# Introduction to Statistical Quality Control_ 4th Edition_16_

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```					        Chapter 7
Process and Measurement System
Capability Analysis

Introduction to Statistical Quality Control,
4th Edition
7-1. Introduction
• Process capability refers to the uniformity of the process.
• Variability in the process is a measure of the uniformity of
output.
• Two types of variability:
– Natural or inherent variability (instantaneous)
– Variability over time
• Assume that a process involves a quality characteristic that
follows a normal distribution with mean , and standard
deviation, . The upper and lower natural tolerance limits
of the process are
UNTL =  + 3
LNTL =  - 3
Introduction to Statistical Quality Control,
4th Edition
7-1. Introduction

• Process capability analysis is an
engineering study to estimate process
capability.
• In a product characterization study, the
distribution of the quality characteristic is
estimated.

Introduction to Statistical Quality Control,
4th Edition
7-1. Introduction
Major uses of data from a process capability analysis

1.    Predicting how well the process will hold the tolerances.
2.    Assisting product developers/designers in selecting or
modifying a process.
3.    Assisting in establishing an interval between sampling
for process monitoring.
4.    Specifying performance requirements for new
equipment.
5.    Selecting between competing vendors.
6.    Planning the sequence of production processes when
there is an interactive effect of processes on tolerances
7.    Reducing the variability in a manufacturing process.
Introduction to Statistical Quality Control,
4th Edition
7-1. Introduction

Techniques used in process capability analysis

1. Histograms or probability plots
2. Control Charts
3. Designed Experiments

Introduction to Statistical Quality Control,
4th Edition
7-2. Process Capability Analysis
Using a Histogram or a
Probability Plot
7-2.1 Using a Histogram
• The histogram along with the sample mean and
sample standard deviation provides information
–   The process capability can be estimated as x  3s
–   The shape of the histogram can be determined
–   Histograms provide immediate, visual impression of
process performance

Introduction to Statistical Quality Control,
4th Edition
Example 7-1
• Pgs. 353-354
• This procedure works if data are distributed
normally

Introduction to Statistical Quality Control,
4th Edition
Reasons for poor process capability

• See Fig. 7-3
– Poor process centering
• Assume that this can be corrected
– Excess process variability
• Harder to correct

Introduction to Statistical Quality Control,
4th Edition
7-2.2 Probability Plotting

•   Probability plotting is useful for
–   Determining the shape of the distribution
–   Determining the center of the distribution
–   Determining the spread of the distribution.
•   Recall normal probability plots (Chapter 2)
–   The mean of the distribution is given by the 50th
percentile
–   The standard deviation is estimated by
  84th percentile – 50th percentile
ˆ

Introduction to Statistical Quality Control,
4th Edition
7-2.2 Probability Plotting

Cautions in the use of normal probability plots
• If the data do not come from the assumed
drawn from the plot may be in error.
• Probability plotting is not an objective procedure
(two analysts may arrive at different
conclusions).

Introduction to Statistical Quality Control,
4th Edition
Example
•   See Fig. 7-4
•   First, est = 260 psi
•   Then, est = 298 – 260 = 38 psi
•   Can also use normal probability plot to
estimate fallout
– If LSL = 200 psi
• Then, from Fig 7-4, about 5% will be below that
value

Introduction to Statistical Quality Control,
4th Edition
7-3. Process Capability Ratios

7-3.1 Use and Interpretation of C p
• Recall
USL  LSL
Cp 
6
where LSL and USL are the lower and upper
specification limits, respectively.

Introduction to Statistical Quality Control,
4th Edition
7-3.1 Use and Interpretation of Cp

The estimate of Cp is given by

ˆ  USL  LSL
Cp
6ˆ
Where the estimate  can be calculated using the sample
ˆ
standard deviation, S, or R / d 2

Introduction to Statistical Quality Control,
4th Edition
7-3.1 Use and Interpretation of Cp

Piston ring diameter in Example 5-1
• The estimate of Cp is

ˆ  74.05  73.95
Cp
6(0.0099)
 1.68

Introduction to Statistical Quality Control,
4th Edition
7-3.1 Use and Interpretation of Cp

One-Sided Specifications

USL  
C pu 
3
  LSL
C pl 
3
These indices are used for upper specification and
lower specification limits, respectively

Introduction to Statistical Quality Control,
4th Edition
Example 7-2
• Pg. 359

Introduction to Statistical Quality Control,
4th Edition
Table 7-3
• Process fallout for one- and two-sided
specifications

Introduction to Statistical Quality Control,
4th Edition
7-3.1 Use and Interpretation of Cp

Assumptions
The quantities presented here (Cp, Cpu, Clu) have some very
critical assumptions:
1. The quality characteristic has a normal distribution.
2. The process is in statistical control
3. In the case of two-sided specifications, the process mean
is centered between the lower and upper specification
limits.
If any of these assumptions are violated, the resulting
quantities may be in error.

Introduction to Statistical Quality Control,
4th Edition
Table 7-4
• Recommended minimum values of the PCR
• For example, a new process with two-sided
specifications has a recommended C p of
1.50
– This implies that process fallout would be 7
ppm
• Six  would result in a Cp of 2.0

Introduction to Statistical Quality Control,
4th Edition
7-3.2 Process Capability Ratio on
Off-Center Process

•   Cp does not take into account where the
process mean is located relative to the
specifications.
•   A process capability ratio that does take
into account centering is Cpk defined as
Cpk = min(Cpu, Cpl)

Introduction to Statistical Quality Control,
4th Edition
Figure 7-7
• All of the panels in the figure have Cp = 2.0
• But, when the process mean shifts, the
capability of the process can change
– Note that  does not shift

Introduction to Statistical Quality Control,
4th Edition
Figure 7-7, cont.
• For panel b, N(53, 22)
– Cpk = min(Cpu, Cpl)
• Cpu = (62-53)/[3(2)] = 1.5
• Cpl = (53-38)/[3(2)] = 2.5
– Cpk = 1.5

Introduction to Statistical Quality Control,
4th Edition
7-3.3 Normality and the Process
Capability Ratio
•   The normal distribution of the process
output is an important assumption.
•   If the distribution is nonnormal, Luceno
(1996) introduced the index, Cpc, defined
as                 USL  LSL
C pc 

6   EXT
2

Introduction to Statistical Quality Control,
4th Edition
Example
• USL = 90, LSL = 80
– So, T = (90 + 80)/2 = 85
• (T = target value)
– Let X = 84
– Then, Cpc = 1.33
• (Be careful with this result…I don’t trust it!)

Introduction to Statistical Quality Control,
4th Edition
7-3.3 Normality and the Process
Capability Ratio
•   A capability ratio involving quartiles of
the process distribution is given by
USL  LSL
C p (q ) 
x 0.99865  x 0.00135

•   In the case of the normal distribution
Cp(q) reduces to Cp
Introduction to Statistical Quality Control,
4th Edition
Why does it reduce to Cp?
• In the case of the normal distribution
– x.00135 =  – 3
– x.99865 =  + 3

Introduction to Statistical Quality Control,
4th Edition
7-4. Process Capability Analysis
Using a Control Chart

•   If a process exhibits statistical control, then the
process capability analysis can be conducted.
•   A process can exhibit statistical control, but may
not be capable.
•   PCRs can be calculated using the process mean
and process standard deviation estimates.

Introduction to Statistical Quality Control,
4th Edition
Example
• Pgs. 373-375

Introduction to Statistical Quality Control,
4th Edition
7-5. Process Capability Analysis
Designed Experiments

•   Systematic approach to varying the
variables believed to be influential on the
process. (Factors that are necessary for
the development of a product).
•   Designed experiments can determine the
sources of variability in the process.

Introduction to Statistical Quality Control,
4th Edition
Example
• Machine that fills bottles with a soft-drink
beverage
– Each machine has many filling heads that are
– Quality characteristic measured is syrup content
in degrees brix
– Three possible causes of variabililty

Introduction to Statistical Quality Control,
4th Edition
Example, cont.
•   Variability is B2 = M2 + H2 + A2
•   Conduct an experiment
•   Say the result is as shown in Fig. 7-12
– Improve the process by reducing this variance

Introduction to Statistical Quality Control,
4th Edition
7-7: Setting spec limits on
discrete components
• Setting specifications to insure that the final
product meets specifications
• As discussed previously, if normally
distributed variables are linked, the result is
normally distributed with mean the sum of
the individual means, and variance the sum
of the individual variances

Introduction to Statistical Quality Control,
4th Edition
Example 7-9
• Pgs. 388-389

Introduction to Statistical Quality Control,
4th Edition
Example 7-10
• Pgs. 389-390

Introduction to Statistical Quality Control,
4th Edition
Example 7-11
• Pgs. 391-392

Introduction to Statistical Quality Control,
4th Edition
7-8: Estimating tolerance limits
• Confidence limits
– Provide an interval estimate of the parameters
of a distribution
• Tolerance limits
– Indicate the limits between which we can
expect to find a specified proportion of a
population

Introduction to Statistical Quality Control,
4th Edition
7-8.1: Tolerance limits based on
the normal distribution
• Suppose x~N(, 2), both unknown
• Take a sample of size n and compute xbar and S2
• Natural tolerance limits might be estimated using:
– Xbar + Za/2S
• Since xbar and S are only estimates, the interval
may or may not always contain 100(1-a)% of the
distribution

Introduction to Statistical Quality Control,
4th Edition
7-8.1: Tolerance limits based on
the normal distribution
• However, we may use a constant K such
that in a large number of samples a fraction
g of the intervals xbar + KS will include at
least 100(1-a)% of the distribution
• Values of K are tabulated in Appendix
Table VII
– 2<n<1000, g = .90, .95, .99, and a = .10, .05, .01

Introduction to Statistical Quality Control,
4th Edition
Example 7-13
• Pg. 396

Introduction to Statistical Quality Control,
4th Edition
Assignment
• Work odd-numbered exercises on the topics
covered in class

Introduction to Statistical Quality Control,
4th Edition
End

Introduction to Statistical Quality Control,
4th Edition

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