Steps in Using the
X and R Chart
1. Decisions preparatory to the control charts
– Objective:
• Is specification met?
• Does the process need any change?
• Does the process need any corrective action?
• Set the standard for acceptance/rejection of
manufactured or purchased product
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Steps in Using the
X and R Chart
– Variable
• Choose the right variable
– Cause and effect diagram (Figure 9.8)
• measurable?
• economic?
– Homogeneous?
– Size (minimize variation within group)
– Frequency (hourly? daily?)
– Forms and methods (Figures 2.7 and 2.8)
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Steps in Using the
X and R Chart
2. Starting the control chart
– Recording, finding average and range, plotting
– For sample size n and number of samples k
n
Xi  
j 1
X ij
k
Xi
,X  
n
i 1 k
Ri  X i,max
k
Ri
 X i,min , R  
i 1 k
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Steps in Using the
X and R Chart
3. Determining trial limits
– Determine a minimum number of samples
UCLR  D4 R,LCLR  D3 R
UCL X  X  A2 R,LCLX  X  A2 R
4. Preliminary conclusions
– Has the mean shifted?
– Is the process producing too many defective
items?
5. Revision and use of the control
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Control Chart for Attributes
Topic
• Control charts for attributes
• The p and np charts
• Variable sample size
• Sensitivity of the p chart
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Control Chart for Attributes
• In a control chart for variables, quality characteristic
is expressed in numbers. Many quality characteristics
(e.g., clarity of glass) can be observed only as
attributes, i.e., by classifying into defectives and nondefectives.
• If many quality characteristics are measured, a
separate control chart for variable will be needed for
each quality characteristic. However, the use of one
control chart for all the characteristics may be
cheaper.
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Control Chart for Attributes
• The cost of collecting data for attributes is less that
for the variables
• There are various types of control charts for
attributes:
– The p chart for the fraction rejected
– The np chart for the total number rejected
– The c chart for the number of defectives
– The u chart for the number of defectives per unit
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Control Chart for Attributes
• Poisson Approximation:
– Occurrence of defectives may be approximated by
Poisson distribution
– Given  np , the expected number of defectives in
sample, Probability of c or fewer defectives is
given in Table G
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The p and np Charts
• The 3-sigma limits for the p chart, for sample I
p(1  p )
p(1  p )
UCLp  p  3
, LCLp  p  3
ni
ni
UCLnp  n p  3 n p(1  p ), LCLnp  n p  3 n p(1  p )
• Note: p is the observed mean, p is the expected
mean and p0 is the target mean.
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Variable Sample Size
Problems and Solution Approaches
•
•
•
•
Control limits change with the sample size.
If sample size decreases, the control limits get wider.
If the sample size increases, the control limits get tighter.
Three solutions:
– Compute new control limits for every sample
– Estimate the average sample size for the immediate
future. Compute control limits and use it unless the
sample size is substantially different.
– Draw several sets of control limits. A plan is to use 3
sets of limits - one for expected sample size, one for
minimum and one for maximum.
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Variable Sample Size
Choice Between the p and np Charts
• If the sample size varies, p chart, is more appropriate
• If the sample size is constant, np chart, may be used
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Sensitivity of the p Chart
• Smaller samples are less sensitive
• Smaller samples may not be useful at all e.g., if only
0.1% of the product is rejected
• If a control chart is required for a single measurable
characteristic, X chart will give useful results with a
much smaller sample.
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Problem 6.4: A manufacturer purchases small bolts in
cartons that usually contain several thousand bolts.
Each shipment consists of a number of cartons. As part
of the acceptance procedure for these bolts, 400 bolts
are selected at random from each carton and are
subjected to visual inspection for certain nonconformities. In a shipment of 10 cartons, the
respective percentages of rejected bolts in the samples
from each carton are 0, 0, 0.5, 0.75, 0,2.0, 0.25, 0,
0.25, and 1.25. Does this shipment of bolts appear to
exhibit statistical control with respect to the quality
characteristics examined in this inspection?
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Problem 6.5: An item is made in lots of 200 each. The
lots are given 100% inspection. The record sheet for
the first 25 lots inspected showed that a total of 75
items did not conform to specifications.
a. Determine the trial limits for an np chart.
b. Assume that all points fall within the control limits.
What is your estimate of the process average fraction
nonconforming  p ?
c. If this  p remains unchanged, what is the probability
that the 26th lot will contain exactly 7 nonconforming
units? That it will contain 7 or more nonconforming
units? (Hint: use Poisson approximation and Table G)
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