Chapter 7
1
Chapter 7
2
Learning Objectives
Chapter 7
3
P Control Chart
• The most versatile and widely used attributes
control chart
-- used when the subgroup size is not constant
• Used to evaluate fraction defective
• Control limits are based on Binomial
Distribution
Chapter 7
4
P Control Chart
P CONTROL CHART ALCAS PEFLOW SOLDER MACHINE
90
PERCENT DEFECTIVE
80
70
UCL PERCENT
60
50
40
30
20
10
0
2
4
6
8
10
12
14
16
18
20
22
24
26
SUBGROUP NUMBER
Chapter 7
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P Chart
• Compute the control limits

P (1  P )
UCL

P

3

n


P (1  P )

LCL  P  3

n

• The UCL and LCL are not straight lines, they
rise and fall with respect to the subgroup size
Chapter 7
6
The P Chart
• When to use:
– Only when variable data cannot be obtained.
– When charting fraction rejected as
nonconforming from a varying sample size.
– When screening multiple characteristics for
potential monitoring on variable control
charts.
– When tracking the quality level of a process
before any rework is performed.
Chapter 7
7
• How
The P Chart
– By counting the number of defective items from a sample
and then plotting the percent that are defective.
• Conditions
– In order to be of help, there should be some rejects in each
observed sample.
– The higher the quality level, the larger the sample size must
be to contain rejects. For example, if 20% of a product is
rejectable, a sample size of 5 will be needed. However, a
sample of 1,000 will give an average of only one reject per
sample if 0.1% of the product is rejectable.
Chapter 7
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The P Chart
Table 1 Formulas for the P Chart
Chart
p
Chapter 7
Control
limits
UCL  P  3
P (1  P )
n
LCL  P  3
P (1  P )
n
Centerline
Summation of
plot po int s
p
Number of
plot po int s
Plot point
Sample
size
Number of
defective units
in a sample
p
Sample size(n)
Varying
9
The P Chart
• To calculate Plot Points
– The p plot point is the fraction defective in a
sample. The centerline is the average fraction
defective in series of samples. Figure 1 is a cross
section showing countersunk holes for rivet
installation.
Chapter 7
10
The P Chart
• To calculate Plot Points
– In a sheet metal assembly shop a common process
is bucking rivets. Because of the combined
variation in the rivets, the drilled holes, and the
bucking process, there are quality problems. After
the rivets in an assembly have been bucked into
place they are checked for nonconformity. A P
chart is used to track the first-time-through fraction
defective.
Chapter 7
11
Figure 1
• Cross section of sheet metal plates with
countersunk holes for rivet installation
Chapter 7
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P Chart
Chapter 7
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P Chart
TOTALS
Chapter 7
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X Relay
Chapter 7
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Fluorocarbon Leak Test
FLUOROCARB. LEAK TEST
P Chart (%)
7
6
5
4
3
2
1
UCL
1 2 3 4 5 6 8 9 14 15 16 17 18 19 20 21 22 23 24 25 28 29
Chapter 7
16
NP Control Chart
• Number of defectives
• Used when subgroup size is constant (n)
• The actual number of defects is represented by
pn (or np)
Chapter 7
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NP Control Chart
• NP Control Chart plating defects of assembled
parts
15
14
13
12
11
UCL
10
9
8
7
6
5
4
3
2
1
2
4
6
8 10
12 14 16 18 20 22 24 26 28 30
Chapter 7
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LOT NUMBER
NP Control Charts
• Computations
– Control limits are based on the Binomial
Distribution
1) Central line: nP   Pn
K
2) UCL  nP  3 nP(1  P)
3) LCL  nP  3 nP (1  P )
Chapter 7
Where
Pn

P
n
19
NP Charts
Chapter 7
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4 Steps in control chat for Fraction
Rejected
I
II
III
IV
Chapter 7
Preparatory Decisions
Starting the control chart
Continuing the control chart
Reports and Action based on control chart
21
I. Preparatory Decisions
•
•
•
•
•
Purpose
Select Quality Characteristic
Selection of subgroup
P or NP chart
Control limit calculated
Chapter 7
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II. Starting the control chart
• Recording data
r
r

• p  n , p  n Calculation

• Trial limits
• Plotting points
i
i
i
i
i
Chapter 7
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III Continuing the control chart
P0  P
•
•
•
•
•
Chapter 7
If chart in control
UCL  P0 
3 P0 1  P0 
ni
Selection of P0
Calculation of control limits
Plotting the points/limits
Interpretation of lack of control
Periodic Review/Revision of Pi
24
IV Reports and Action based on
control chart
• Action to bring process into control-Pareto, high
spot/low spot.
• Review of Design and specification
• Information to Management(Quality level)
• Sensitivity of p chart : 0.1% requires 1000s
Chapter 7
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Example of Attribute control chart
LOT
1
2
3
4
5
Total
Chapter 7
Number Number of
inspecting defectives
500
27
50
12
800
12
100
14
150
15
1600
80
P
0.054
0.240
0.015
0.140
0.100
26
Example of Attribute Control Chart
Subgrou
p
Subgrou
p Size n
# of
Defectiv
es
Percent
Defectiv
e P (%)
UCL(%)
LCL
1
115
15
13.0
17.7
1.
2
220
18
8.2
15.4
3.
3
210
23
10.9
15.6
3.
4
220
22
10.0
15.4
3.
5
255
18
7.0
15.0
4.
6
365
15
4.1
7
255
44
8
300
13
4.3
9
280
33
11.7
14.8
10
330
42
12.7
14..3
Total
2550
243
P
Chapter 7
4.
15.0
Totlal defective

Total inspected
4.
4.
(100) 
%
LCL  P 
3
n
P 100 P 
UCL  P 
3
n
P 10027 P 
• Purpose of maintaining attribute charts is to
continuously improve the processes for defect
free production by highlighting the key problems.
• How to work on getting desirable pattern
Recommend working on 3 top problems for the day
or week of month as the time permits and solve
by a systematic problem solving method namely
(1) Define problems
(2) Find key causes
(3) Solution to cure the key causes demonstrated by
statistics
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Design of the Fraction Nonconforming Chart
•
Three parameters must be specified
1. The sample size
2. The frequency of sampling
3. The width of the control limits
•
•
Chapter 7
Common to base chart on 100% inspection
of all process output over time
Rational subgroups may also play role in
determining sampling frequency
41
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Average sample size approach
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Chapter 7
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Charts for Nonconformities
C cart and U chart
• Often times there is interest in knowing how
many defects an item has
-- C charts: total number of nonconformities in
subgroups of fixed size (defects per square
yard of cloth)
-- U charts: average number of nonconformities
per unit (defects per TV set)
Chapter 7
56
Charts for Nonconformities
• Used during inspection of complex assemblies
• Control limits based on Poisson Distribution
Chapter 7
57
C Control Chart
• UCL and LCL are straight lines because of
fixed sample sizes
CL= c
UCL= c  3 c
LCL= c  3 c
Where c is the central line and equal to
Total Defects
Num berof Subgroups
Chapter 7
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The data in table 2 represent the types of defects
found on the first two boxes. Box 1 has nine
defects and box 2 has twelve.
Notice that the types and quantity of defects are
different for the two boxes; nevertheless, the
total number of defects is plotted on the C chart.
Chapter 7
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Note:
A C chart (as well as any other attribute chart)
should only be used when there is absolutely
no way to obtain variable data from the
characteristic in which measurable data is
available.
To get started, this data can be analyzed on an
attribute chart to get ideas (using the Pareto
analysis) on the characteristics.
Chapter 7
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U Control Chart
• As the subgroup size varies, UCL and LCL varies
CL= u
UCL=
LCL=
3 u
u
n
3 u
u
n
• where u is the central line and equal to
Total Defects
Total Units
Chapter 7
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The U Chart
When to use:
• Only when variable data cannot be
obtained.
• When plotting the average number of
defects found per unit.
• When screening multiple characteristics for
potential monitoring on variable control
charts.
How:
• Each unit is examined and the average
number of defects found are plotted.
Chapter 7
62
The U Chart
Conditions:
• Constant unit size, but any convenient number of
units per plot point. Unit size is different from
samples size. For example, one unit could be
defined as 1 square-foot of material.
• On a particular day, 12.3 square-feet of material
is inspected. The plot point would represent the
average number of defects per unit, but the
sample size would be 12.3.
Chapter 7
63
The U Chart
• There are potentially several different types
of defects per unit, but none of which would
necessarily render the part a defective.
• For example: paint blemishes on a skin
panel, or various electrical faults on a
circuit board. A unit can be single part, an
assembly of several parts, an area of
material, or any rational grouping in which
the likelihood of defect(s) is constant from
unit to unit.
Chapter 7
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• In order for this type of analysis to be of
help, there should be some defects in each
observed unit.
Chapter 7
65
The U Chart
• To Calculate Plot Points:
• The u plot point is the average number of
defects per unit in a sample of n units. The
centerline is the average of all the plot points
on the chart. Figure 6-17 shows a roll of
composite material/dyed cloth with potential
multiple
defects.
Chapter 7
66
The U Chart
• Composite materials are generally made up of
several layers of materials that are bonded to
form the desired characteristics.
• Prior to lay up, are used each day, the number
of rolls inspected for defects. Since different
amounts of the material are used each day, the
number of rolls inspected also changes
daily.
Chapter 7
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Attribute Control Charts
Chart
type
Names/remarks
Central
line
Control
Limits
p 1  p 
n
p 1  p 
LCL p  p  3
n
UCLp  p  3
P
Fraction defective
P
np
Number defectives
np
c
u
Chapter 7
Number of defects
per subgroup
Number of defects
per inspection unit
C
u
UCLnp  np  3 np 1  p 
LCLnp  np  3 np 1  p 
UCLc  c  3 c
LCLc  c  3 c
UCLu  u  3 u
n
LCLu  u  3 u
n
68
Attribute
data
Variable data
N>50
N not constant
Fraction
defective
N>50
N is constant
Fraction
defective
n1
n not constant
Defects
per unit
X-bar chart
R chart
P chart
np chart
n<6
n>6
n>25
X-bar chart
S chart
X-bar chart
S2 chart
u chart
n=1
n1
Is constant
Defects per unit
Individual
chart
c chart
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Low Defect Levels
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Actions taken to improve a process
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Learning Objectives
Chapter 7
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