Technical Note 8 Process Capability and Statistical Quality Control

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Technical Note 8Process Capability and
Statistical Quality Control

• Process Variation
• Process Capability
• Process Control Procedures
• Variable data
• Attribute data
• Acceptance Sampling
• Operating Characteristic Curve

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Basic Causes of Variation

• Assignable causes
• Common causes
• Key

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Types of Control Charts

• Attribute (Go or no-go information)
• Defectives
• p-chart application
• Variable (Continuous)
• Usually measured by the mean and the standard
  deviation.
• X-bar and R chart applications

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Types of Statistical Quality Control
Statistical
Statistical
Quality Control
Quality Control
Process
Acceptance
Process
Acceptance
Control
Sampling
Control
Sampling
Variables
Attributes
Variables
Attributes
Variables
Attributes
Variables
Attributes
Charts
Charts
Charts
Charts
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Statistical Process Control (SPC) Charts
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Control Limits

• We establish the Upper Control Limits (UCL) and
  the Lower Control Limits (LCL) with plus or minus
  3 standard deviations. Based on this we can
  expect 99.7 of our sample observations to fall
  within these limits.

99.7
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Example of Constructing a p-Chart Required Data
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Statistical Process Control FormulasAttribute
Measurements (p-Chart)
Given
Compute control limits
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Example of Constructing a p-chart Step 1
1. Calculate the sample proportions, p (these
are what can be plotted on the p-chart) for each
sample.
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Example of Constructing a p-chart Steps 23
2. Calculate the average of the sample
proportions.
3. Calculate the standard deviation of the sample
proportion
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Example of Constructing a p-chart Step 4
4. Calculate the control limits.
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Example of Constructing a p-Chart Step 5
5. Plot the individual sample proportions, the
average of the proportions, and the control
limits
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R Chart

• Type of variables control chart
• Interval or ratio scaled numerical data
• Shows sample ranges over time
• Monitors variability in process
• Example Weigh samples of coffee compute ranges
  of samples Plot

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R Chart Control Limits
From Table (function of sample size)
Sample Range in sample i
Samples
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R Chart Example

• Youre manager of a 500-room hotel. You want to
  analyze the time it takes to deliver luggage to
  the room. For 7 days, you collect data on 5
  deliveries per day. Is the process in control?

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R Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32

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R Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

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R Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.7
  0 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20
  5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70
  2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10
  .10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.
  90 6.90 9.30 6.79 4.22

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R Chart Control Limits Solution
From Table (n 5)
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R Chart Control Chart Solution
UCL
R-bar
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?X Chart

• Type of variables control chart
• Interval or ratio scaled numerical data
• Shows sample means over time
• Monitors process average
• Example Weigh samples of coffee compute means
  of samples Plot

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?X Chart Control Limits
From Table
Mean of sample i
Range of sample i
Samples
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?X Chart Example

• Youre manager of a 500-room hotel. You want to
  analyze the time it takes to deliver luggage to
  the room. For 7 days, you collect data on 5
  deliveries per day. Is the process in control?

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X Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.7
  0 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20
  5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70
  2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10
  .10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.
  90 6.90 9.30 6.79 4.22

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?X Chart Control Limits Solution
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?X ChartControl Chart Solution
UCL
X-bar
LCL
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X AND R CHART EXAMPLEIN-CLASS EXERCISE

• The following collection of data represents
  samples of the amount of force applied in a
  gluing process
• Determine if the process is in control
• by calculating the appropriate upper and lower
• control limits of the X-bar and R charts.

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X AND R CHART EXAMPLEIN-CLASS EXERCISE
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Example of x-bar and R charts Step 1. Calculate
sample means, sample ranges, mean of means, and
mean of ranges.
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Example of x-bar and R charts Step 2. Determine
Control Limit Formulas and Necessary Tabled Values
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Example of x-bar and R charts Steps 34.
Calculate x-bar Chart and Plot Values
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Example of x-bar and R charts Steps 56
Calculate R-chart and Plot Values
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SOLUTIONExample of x-bar and R charts

• 1. Is the process in Control?
• 2. If not, what could be the cause for the
  process being out of control?

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Process Capability

• Process limits -
• Tolerance limits -

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Process Capability

• How do the limits relate to one another?

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Process Capability Measurement
Cp index Tolerance range / Process range What
value(s) would you like for Cp?
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UTL
LTL
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• While the Cp index provides useful information on
  process variability, it does not give information
  on the process average relative to the tolerance
  limits. Note

UTL
LTL
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Cpk Index
s
Together, these process capability Indices show
how well parts being produced conform to design
specifications.
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Since Cp and Cpk are different we can conclude
that the process is not centered, however the Cp
index tells us that the process variability is
very low
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An example of the use of process capability
indices
The design specifications for a machined slot is
0.5 .003 inches. Samples have been taken and
the process mean is estimated to be .501. The
process standard deviation is estimated to be
.001. What can you say about the capability of
this process to produce this dimension?
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Process capability
Machined slot (inches)
0.497 inches LTL
0.503 inches UTL
? 0.001 inches
Process mean 0.501 inches
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Sampling Distributions(The Central Limit Theorem)

• Regardless of the underlying distribution, if the
  sample is large enough (gt30), the distribution of
  sample means will be normally distributed around
  the population mean with a standard deviation of
  

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Computing Process Capability Indexes Using
Control Chart Data

• Recall the following info from our in class
  exercise
• Since A2 is calculated on the assumption of three
  sigma limits

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• From the Central Limit Theorem
• So,
• Therefore,

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• Suppose the Design Specs for the Gluing Process
  were 10.7 ? .2, Calculate the Cp and Cpk
  Indexes
• Answer

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Note, multiplying each component of the Cpk
calculation by 3 yields a Z value. You can use
this to predict the of items outside the
tolerance limits From Appendix E we would
expect non-conforming product from this process
.002 or .2 of the curve
.02 or 2 of the curve
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Capability Index In Class Exercise

• You are a manufacturer of equipment. A drive
  shaft is purchased from a supplier close by. The
  blueprint for the shaft specs indicate a
  tolerance of 5.5 inches .003 inches. Your
  supplier is reporting a mean of 5.501 inches.
  And a standard deviation of .0015 inches.
• What is the Cpk index for the suppliers process?

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• Your engineering department is sent to the
  suppliers site to help improve the capability on
  the shaft machining process. The result is that
  the process is now centered and the CP index is
  now .75. On a percentage basis, what is the
  improvement on the percentage of shafts which
  will be unusable (outside the tolerance limits)?

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To answer this question we must determine the
percentage of defective shafts before and after
the intervention from our engineering department
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Before
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After
Since the process is centered then Cpk Cp Cp
UTL-LTL / 6s, so the tolerance limits are .75
x 6s 4.5s apart each 2.25s from the mean
-4
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So the percentage decrease in defective parts is
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Basic Forms of Statistical Sampling for Quality
Control

• Sampling to accept or reject the immediate lot of
  product at hand
• Sampling to determine if the process is within
  acceptable limits

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Acceptance Sampling

• Purposes
• Advantages

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Acceptance Sampling

• Disadvantages

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Risk

• Acceptable Quality Level (AQL)
• a (Producers risk)
• Lot Tolerance Percent Defective (LTPD)
• ? (Consumers risk)