Chapter 10 - AS Nida

Chapter 10
Specification Limit
Process Capability, and Gage
Measurement

Specification limit
• Specification limit : conformance boundary
specified formal characteristic.
• Type of Specification
– Two-sided : upper and lower limits
– One-sided : either an upper or lower limit

Statistical tolerance limits
• Statistical tolerance limits: limits of the
interval for which it can be stated with a
given level of confidence that it contains at
least a specified proportion of the
population.

Setting Up Statistical Tolerance
Limit
• Use control chart to measure of process
capability to established requirement
• From observations of a given quality
characteristics from one or more samples.

Statistical tolerance limits
• When process is in control, data can be used to
determine statistical tolerance limit.
• Assume that a process involves a quality
characteristic that follows a normal distribution
with mean µ, and standard deviation, σ. The upper
and lower natural tolerance limits or statistical
tolerance limits of the process are
UNTL = µ + 3σ
LNTL = µ -3σ

Statistical Tolerance Limits
• Page 325
• Obtain a sample of n observation
• Check for normality assumption
• Calculate statistics parameters
• Select the desired confidence level and
population proportion to be cover by limits
• Determine k2 • Compute limits

Types of specification conflict
• Type I : distribution too wide for double
specification
• Type II : Distribution not centered correctly
for single specification limit
• Type III : Specification limits too wide for
acceptable product

Type I : distribution too wide for
• Possible action
double specification
– Changing the process
– Changing the specification
– Setting up an inspection/sorting operation to
find, remove, or repair the units of product
falling outside either specification limit
– Adjusting centering of distribution to strike a
balance among relative costs

Type II : Distribution not centered
correctly for single specification limit
• Possible action
– Attempting to center the process at a value far
enough from specification limit
– Changing the specification limit
– Setting up inspection/sorting operation to find
and remove or repair the units of product that
fall out side the specification limit

Type III : Specification limits too
wide for acceptable product
• Possible action
– Conducting ore formal program of process
experimentation to determine what range of
value cause nonconformities
– Tightening the specification limits
– Establishing shop or manufacturing limits on a
temporary basis until final values are
determined

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

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.

Techniques used in process
capability analysis
1. Histograms or probability plots
2. Control Charts
3. Designed Experiments

Process Capability Analysis Using a
Histogram or a Probability Plot
Using a Histogram
• The histogram along with the sample mean and
sample standard deviation provides information
about process capability.
– The process capability can be estimated as
x ±
– The shape of the histogram can be determined (such
as if it follows a normal distribution)
– Histograms provide immediate, visual impression of
process performance.
3 s

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.
σˆ
=

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

Process Capability Ratios
Use and Interpretation of Cp • Recall
USL − LSL
Cp =
6σ
where LSL and USL are the lower and upper
specification limits, respectively.

Use and Interpretation of C p
The estimate of C p is given by
Ĉ p
=
USL − LSL
6σˆ
Where the estimate σˆ can be calculated using the sample
standard deviation, S, or R
/ d
2

Use and Interpretation of C p
One-Sided Specifications
C
C
pu
pl
USL − µ
=
3σ
µ − LSL
=
3σ
These indices are used for upper specification and
lower specification limits, respectively

Use and Interpretation of C p
Assumptions
The quantities presented here (C p, C pu, C lu) 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.

Process Capability Ratio an
Off-Center Process
• Cpdoes 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)

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
C pc
=
6
USL
π
2
E
−
LSL
X
−
T

More About Process Centering
• Cpk should not be used alone as an
measure of process centering.
• Cpk depends inversely on σ and becomes
large as σ approaches zero. (That is, a large
value of Cpk does not necessarily reveal anything
about the location of the mean in the interval
(LSL, USL)

More About Process Centering
• An improved capability ratio to measure process
centering is Cpm .
USL − LSL
Cpm =
6τ
where τ is the squre root of expected squared
deviation from target: T =½(USL+LSL),
τ
2
=
[ ( ) ] 2 2
2
x − T = σ + ( µ T)
E −

More About Process Centering
• Cpm can be rewritten another way: (same as 10.8)
USL − LSL
Cpm
=
2
2
6 σ + ( µ − T)
where
=
C
p
1+
ξ
2
− µ
ξ =
σ
T

More About Process Centering
• A logical estimate of C pm is:
where
Ĉ
V
pm
=
=
Ĉ
1+
T − x
S
p
V
2

pk
Confidence Intervals and Tests
on Process Capability Ratios
C pk
• Ĉ pk is a point estimate for the true C pk , and
subject to variability. An approximate 100(1-α)
percent confidence interval on C pk is
⎡
1 1 ⎤
⎡
⎢1
− Zα
/ 2 + ⎥ ≤ C pk ≤ Ĉ pk ⎢1
+ Zα
⎢
2(
n −1)
⎣
9nĈ
pk ⎥⎦
⎢⎣
/ 2
1
9nĈ
pk
+
1
2(
n
−
1)

Confidence Intervals and Tests on
Process Capability Ratios
Example n = 20, Ĉ pk = 1.33. An approximate 95%
confidence interval on C pk is
⎡ 1 1 ⎤ ⎡ 1 1 ⎤
1.33 ⎢1− 1.96 + ⎥≤Cpk ≤ 1.33 ⎢1+ 1.96 + ⎥
⎣ 9(20)1.33 2(19) ⎦ ⎣ 9(20)1.33 2(19) ⎦
• The result is a very wide confidence interval ranging
from below unity (bad) up to 1.67 (good). Very little has
really been learned about actual process capability
(small sample, n = 20.)

Confidence Intervals and Tests
on Process Capability Ratios
Cpc • Ĉpc is a point estimate for the true Cpc , and
subject to variability. An approximate 100(1-α)
percent confidence interval on Cpc is
where
1+
t
Ĉ
α
, n −1
2
pc
⎡ s
⎢
⎣ c
c
c
n
=
⎤
⎥
⎦
1
n
≤
n
C
pc
≤
1−
t
∑ x − T
i=
1
i
Ĉ
α
, n −1
2
pc
⎡ s
⎢
⎣ c
c
n
⎤
⎥
⎦

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.

Gage and Measurement
System Capability Studies
Control Charts and Tabular Methods
• Two portions of total variability:
– product variability which is that variability
that is inherent to the product itself
– gage variability or measurement variability
which is the variability due to measurement
error
σ
= σ + σ
2
Total
2
product
2
gage

Control Charts and Tabular
Methods
X and R Charts
• The variability seen on the Xchart
can be
interpreted as that due to the ability of the gage
to distinguish between units of the product
• The variability seen on the R chart can be
interpreted as the variability due to operator.

Control Charts and Tabular
Methods
Precision to Tolerance (P/T) Ratio
• An estimate of the standard deviation for
measurement error is
σˆ
=
R
gage
d2
• The P/T ratio is
6σˆ
gage
P/
T =
USL−
LSL

Control Charts and Tabular
Methods
• Total variability can be estimated using
the sample variance. An estimate of
product variability can be found using
σ
S
2
Total
2
σˆ
=
= σˆ
2
product
σ
2
product
= S
2
product
2
+ σˆ
−σˆ
+ σ
2
gage
2
gage
2
gage

Control Charts and Tabular
Methods
Percentage of Product Characteristic Variability
• A statistic for process variability that does not
depend on the specifications limits is the
percentage of product characteristic variability:
σˆ
σˆ
gage ×
product
100

Control Charts and Tabular
Methods
Gage R&R Studies
• Gage repeatability and reproducibility (R&R)
studies involve breaking the total gage
variability into two portions:
– repeatability which is the basic inherent
precision of the gage
– reproducibility is the variability due to
different operators using the gage.

Control Charts and Tabular
Methods
Gage R&R Studies
• Gage variability can be broken down as
σ
2
measurement
error
=
σ
2
gage
2
reproducibility
2
repeatability
• More than one operator (or different conditions)
would be needed to conduct the gage R&R
study.
=
σ
+
σ

Control Charts and Tabular
Methods
Statistics for Gage R&R Studies (The Tabular
Method)
• Say there are p operators in the study
• The standard deviation due to repeatability can be found
as
R
σˆ
repeatabil ity =
d
where
R =
R
1
+ R
2
and d 2 is based on the # of observations per part per
operator.
p
2
+ +
R
p

Control Charts and Tabular
Methods
Statistics for Gage R&R Studies (the Tabular
Method)
• The standard deviation for reproducibility is given as
where
σˆ
=
R
x
x
reproducibility
x
max
min
=
x
max
− x
d 2 is based on the number of operators, p
1
R
d
min
1
x
2
= max( x , x
= min( x , x
2
2
, …x
, …x
p
p
)
)

Methods Based on Analysis
of Variance
• The analysis of variance can be extended to
analyze the data from an experiment and to
estimate the appropriate components of gage
variability.
• For illustration, assume there are a parts and b
operators, each operator measures every part n
times.

Methods Based on Analysis
of Variance
• The measurements, y ijk , could be represented by
the model
⎧ i = 1,
2,...
a
⎪
yijk = µ + τi
+ β j + ( τβ)
ij + εijk
⎨ j = 1,
2,...,
b
⎪
⎩k
= 1,
2,...,
n
where i = part, j = operator, k = measurement.

Methods Based on Analysis
of Variance
• The variance of any observation can be given by
2 2 2 2
στ , σβ,
στβ,
σ
2 2 2 2
ijk ) y ( V σ + σ + σ + σ = τ β τβ
are the variance components.

Methods Based on Analysis
of Variance
• Estimating the variance components can be
accomplished using the following formulas
σˆ
σˆ
σˆ
σˆ
2
2
τβ
2
β
2
τ
=
=
=
=
MS
MS
MS
MS
E
B
A
− MS
n
− MS
an
− MS
bn
AB
AB
AB
E