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Problem 6.15: A manufacturer wishes to maintain a
process average of 0.5% nonconforming product or
less less. 1,500 units are produced per day, and 2
days’ runs are combined to form ashipping lot. It is
decided to sample 250 units each day and use an np
chart to control production.
(a) Find the 3-sigma control limits for this process.
(b) Assume that the process shifts from 0.5 to 4%
nonconforming product. Use Table G to find the
probability that the shift will be detected as the result of
the first day’s sampling after the shift occurs.
(c) What is the probability that the shift described in (b)
will be caught within the first 3 days after it occurs?
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The Control Chart for Nonconformities
The c and u charts
• Defective and defect
– A defective article is the one that fails to conform
to some specification.
– Each instance of the article’s lack of conformity to
specifications is a defect
– A defective article may have one or more defects
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The Control Chart for Nonconformities
The c and u charts
• The np and c charts
– Both the charts apply to total counts
– The np chart applies to the total number of
defectives in samples of constant size
– The c chart applies to the total number of defects
in samples of constant size
• The u chart
– If the sample size varies, the u chart for the
number of defects per unit may be used
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The Control Chart for Nonconformities
The c and u charts
• The 3-sigma limits for the c chart are:
UCLc  c  3 c, LCLc  c  3 c
• Note: c is the observed mean,  c is the expected
mean and c0 is the standard mean.
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The Control Chart for Nonconformities
The c and u charts
• As the Poisson distribution is not symmetrical, the
upper and lower 3-sigma limits do not correspond to
equal probabilities of a point on the control chart
falling outside limits (See Section 7.7). To avoid the
problem with asymmetry, the use of 0.995 and 0.005
limits has been favored
• If the distribution does not follow Poisson law, actual
standard deviation may be greater than c and,
therefore, 3-sigma limit may actually be greater than 3 c
limit obtained from the formula
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The Control Chart for Nonconformities
The c and u charts
• The 3-sigma limits for the u chart, for sample i
u
u
UCLu  u  3
, LCLu  u  3
ni
ni
• Note: u is the observed mean,  u is the expected
mean and u0 is the standard mean. ni is some
measure of the i -th sample; it may be number of
units, number of square feet of area, etc.
•  u is not Poisson distributed but n u is
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Problem 7.3: Use Table G, App. 3, to find 0.995 and
0.005 probability limits for a c chart when c = 5.8. Also
when c = 12.0
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Problem 7.8: A c chart is used to monitor the number of
surface imperfections on sheets of photographic film.
The chart presently is set up based on c of 2.6.
(a) Find the 3-sigma control limits for this process.
(b) Use Table G to determine the probability that a point
will fall outside these control limits while the process is
actually operating at a c of 2.6.
(c) If the process average shifts to 4.8, what is the
probability of not detecting the shift on the first sample
taken after the shift occurs?
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Problem 7.16: A shop uses a control chart on
maintenance workers based on maintenance errors
per standard worker-hour. For each worker, a random
sample of 5 items is taken daily and the statistic c/n is
plotted on the worker’s control chart where c is the
count of errors found in 5 assemblies and n is the total
worker-hours required for the 5 assemblies.
(a) After the first 4 weeks, the record for one worker is
c=22 and n=54. Determine the central line and the
3-sigma control limits.
(b) On a certain day during the 4-week period, the worker
makes 2 errors in 4,3 standard worker-hour. Determine
if the point for this day falls within control limits.
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Reading and Exercises
• Chapter 6:
– pp. 223-234 (Sections 6.1-6.6)
– Problems 6.6, 6.10, 6.16
– Note for Problems 6.6 and 6.10: p0 is obtained by
eliminating out-of-control points and computing mean
value with only the in-control points
• Chapter 7:
– pp. 260-265, 268-275 (Sections 7.1-7.3,7.5-7.7)
– Problems 7.4, 7.11, 7.15, 7.26
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