208
CHAPTER SIX
m
FORMULAS
S
-
Sd
So =
Snew
=
AVERAGE AND RANGE CHARTS
m - md
X chart:
and
m
ΣΧ;
=
m
00=
: so/ca
UCLX = Xo + Ago
LCLX = Xo - Ago
UCL, = B600
LCL, = B500
UCLX = X + AUR
LCLy = X - AZR
R chart:
m
R;
CHAPTER PROBLEMS
R
m
1. Describe the difference between chance and assignable
causes.
UCLR = DAR
LCLR = D3R
Revising the charts:
ΣΧ - Χε
m
2. How would you use variation to manage a group of
people? Why should a manager be aware of assignable
and chance causes?
3. Select one of the tools taught in this chapter and
describe why applying it to problems in your
future
will enhance your ability to solve those problems.
X = Xnew
=
m -
md
UCLX = X. + Ago
LCLy = X. – Ago
0o = Ro/d2
ŹR - Rd
m
Rnew
m - md
UCLR = 1200
LCLR D100
-
AVERAGE AND STANDARD DEVIATION
CHARTS
X AND R CHARTS
4. A large bank establishes X and R charts for the time
required to process applications for its charge cards. A
sample of five applications is taken each day. The first
four weeks (20 days) of data give
X = 16 min
5 = 3min R = 7 min
Based on the values given, calculate the centerline and
control limits for the X and R charts.
5. Steering wheels in many vehicles are outfitted with an
airbag and horn as well as switches for controlling
radio volume, cruise control, and other devices. Con-
necting these devices and switches to a steering wheel
requires excellent positioning control during assembly.
For this reason, hole location is closely monitored in
the X, Y, and Z directions. The data below are X and
R values for 25 samples of size n = 4 for hole location
data for the X-axis dimension.
X chart:
m
m
X;
X =
UCLy = X + Azs
LCLy = X – Az5
s chart:
Subgroup Number
X
Range
5=
1
50.3
0.73
וח
2
49.6
0.75
=
B45
3
50.8
0.79
UCL
LCL, = B35
Revising the charts:
4
50.9
0.74
0.72
5
49.8
m
X - Xd
6
50.5
0.73
Xo = #new
7
50.2
0.71
m - md
Variables Control Charts
209
Subgroup Number
X
Range
8
0.70
0.65
9
49.9
50.0
50.1
50.2
10
0.67
11
0.65
12
50.5
0.67
13
50.4
0.68
14
50.8
0.70
15
50.0
0.65
16
49.9
0.66
17
50.4
0.67
18
50.5
0.68
Set
up the X and R charts on this process. Does the
process seem to be in control? Why or why not? If
necessary, assume assignable causes and revise the trial
control limits.
7. When studying a process control chart tracking one
variable, what is meant by the statement, “The
process
is in a state of statistical control”?
8. Describe how both an X and R or s chart would look
if they were under normal statistical control.
9. X charts describe the accuracy of a process, and R and
s charts describe the precision. How would accuracy
be recognized on an X chart? How would precision be
recognized on either an R or s chart?
10. Why is the use and interpretation of an R or s chart so
critical when examining an X chart?
11. Create an X and R chart for the clutch plate informa-
tion in Table 6.1. You will need to calculate the range
values for each subgroup. Calculate the control limits
and centerline for each chart. Graph the data with the
calculated values. Beginning with the R chart, how
does the process look?
12. RM Manufacturing makes thermometers for use in
the medical field. These thermometers, which read in
degrees Celsius, are able to measure temperatures to
a level of precision of two decimal places. Each hour,
RM Manufacturing tests eight randomly selected
thermometers in a solution that is known to be at a
temperature of 3°C. Use the following data to create
and interpret an X and R chart. Based on the desired
thermometer reading of 3º, interpret the results of your
plotted averages and ranges.
19
50.7
0.70
20
50.2
0.65
21
49.9
0.60
22
50.1
0.64
23
49.5
0.60
24
50.0
0.62
25
50.3
0.60
Set up an X and R chart on this process. Interpret the
chart. Does the process seem to be in control? If nec-
essary, assume assignable causes and revise the trial
control limits. If the hole location is to be centered at
50.0, how does this process compare?
6. The data below are X and R values for 12 samples
of size n = 5. They were taken from a process pro-
ducing bearings. The measurements are made on the
inside diameter of the bearing. The data have been
coded from 0.50; in other words, a measurement of
0.50345 has been recorded as 345. Range values are
coded from 0.000; that is, 0.00013 is recorded as 13.
Subgroup
Average Temperature
3.06
Range
0.10
1
2
3.03
0.09
3
3.10
0.12
Subgroup Number
X
Range
4
st
3.05
0.07
1
5
345
13
2.98
0.08
6
2
347
14
3.00
0.10
7
3.01
3
0.15
350
12
346
8
4
11
3.04
0.09
9
5
350
15
3.00
0.09
6
3.03
345
16
10
0.14
11
7
349
14
2.96
0.07
12
8
348
13
2.99
0.11
13
3.01
9
348
12
0.09
10
354
15
14
2.98
0.13
11
352
13
15
3.02
0.08
12
355
16
210
CHAPTER SIX
Month
Average Number Leaving
Range
January
February
13. Interpret the X and R charts in Figure P6.1.
14. Interpret the X and R charts in Figure P6.2.
15. The variables control chart seen in Figure P6.3 is
monitoring the main score residual for a peanut
canister pull top. The data are coded from 0.00
(in other words, a value of 26 in the chart is actu-
ally 0.0026). Finish the calculations for the sum,
averages, and range. Create an X and R chart, cal-
culate the limits, plot the points, and interpret the
chart.
16. Working at a call center isn't an easy job. As custom-
ers contact the center, the employee must access a
variety of computer screens in order to answer cus-
tomer questions. Many employees don't stay with the
job very long. One call center uses X and R charts to
track voluntary quits over time. The averages compiled
below are from three call centers. Use the data below
to create an X and R chart. Describe how the process
is performing
March
April
May
June
July
August
September
October
November
December
January
February
March
April
May
June
29
29
30
32
33
34
40
39
38
32
34
31
31
32
42
41
49
50
NWNWNNWN ONWNW
Capability Data Set
X-bar UCL = 45.86, Mean = 45.25,
LCL = 44.65 (n = 5)
46.0
UCL
UCLX
45.5
Mean
ܛܓ
45.0
LCL
LCLX
Range UCL = 2.22, Mean = 1.05,
LCL = 0 (n = 5)
AM
M - ਭਾਈ
UCL
UCLR
2.0
1.5
Mean
1.0
الما
0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
FIGURE P6.1
Problem 12
354
CHAPTER TEN
16
150
7
0.047
17
150
4
0.027
18
150
3
0.020
6. The following table gives the number of nonconform-
ing product found while inspecting a series of 12 con-
secutive lots of galvanized washers for finish defects
such as exposed steel, rough galvanizing, and discol-
oration. A sample size of n = 200 was used for each
lot. Find the centerline and control limits for a fraction
nonconforming chart. If the manufacturer wishes to
have a process capability of p = 0.005, is the process
capable?
19
150
9
0.060
20
150
8
0.053
21
150
8
0.053
22
150
6
0.040
23
150
2
0.013
Sample
Size
Sample
Size
24
150
9
0.060
Nonconforming
Nonconforming
25
150
7
0.047
200
0
200
0
26
150
3
0.020
200
1
200
0
27
150
4
0.027
200
2
200
1
28
150
6
0.040
200
0
200
0
29
150
5
0.033
200
1
200
3
30
150
4
0.027
200
1
200
1
Total
4,500
173
7. Thirst-Quench, Inc. has been in business for more
than 50 years. Recently Thirst-Quench updated their
machinery and processes, acknowledging their out-of-
date style. They have decided to evaluate these changes.
The engineer is to record data, evaluate those data, and
implement strategy to keep quality at a maximum. The
plant operates eight hours a day, five days a week, and
produces 25,000 bottles of Thirst-Quench each day.
Problems that have arisen in the past include partially
filled bottles, crooked labels, upside down labels, and
no labels. Samples of size 150 are taken each hour.
Create a p chart.
8. Nearly everyone who visits a doctor's office is covered
by some form of insurance. For a doctor, the process-
ing of forms in order to receive payment from an insur-
ance company is a necessary part of doing business.
If a form is filled out incorrectly, the form cannot be
processed and is considered nonconforming (defec-
tive). Within each office, an individual is responsible
for inspecting and correcting the forms before filing
them with the appropriate insurance company. A local
doctor's office is interested in determining whether
errors on insurance forms are a major problem. Every
week they take a sample of 20 forms to use in creating
a p chart. Use the following information to create a p
chart. How are they doing?
Number
Subgroup Inspected
Number
Number Non-
conforming
np
Proportion
Noncon-
forming p
n
1
150
6
0.040
Nonconforming
2
150
3
0.020
1
2
3
150
9
0.060
2
5
4
150
7
0.047
3
8
5
150
9
0.060
4
6
150
2
0.013
5
7
150
3
0.020
6
8
150
5
0.033
7
9
150
6
0.040
8
10
150
8
0.053
9
aw sa vw a v A5
11
150
9
0.060
10
12
150
7
0.047
11
13
150
7
0.047
12
14
150
2
0.013
13
15
150
5
0.033
14
Quality Control Charts for Attributes
353
CHAPTER PROBLEMS
60
3
60
6
60
2
60
5
1. Review the control charts covered in this chapter.
Describe and discuss how one of these charts could
be applied to a situation you face at work or in your
personal life.
2. How would you determine that a process is under con-
trol? How is the interpretation of a p, u, or c chart
different from that of an X and R chart?
60
7
60
4
60
3
60
4
60
1
60
0
60
5
60
3
60
4.
60
3
60
7
CHARTS FOR FRACTION
NONCONFORMING
3. Bottled water is very popular with consumers. Manu-
facturers recognize that customers don't want bottles
with caps that aren't sealed correctly. Faulty seals
make the consumer wonder about the purity of the
water within, which creates an immediate lost sale and
affects the decision to purchase a similar product in
the future. For this reason, Local Water Bottling Co.
inspects 250 water bottles per hour (out of a total pro-
duction of 5,000 bottles per hour) to check the caps.
Any bottle inappropriately capped is unusable and will
be destroyed. Use the following production informa-
tion to create the centerline and control limits for a
fraction nonconforming control chart to monitor their
recent process. Graph and interpret this chart. What
level of performance is their process capable of?
60
4
60
6
60
8
60
4.
60
2
60
3
60
6
n
Defectives
250
1
5. Given the following information about mistakes made
on tax forms, make and interpret a fraction noncon-
forming chart.
250
1
250
0
250
Sample
Size
Sample
Size
250
0
Nonconforming
Nonconforming
250
0
20
0
20
10
250
1
20
0
20
2
250
0
0
20
1
250
0
2
20
0
250
1
0
1
1
0
ô ô ô ô ô ô ô ô ô
6
0
0
4. When producing IC chips, the quality inspectors check
the functionality of the chip, enabling them to determine
whether the chip works or doesn't work. Create the
centerline and control limits for today's fraction defec-
tive chart. Graph and interpret the chart. How should
they expect the process to perform in the near future?
1
0
1
ô ô ô ô ô ô ô ô ô ô ô ô
2
0
3
2
n
np
20
1
0
60
2
20
1
4
60
4
20
0
1
60
5
20
2
20
0
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