Statistical Process Control - Northern Arizona University
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Transcript Statistical Process Control - Northern Arizona University
Statistical Process
Control
Overview
Variation
Control charts
R charts
X-bar (x ) charts
P charts
Statistical Quality Control (SPC)
Measures performance of a process
Primary tool - statistics
Involves collecting, organizing, & interpreting
data
Used to:
Control the process as products are produced
Inspect samples of finished products
Bottling Company
Machine automatically fills a 20 oz bottle.
Problem with filling too much? Problems with
filling to little?
So Monday the average is 20.2 ounces.
Tuesday the average is 19.6 ounces.
Is this normal? Do we need to be concerned?
Wed is 19.4 ounces.
Natural Variation
Machine can not fill every
bottle exactly the same
amount – close but not
exactly.
Natural variation
Ounces
Bottle
1
2
3
4
5
21.2
21.0
20.8
20.6
20.4
20.2
20.0
19.8
1
2
3
Bottle
4
5
Amount
19.9
20.2
20.1
20.0
19.9
Bottle
1
2
3
4
5
Assignable variation
A cause for part of the
variation
Assignable variation
Ounces
21.2
21.0
20.8
20.6
20.4
20.2
20.0
19.8
1
2
3
Bottle
4
5
Amount
20.9
21.0
21.0
20.8
20.9
SPC
Objective: provide statistical signal when
assignable causes of variation are present
Control Chart Types
Continuous
Numerical Data
Categorical or Discrete
Numerical Data
Control
Charts
Variables
Charts
R
Chart
Attributes
Charts
X
Chart
P
Chart
C
Chart
Measuring quality
Attributes
Variables
Characteristics that
you measure, e.g.,
weight, length
May be in whole or in
fractional numbers
Continuous random
variables
Characteristics for which
you focus on defects
Classify products as
either ‘good’ or ‘bad’, or
count # defects
e.g., radio works or not
Categorical or discrete
random variables
Control Chart Purposes
Show changes in data pattern
e.g., trends
Make corrections before process is out of control
Show causes of changes in data
Assignable causes
Data outside control limits or trend in data
Natural causes
Random variations around average
Figure S6.7
Steps to Follow When Using Control
Charts
TO SET CONTROL CHART LIMITS
1. Collect 20-25 samples of n=4 or n=5 a stable
process
compute the mean of each sample.
2. Calculate control limits
Compute the overall means
Calculate the upper and lower control limits.
Steps to Follow When Using Control Charts continued
TO MONITOR PROCESS USING THE CONTROL CHARTS:
1.
Collect and graph data
Graph the sample means and ranges on their respective
control charts
Determine whether they fall outside the acceptable limits.
2.
Investigate points or patterns that indicate the process is out of
control. Assign causes for the variations.
3.
Collect additional samples and revalidate the control limits.
R Chart
Monitors variability in process
Variables control chart
Interval or ratio scaled numerical data
Shows sample ranges over time
Difference between smallest & largest values in
inspection sample
R Chart
Control Limits
UCL R D 4 R
From Table S6.1
LCL R D 3 R
s
R
Ri
i 1
s
Sample Range at
Time i
# Samples
Control Charts
for Variables
West Allis Industries
The management of West Allis Industries is
concerned about the production of a special
metal screw ordered by several of their
largest customers. The diameter of the
screw is critical.
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
Sample
2
3
4
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5047
Should be at least 20 samples of size 4 to
calculate the control limits.
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5039
R
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5039
R
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
4
0.5022 0.5009 0.5027
0.5041 0.5024 0.5020
0.5026 0.5035 0.5023
0.5027
– 0.5009
0.5034
0.5024
0.5015
0.5056 0.5034 0.5039
R
= 0.0018
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
4
R
0.5022 0.5009 0.5027 0.0018
0.5041 0.5024 0.5020
0.5026 0.5035 0.5023
0.5027
– 0.5009
= 0.0018
0.5034
0.5024
0.5015
0.5056 0.5034 0.5039
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
4
R
0.5022 0.5009 0.5027 0.0018
0.5041 0.5024 0.5020
0.5026 0.5035 0.5023
0.5027
– 0.5009
= 0.0018
0.5034
0.5024
0.5015
0.5056 0.5034 0.5039
0.5041 - 0.5020
= 0.0021
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5047
R
0.0018
0.0021
0.0017
0.0026
0.0022
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5047
R=
R
0.0018
0.0021
0.0017
0.0026
0.0022
0.0021
Control Charts
for Variables
Control Charts – Special Metal Screw
R-Charts
UCLR = D4R
LCLR = D3R
R = 0.0021
Control Charts
for
Variables
Control Chart
Factors
Size of
Sample
(n)
Factor for UCL
and LCL for
x-Charts
(A2)
Factor for
LCL for
R-Charts
(D3)
Factor
UCL for
R-Charts
(D4)
2
3
4
5
6
7
1.880
1.023
0.729
0.577
0.483
0.419
0
0
0
0
0
0.076
3.267
2.575
2.282
2.115
2.004
1.924
Control Charts
for
Variables
Control Chart
Factors
Factor for UCL Factor for
Factor
Control
- Special
Metal
Screw UCL for
Size of Charts
and LCL
for
LCL for
Sample
R-Charts
R-Charts
R = 0.0020
D4 = 2.2080
R - Charts x-Charts
(n)
(A2)
(D3)
(D4)
2
3
4
5
6
7
1.880
1.023
0.729
0.577
0.483
0.419
0
0
0
0
0
0.076
3.267
2.575
2.282
2.115
2.004
1.924
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
UCLR = D4R
LCLR = D3R
R = 0.0021
D4 = 2.282
D3 = 0
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = 0.0021
D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
UCLR = 2.282 (0.0021) = 0.00479 in.
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = 0.0021
D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
UCLR = 2.282 (0.0021) = 0.00479 in.
LCLR = 0 (0.0021) = 0 in.
Control Charts
for Variables
Control Charts—Special Metal Screw
R-Charts
R = 0.0021
D4 = 2.282
D3 = 0
UCLR = D4R
LCLR = D3R
UCLR = 2.282 (0.0021) = 0.00479 in.
LCLR = 0 (0.0021) = 0 in.
Range Chart Special Metal Screw
X Chart
Monitors process average
Variables control chart
Interval or ratio scaled numerical data
Shows sample means over time
X Chart
Control Limits
UCL x x A R
From
Table S6.1
LCL x x A R
s
x
xi
i 1
s
Sample
Range at
Time i
Sample
Mean at
Time i
s
R
# Samples
Ri
i 1
s
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5047
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5039
R
_
x
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5039
R
_
x
Control Charts
for Variables
Special Metal Screw
Sample
Sample
Number
1
2
3
1
0.5014 0.5022 0.5009
2
0.5021 0.5041 0.5024
3
0.5018 0.5026 0.5035
4
0.5008 0.5034 0.5024
+ 0.5022
5(0.5014
0.5041
0.5056+ 0.5009
0.5034
_
4
R
x
0.5027 0.0018 0.5018
0.5020
0.5023
0.5015
+0.5039
0.5027)/4 = 0.5018
Control Charts
for Variables
Special Metal Screw
Sample
Sample
_
Number
1
2
3
4
R
x
1
0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2
0.5021 0.5041 0.5024 0.5020
3
0.5018 0.5026 0.5035 0.5023
(0.5021
+0.5008
0.50410.5034
+ 0.5024
+ 0.5020)/4
0.5027
4
0.5024
0.5015 =
5
0.5041 0.5056 0.5034 0.5039
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5047
R
0.0018
0.0021
0.0017
0.0026
0.0022
_
x
0.5018
0.5027
0.5026
0.5020
0.5045
Control Charts
for Variables
Special Metal Screw
Sample
Number
1
2
3
4
5
1
0.5014
0.5021
0.5018
0.5008
0.5041
Sample
2
3
0.5022 0.5009
0.5041 0.5024
0.5026 0.5035
0.5034 0.5024
0.5056 0.5034
4
0.5027
0.5020
0.5023
0.5015
0.5047
R=
R
0.0018
0.0021
0.0017
0.0026
0.0022
0.0021
=
x=
_
x
0.5018
0.5027
0.5026
0.5020
0.5045
0.5027
Control Charts
for Variables
Control Charts—Special Metal Screw
X-Charts
R = 0.0021
x= = 0.5027
=
UCLx = x + A2R
LCL = x= - A R
x
Example 7.1
2
Control Charts
for
Variables
Control Chart
Factors
Factor for UCL Factor for
Factor
Control
- Special
Metal
Screw UCL for
Size of Charts
and LCL
for
LCL for
Sample
R-Charts
R-Charts
R = 0.0020
x - Charts x-Charts
(n)
(A2) x = 0.5025 (D3)
(D4)
2
1.880
UCL
=
x
+
A
x
2R
3
1.023
LCL
2R
4 x = x - A0.729
5
6
7
Example 7.1
0.577
0.483
0.419
0
0
0
0
0
0.076
3.267
2.575
2.282
2.115
2.004
1.924
Control Charts
for Variables
Control Charts—Special Metal Screw
x- Charts
R = 0.0021
x= = 0.5027
=
UCLx = x + A2R
LCL = x= - A R
x
2
A2 = 0.729
Control Charts
for Variables
Control Charts—Special Metal Screw
x-Charts
R = 0.0021
x= = 0.5027
A2 = 0.729
=
UCLx = x + A2R
LCL = x= - A R
x
2
UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.
Example 7.1
Control Charts
for Variables
Control Charts—Special Metal Screw
x-Charts
R = 0.0021
x= = 0.5027
A2 = 0.729
=
UCLx = x + A2R
LCL = x= - A R
x
2
UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.
LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in.
x-Chart Special Metal Screw
x-Chart Special Metal Screw
x-Chart Special Metal Screw
Investigate Cause
p Chart
Shows % of nonconforming items
Attributes control chart
Nominally
e.g.,
scaled categorical data
good-bad
p Chart Control Limits
UCLp p z
p (1 p )
n
z = 2 for 95.5% limits;
z = 3 for 99.7% limits
LCL p p z
s
p
p (1 p )
n
# Defective Items in
Sample i
xi
i 1
s
n
i 1
i
Size of sample i
Hometown Bank
HOMETOWN BANK
The operations manager of the booking
services department of Hometown Bank
is concerned about the number of wrong
customer account numbers recorded by
Hometown personnel. Each week a
random sample of 2,500 deposits is
taken, and the number of incorrect
account numbers is recorded. The
records for the past 12 weeks are shown
in the following table. Is the process out
of control? Use 3-sigma control limits.
Control Charts for Attributes
Sample
Number
Wrong
Account Number
1
2
3
4
5
6
7
8
9
10
11
12
15
12
19
2
19
4
24
7
10
17
15
3
UCLp = p + zp
LCLp = p - zp
p =
Total
p(1 - p)/n
147
Hometown Bank
n = 2500
Total defectives
p = Total observations
Sample
Number
1
2
3
4
5
6
7
8
9
10
11
12
Control Charts
Wrong
for Attributes
Account Number
15
12
19
2
19
4
24
7
10
17
15
3
Hometown Bank
n = 2500
UCLp = p + zp
LCLp = p - zp
p =
Total
p(1 - p)/n
147
147
p=
12(2500)
Sample
Number
1
2
3
4
5
6
7
8
9
10
11
12
Control Charts
Wrong
for Attributes
Account Number
15
12
19
2
19
4
24
7
10
17
15
3
Hometown Bank
n = 2500
UCLp = p + zp
LCLp = p - zp
p =
Total
p(1 - p)/n
147
p = 0.0049
Control Charts
for Attributes
Hometown Bank
n = 2500 p = 0.0049
UCLp = p + zp
LCLp = p – zp
p =
p(1 – p)/n
Control Charts
for Attributes
Hometown Bank
n = 2500 p = 0.0049
UCLp = p + zp
LCLp = p – zp
p =
0.0049(1 – 0.0049)/2500
Control Charts
for Attributes
Hometown Bank
n = 2500 p = 0.0049
UCLp = p + zp
LCLp = p – zp
p = 0.0014
Control Charts
for Attributes
Hometown Bank
n = 2500 p = 0.0049
UCLp = 0.0049 + 3(0.0014)
LCLp = 0.0049 – 3(0.0014)
p = 0.0014
Control Charts
for Attributes
Hometown Bank
n = 2500 p = 0.0049
UCLp = 0.0049 + 3(0.0014)
LCLp = 0.0049 – 3(0.0014)
p = 0.0014
Why 3?
3-sigma limits
Also to within 99.7%
Control Charts
for Attributes
Hometown Bank
n = 2500 p = 0.0049
UCLp = 0.0091
LCLp = 0.0007
p = 0.0014
p-Chart
Wrong Account Numbers
p-Chart
Wrong Account Numbers
p-Chart
Wrong Account Numbers
Investigate Cause
Figure S6.7
Which control chart is appropriate?
Webster Chemical Company produces mastics and
caulking for the construction industry. The product
is blended in large mixers and then pumped into
tubes and capped.
Webster is concerned whether the filling process for
tubes of caulking is in statistical control. The
process should be centered on 8 ounces per tube.
Several samples of eight tubes are taken and each
tube is weighed in ounces.
Which control chart is appropriate?
Webster Chemical Company produces mastics and
caulking for the construction industry. The product
is blended in large mixers and then pumped into
tubes and capped.
Webster is concerned whether the filling process for
tubes of caulking is in statistical control. The
process should be centered on 8 ounces per tube.
Several samples of eight tubes are taken and each
tube is weighed in ounces.
X-bar and R charts
Which control chart is appropriate?
A sticky scale brings Webster’s attention to
whether caulking tubes are being properly
capped. If a significant proportion of the
tubes aren’t being sealed, Webster is placing
their customers in a messy situation. Tubes
are packaged in large boxes of 144. Several
boxes are inspected. The number of leaking
tubes in each box is recorded.
Which control chart is appropriate?
A sticky scale brings Webster’s attention to
whether caulking tubes are being properly
capped. If a significant proportion of the
tubes aren’t being sealed, Webster is placing
their customers in a messy situation. Tubes
are packaged in large boxes of 144. Several
boxes are inspected. The number of leaking
tubes in each box is recorded.
P charts