SMU
EMIS 7364
NTU
TO-570-N
Statistical Quality Control
Dr. Jerrell T. Stracener,
SAE Fellow
Control Charts for Attributes Data
Updated: 3-17-04
1
Attributes Data
Definition - Attributes are quality characteristics for
which each inspected item can be classified as
conforming or nonconforming to the specification
on that quality characteristic
Types
• Fraction nonconforming
• Number nonconforming
• Number of nonconformities per unit
• Average number of nonconformities per unit
2
Attributes Data
• Are best used where subjective characteristics
must be checked (presence or absence of nicks,
for example)
• Tend to emphasize defect reduction as the end
goal (vs. variability reduction)
defects (count)
percentage non-conforming (or percentage)
defect per unit
• Measurements that depend on counting are
called attributes measurements.
3
Fraction Nonconforming
Statistical Basis - For a stable process producing
identical and independent items with probability p
that an item will not conform to spec where D is
the number of items nonconforming in a random
sample of size n,
 n  x n x
b( x )  P(D  x )   p q , x  0 , 1 , . . . , n
x
and
 D  np
σ D  npq
4
Sample Fraction Nonconforming
The sample fraction non-conforming is defined as
the ratio of the number of nonconforming in the
sample, D, to the sample size n, i.e.
D
p
n
^
and
p(1  p)
^ 
p
n
5
Control Chart for Fraction Nonconforming p-chart
p(1  p)
UCL  p  3
n
Center Line  p
p(1  p)
LCL  p  3
n
6
Control Chart for Fraction Nonconforming p-chart
where
Di Fraction nonconforming in the
pi 
 ith sample for i = 1, 2, . . . , m
n
^
1 m ^
1
p  p 
m i 1 i mn
m
D
i 1
i
7
Test for Shift in Process Fraction Nonconforming
• Hypothesis:
H0: p1 = p2
H2: p1 > p2
• Test statistic:
^
Z
^
p1  p 2
 ^  1 1 
p1  p   

 n1 n 2 
^
8
Test for Shift in Process Fraction Nonconforming
where
f1
p1 
n1
^
f2
p2 
n2
^
^
^
n1 p1  n 2 p 2
p
n1  n 2
^
• Decision rule:
reject H0 if Z > Z;
otherwise accept H0
9
The p-chart - instructions
1. Obtain a series of samples of some appropriate
size. Convenient sample sizes are 50 and 100.
The ‘sample’ may actually be the complete lot if
the entire lot has been checked. Have 20 or
more groups if possible, but not less than 10
groups.
2. Count the number of defective units (warped,
undersize, oversize, or whatever the
characteristics may be in which you are
interested). Calculate the value of p for each
sample.
10
The p-chart - instructions
3. Calculate p (the average percentage defective).
This is the centerline for the p-chart.
4. Calculate upper and lower control limits for the
p-chart.
11
Uses of a p-chart
• Characteristics on which it is difficult or
impractical to obtain variables measurement.
• Studies of defects produced by machines or
operators which are directly under the
machine operators control.
• Direct studies of the amount of dropouts,
shrinkage, or scrap at specific operations
12
Uses of a p-chart - continued
• Can cover all defects and all characteristics,
• Can be a valuable capability study in itself
• Will also provide a good measure of the
effectiveness of changes, corrections or
improvements which have been made as a
result of other studies
13
p-charts vs. x - R-charts
• The p-chart is less powerful than x and R charts.
It provides less information
• With the x - R chart, we can study the process
without regard to the specifications - the p-chart
requires specs.
• The p-chart cannot tell us whether nonconformances are caused by poor centering,
excessive variability, or out-of-control conditions.
14
p-charts vs. x - R-charts
• The p-chart cannot warn of trends or shifts
unless they are so pronounced that they actually
resulted in a change in the number of defective
units produced.
15
Number of Nonconformities
c - the total number of nonconformities in a unit
16
Number of Nonconformities
Statistical Basis - The number of opportunities for
nonconformities is infinitely large and the
probability of occurrence of a nonconformity at
any location is small and constant. Samples are
of a constant size and the inspection unit is the
same for each sample.
X = number of nonconformities in the sample
c x ec
p( x )  P ( X  x ) 
, x  0 ,1, ... , n
x!
where c = expected number of nonconformities
17
Control Chart for Nonconformities - The c-chart
C - is the count of defects in a sample
c = count
c = average of the counts
total nonconform ities

total samples
UCL  c  3 c
Center Line  c
LCL  c  3 c
18
Average Number of Nonconformities Per Unit - u
c
u
n
Where c is the total nonconformities in a sample
of n inspection units
19
Control Chart - The u-chart
Occasionally, it is useful to work with defects per
unit, particularly when the inspection unit consists
of several physical units of product. Then if
n = sample size, defined as a consistent area of
opportunity
U
c total number of defects
 
n
area of opportunit y
20
Control Chart - The u-chart
U
UCL  U  3
n
Center Line  U
U
UCL  U  3
n
n
where
U
u
i 1
n
i
21