Download Document

Control Charts
Control Charts for Attributes

For variables that are categorical

Good/bad, yes/no, acceptable/unacceptable

Measurement is typically counting defectives

Charts may measure

Percent defective (p-chart)

Number of defects (c-chart)
Control Limits for p-Charts
Population will be a binomial distribution, but applying the
Central Limit Theorem allows us to assume a normal
distribution for the sample statistics
UCLp = p + z sp^
LCLp = p – z s^p
where
p=
z=
sp^ =
n=
s^p =
p (1 - p)
n
mean fraction defective in the sample
number of standard deviations
standard deviation of the sampling distribution
sample size
p-Chart - Example
Clerks at Mosier Data systems key in thousands of insurance
records each day for a variety of client firms. The CEO wants to set
control limits to include 99.73% of the random variation in the data
entry process when it is in control.
Sample of the work of 20 clerks are gathered and shown in the
following table (Next slide). Mosier carefully examined 100 records
entered by each clerk and counts the number of errors. She also
computes the fraction defective in each sample.
Calculate the control limits….
p-Chart – Example
Sample
Number
1
2
3
4
5
6
7
8
9
10
Number
of Errors
Sample
Number
Number
of Errors
6
5
0
1
4
2
5
3
3
2
11
12
13
14
15
16
17
18
19
20
6
1
8
7
5
4
11
3
0
4
p-Chart – Example
Sample
Number
1
2
3
4
5
6
7
8
9
10
Number
of Errors
Fraction
Defective
6
5
0
1
4
2
5
3
3
2
.06
.05
.00
.01
.04
.02
.05
.03
.03
.02
Sample
Number
Number
of Errors
11
6
12
1
13
8
14
7
15
5
16
4
17
11
18
3
19
0
20
4
Total = 80
Fraction
Defective
.06
.01
.08
.07
.05
.04
.11
.03
.00
.04
p-Chart – Example
Sample
Number
Number
of Errors
Fraction
Defective
1
2
3
4
5
6
7
8
9
10
6
5
0
1
4
2
5
3
3
2
.06
.05
.00
.01
.04
.02
.05
.03
.03
.02
p=
80
= .04
(100)(20)
Sample
Number
Number
of Errors
Fraction
Defective
11
6
12
1
13
8
14
7
15
5
16
4
17
11
18
3
19
0
20
4
Total = 80
s ^p =
(.04)(1 - .04)
100
.06
.01
.08
.07
.05
.04
.11
.03
.00
.04
= .02
p-Chart - Example
UCLp = p + z s ^p = .04 + 3(.02) = .10
Fraction defective
LCLp = p – z s p^ = .04 - 3(.02) = 0
.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
–
–
–
–
–
–
–
–
–
–
–
–
UCLp = 0.10
p = 0.04
|
|
|
|
|
|
|
|
|
|
20
18
16
14
12
10
8
6
4
2
Sample number
LCLp = 0.00
p-Chart - Example
UCLp = p + z s ^p = .04 + 3(.02) = .10
Fraction defective
LCLp = p – z s p^ = .04 - 3(.02) = 0
.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
–
–
–
–
–
–
–
–
–
–
–
–
Possible
assignable
causes present
UCLp = 0.10
p = 0.04
|
|
|
|
|
|
|
|
|
|
20
18
16
14
12
10
8
6
4
2
Sample number
LCLp = 0.00
Control Limits for c-Charts
Population will be a Poisson distribution, but applying the
Central Limit Theorem allows us to assume a normal
distribution for the sample statistics
UCLc = c + z c
where
LCLc = c – z
c = mean number of defects per unit
c
c-Chart for Cab Company
Red Top Cab company receives several complaints per day about
the behavior of its drivers. Over a 9-day period (where days are the
units of measure), the owner received the following number of calls
from irate passengers: 3, 0, 8, 9, 6, 7, 4, 9, 8 for a total of 54
complaints. The owner wants to compute 99.73% control limits.
c-Chart for Cab Company
UCLc
=c+3 c
=6+3 6
= 13.35
LCLc = c - 3 c
=6-3 6
=0
Number defective
c = 54 complaints/9 days = 6 complaints/day
14
12
10
8
6
4
2
0
UCLc = 13.35
–
–
–
–
–
c= 6
–
–
LCLc = 0
|
|
|
|
|
|
|
|
|
–
1
2
3
4
5
Day
6
7
8
9
Which Control Chart to Use
Variables Data Using an x-Chart and R-Chart
1. Observations are variables
2. Collect 20 - 25 samples of n = 4, or n = 5, or more, each
from a stable process and compute the mean for the x-chart
and range for the R-chart
3. Track samples of n observations each.
Which Control Chart to Use
Attribute Data Using the p-Chart
1. Observations are attributes that can be categorized as good
or bad (or pass–fail, or functional–broken), that is, in two
states.
2. We deal with fraction, proportion, or percent defectives.
3. There are several samples, with many observations in each.
For example, 20 samples of n = 100 observations in each.
Which Control Chart to Use
Attribute Data Using a c-Chart
1. Observations are attributes whose defects per unit of output
can be counted.
2. We deal with the number counted, which is a small part of
the possible occurrences.
3. Defects may be: number of blemishes on a desk; complaints
in a day; crimes in a year; broken seats in a stadium; typos in
a chapter of this text; or flaws in a bolt of cloth.