Distribution
Population (individual)
Distribution
Standard Error of
the Mean
μ
pop.
x 
n
12
X-Bar and R Example
Rational
Subgroup
Subgroup Interval
1
.164
.162
.161
.163
.163
.166
2
.168
.164
.167
.166
.164
.165
3
.165
.164
.164
.166
.161
.165
4
.169
.164
.164
.163
.167
.167
5
.167
.168
.165
.162
.164
.168
X-Double Bar
X-Bar
.1666
.1664
.1642
.1640
.1638
.1662
.16487
R
.005
.006
.006
.003
.006
.003
R-Bar
.00483
13
X-Bar and R Control Chart Limits
X  A2 R
.16487 + (.577 x .00483) = .1676
LCLx-Bar
X  A2 R
.16487 - (.577 x .00483) = .1621
UCLR
D4 R
2.114 x .00483 = .0102
UCLx-Bar
n
A2
D4
d2
2
1.880
3.268
1.128
3
1.023
2.574
1.693
4
.729
2.282
2.059
5
.577
2.114
2.326
6
.483
2.004
2.534
14
15
Attribute Control Chart Limits
Defectives
Changing
Sample Size
Fixed
Sample Size
p(1  p)
p3
n
np  3 np(1  p)
Defects
u3
u
n
c3 c
16
p-Chart Example
n
235
250
200
250
260
225
270
269
237
240
*n-bar =
243.6
p
.0766
.0600
.1100
.0200
.0462
.0667
.0815
.0409
.0820
.0417
p-bar=
.06238
*Note: Use n-bar if all n’s are within 20% of n-bar
UCLp
p3
p(1  p )
n
.06238  3
.06238(1  .06238)
 .1089
243.6
LCLp
p3
p(1  p )
n
.06238  3
.06238(1  .06238)
 .0159
243.6
17
18
The α and β on Control Charts
β
α=
.00135
+3σ
Average
β
-3σ
β
α=
.00135
19
Out of Control Patterns
2 of 3 successive
points outside 2 
8 successive points
same side of centerline
3
2
1
Average
-1
-2
-3
4 of 5 successive points
outside 1 
20
Control Chart Patterns
Instability
“Freaks”
Gradual Trend
Sudden Shifts
Cycles
“Hugging” Centerline
“Hugging Control Limits”
21
Six Sigma Process Capability
LSL
USL
1.5
Cp = 2.0
Cpk = 1.5
.54 ppm
3.4 ppm
22
Cause and Effect Diagram
a.k.a. Ishikawa Diagram, Fishbone Diagram
Person
Procedures
A
B
C
Process
Material
Equipment
23
Pareto Chart
a.k.a. 80/20 Rule
Vital Few
Trivial (Useful) Many
24
25
26
27
28
Taguchi Loss Function
The Taguchi Loss Function:
L (x) = k (x-T)2
Loss
($)
.480
.500
.520
Traditional Loss Function:
Loss
($)
.480
.500
.520
29
Response Curves
Most “Robust” Setting
30