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Chapter 6 Statistical Process Control (SPC). Descriptive Statistics. 1. Measures of Central Tendencies (Location) Mean Median = The middle value Mode - The most frequent number 2. Measures of Dispersion (Spread) Range R=Maximum-Minimum Standard Deviation Variance. x. x. x. - PowerPoint PPT Presentation
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1
Chapter 6Chapter 6
Statistical Statistical Process Control Process Control
(SPC)(SPC)
2
Descriptive Statistics
1. Measures of Central Tendencies (Location)
• Mean
• Median = The middle value
• Mode - The most frequent number
2. Measures of Dispersion (Spread)
• Range R=Maximum-Minimum
• Standard Deviation
• Variance
N
xi
N
)x( 2i
2
1 2 3 4 5 6 7 8
xxxx x
µ
(x-µ)
N
xi2)(
The Standard Deviation
River Crossing Problem
River A B C
1 1 1
2 1 1
3 3 6
3 3 1
3 3 6
3 3 1
3 6 6
3 2 1
2 2 1
2 1 1
Average 2.5 2.5 2.5
Range 2 5 5
St Dev 0.7071 1.5092 2.4152
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Inferential Statistics
Population (N) Parameters
Samples (n)
Statistics
1. Central Tendency: 1. Central Tendency:
2. Dispersion: 2. Dispersion:
x
iN n
xX i
n
ix
N
( )2
s
x X
n
n
i
1
2
1
( )
6
The Normal (Gaussian) Curve
-3 -2 -1 +1 +2 +3
68.26%
95.46%
99.73%
7
0
2
4
6
8
10
12
14
16
18
20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Series1
Series2
Series3
Series4
Series5
Red Bead Experiment
8
Types of Control Charts
Quality Characteristic
n>6
Variable Attribute
Type of Attribute
Constantsamplesize?
Constantsampling
unit?
p-chart
np-chart
u-chart
c-chart
X and MR
chart
X-bar and R chart
X-bar and s chart
Defective DefectYes
Yes
YesYes
No
No
No
No
n>1
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Data
Information
1. Central Tendency
2. Dispersion
3. Shape
Action
Stats
Decision No Action
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The Shape of the Data Distribution
mean = median = mode
modemean
median
Skewed to the right (positively skewed)
medianmode
mean
Skewed to the left (negatively skewed)
)(3 medianmean
SK
• “Box-and-Whisker” Plot
• Pearsonian Coefficient of Skewness
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Control Charts
+3+3σσ
AveragAveragee
-3-3σσ
Common Cause(Chance or Random)
Special Cause(Assignable)
Special Cause(Assignable)
12
Central Limit Theorem
Standard Error of the Mean
xpop
n .
Population (individual) Distribution
Sample (x-bar) Distribution
X
μ
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X-Bar and R Example
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-Bar
R .005 .006 .006 .003 .006 .003 .00483
Rational Subgroup
Subgroup Interval
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X-Bar and R Control Chart Limits
RAX 2
RD4
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
RAX 2
UCLx-Bar .16487 + (.577 x .00483) = .1676
LCLx-Bar.16487 - (.577 x .00483) = .1621
UCLR2.114 x .00483 = .0102
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Attribute Control Chart Limits
Defectives Defects
ChangingSample Size
FixedSample Size
n
ppp
)1(3
)1(3 pnpnp
n
uu 3
cc 3
17
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
n
ppp
)1(3
p-Chart Example
UCLp
LCLp
n
ppp
)1(3
1089.6.243
)06238.1(06238.306238.
0159.6.243
)06238.1(06238.306238.
*Note: Use n-bar if all n’s are within 20% of n-bar
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19
The α and β on Control Charts
+3+3σσ
AveragAveragee
-3-3σσ
α = .00135
α = .00135
β
β
β
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Out of Control Patterns
2 of 3 successive points outside 2
4 of 5 successive points outside 1
8 successive points same side of centerline
-3
3
2
1
-1
-2
Average
21
Control Chart Patterns
Gradual Trend
“Freaks”
Sudden Shifts
Cycles
Instability
“Hugging” Centerline “Hugging Control Limits”
22
Six Sigma Process Capability
Cpk = 1.5 3.4 ppm
USLLSL1.5
Cp = 2.0 .54 ppm
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Cause and Effect Diagrama.k.a. Ishikawa Diagram, Fishbone Diagram
Process
Person Procedures
Material Equipment
B CA
24
Pareto Charta.k.a. 80/20 Rule
Vital Few
Trivial (Useful) Many
25
26
27
28
29
Taguchi Loss Function
.500 .520.480
The Taguchi Loss Function: L (x) = k (x-T)2
Los
s ($
)
.500 .520.480
Traditional Loss Function:
Los
s ($
)
30
Response Curves
Most “Robust” Setting