Transcript MER035 Lecture 1

MER301: Engineering
Reliability
LECTURE 16:
Measurement System Analysis and
Uncertainty Analysis-Part 1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
1
Measurement as a Process
We must submit the output from our design process
to a second (measurement) process
Parts
(Example)
Inputs
Process
Outputs
Inputs
Measurement
Process
L Berkley Davis
Copyright 2009
Outputs
MER301: Engineering Reliability
Lecture 16
• Measurements
2
Measurement System Concerns..






How big is the measurement error?
What are the sources of measurement error?
Are the measurements being made with units which
are small enough to properly reflect the variation
present?
Is the measurement system stable over time?
How much uncertainty should be attached to a
measurement system when interpreting data from
it?
How do we improve the measurement system?
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
3
Measurement System Analysis


Total Error in a measurement is defined as the difference
between the Actual Value and Observed Value of Y
Two general categories of error – Accuracy or Bias and
Precision
Accuracy or Bias of Measurement System is defined as the difference
between a Standard Reference and the Average Observed
Measurement
Precision of a Measurement System is defined as the standard
deviation of Observed Measurements of a Standard Reference
Total Error = Bias Error + Precision Error for independent random
variables




Measurement System Error is described by Average Bias
Error (Mean Shift)and a statistical estimate of the
Precision Error (Variance)
Measurement System Analysis is a Fundamental
Part of Every Experiment
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
4
Precision and Accuracy
Not Accurate, Not Precise
Not Accurate, Precise
L Berkley Davis
Copyright 2009
Accurate, Not Precise
Accurate, Precise
MER301: Engineering Reliability
Lecture 16
5
Measurement System Analysis
Yobserved  Yactual   bias  Ymeasuremen t
 observed   actual   bias  0
2
2
2
 observed
  actual
 0   measuremen
t


Bias or Accuracy error is a constant value and is dealt
with by calibrating the measurement system
Variation or Precision error is a random variable which
depends on the measurement equipment(the instruments
used) and on the measurement system repeatability and
reproducibility. Instrument Capability Analysis, Test/retest
(repeatability)and Gage R&R studies are used to quantify
the size of these errors.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
6
Impact of Measurement System Variation on
Variation in Experimental Data
 actual = Product Std. Dev.
 actual = Product Mean
Actual Defects
LSL
 actual
USL
2
 obs   actual
  m2
Product
variance
LSL
L Berkley Davis
Copyright 2009
 obs
act  actual
Measurement system
variance
Observed Defects
m  m easurem en
t
obs  observed
USL
MER301: Engineering Reliability
Lecture 16
7
Example 16.1-Effect of Measurement System
Variation
 Calculate the effect of measurement system
variation on the acceptance rate for a part
with USL and LSL at Z= +/-1.96 respectively.
2
2

If
m1   act / 2 then what is the percentage of
acceptable parts that will be rejected? If on
2 / 20 what is the
the other hand  m2 2   act
percentage of acceptable parts that will be
rejected?
LSL
USL
Process
T
Failed Good
units
Gauge
Passed Bad
Units
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
Gauge
8
Impact of Measurement System Variation on
Variation in Experimental Data…
LSL
USL
Process
T
Failed Good
units
Gauge
Passed Bad
Units
L Berkley Davis
Copyright 2009
Gauge
MER301: Engineering Reliability
Lecture 16
9
2
2 /2

Example 16.1(con’t: m   act
)
1
 For the product, the spec limits of +/-1.96 mean that
the 2.5% of parts in each tail are out of spec. Thus
Z usl  1.96 
 For
X usl   act
 act
X usl  1.96   act   act
2 / 2 the observed standard deviation is
 m2   act
1
2  2 
2  2 /2  
 obs   act
 act
3/ 2
act
act
m
1
and
Z usl ,Observed 
(1.96 act   act )   act
 act 3 / 2

1.96
3/ 2
 1.60
Then the acceptable parts now rejected are
21.96  1.60  20.975 0.945  0.060
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
10
2 / 20
Example 16.1(con’t:  m2   act
)
2
 For the product, the spec limits of +/-1.96 mean that
the 2.5% of parts in each tail are out of spec. Thus
Z usl  1.96 
2
 For m
2
X usl   act
 act
X usl  1.96   act   act
2 / 20the observed standard deviation is
  act
2   2 / 20  
 obs   act
act
act 1  1 / 20  1.0247  act
and
Z usl ,Observed
(1.96 act   act )   act
1.96


 1.913
1.0247  act
1.0247
Then the acceptable parts now rejected are
21.96  1.913  20.975 0.972  0.006
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
11
Example 16.1(con’t)
 Observed
 Measure  2
Process
2
2
 Measure
Unacceptable
 Process
 Measure


 Observed
2
Process
2
Measure
 20
Acceptable
 Process
Set Measurement System Requirements Based on
the Process Variation
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
12
Gage Repeatability & Reproducibility
---GRR or GR&R---
%GRR 
5.15 m ea su rm en t
Tolerance
100%
 Gage Repeatability & Reproducibility compares
measurement system variation and product variation
 The 5.15   measuremen t term is the size of an interval
containing 99% of the measured values made on a
specific item

 The Tolerance- often equal to 6  actual - is the size of
the interval where a product has acceptable
dimensions, performance, or other characteristics
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
13
Gage Performance relative to
required Tolerance Band
%GRR 
5.15 m ea su rm en t
Tolerance
100%
 R&R less than 10% - Measurement system is
acceptable.
 R&R 10% to 30% - Maybe acceptable - make
decision based on classification of
characteristic, hardware application, customer
input, etc.
 R&R over 30% - Not typically acceptable. Find
the problem using root cause analysis(fishbone),
remove root causes
GRR is a measure of “noise” in the data
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
14
Effect of Gage R&R on Variation
5.15   measuremen t
%GRR 
 100%
6   actual
2
2
2
 observed
  actual
  measuremen
t
 observed /  actual  1  (6  GRR / 5.15) 2
 GRR <10% means < 0.7% of the variation in the
experimental data is from the measurement
system
 GRR> 30% means that > 5.9% of the variation in
the experimental data is from the measurement
system
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
15
GRR Example 16.2

The GRR values for the previous Example 16.1 are
1
2
 m2   act
2

5.15   m1
5.15   act / 2 5.15 / 2

 0.607
6   act
6   act
6
5.15   m2 5.15   act / 20 5.15 / 20
/ 20 GRR2 


 0.192
6   act
6   act
6
2 /2
 m2   act
GRR1 

The capabilities of two (or more) measurement systems
can be compared by comparing the GRR’s for each. Since
GRR2<GRR1 , the second measurement system is more
capable than the first. The observed standard deviations
quantify how much better….
 obs /  act  1.225
 obs /  obs  0.8365
 obs /  act  1.0247
 obs /  obs  1.195
1
2
L Berkley Davis
Copyright 2009
2
MER301: Engineering Reliability
Lecture 16
1
1
2
16
GRR Example 16.2


2
m1
The GRR values for the previous Example 16.1 are at best
marginally acceptable(GRR2 ) or not acceptable(GRR1 )

2
act
/2
2 / 20
 m2   act
2

GRR1 
GRR2 
5.15   m1
6   act
5.15   m2
6   act
5.15   act / 2 5.15 / 2


 0.607
6   act
6

5.15   act / 20 5.15 / 20

 0.192
6   act
6
For a GRR value equal to 10% (0.10) there results
5.15   m
2  (0.10  6 / 5.15) 2  1 / 73.67  1 / 74
GRR  0.10 
  m2 /  act
6   act
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
17
2 / 74)
Example 16.2 ( GRR  0.10 so  m2   act
 For the product, the spec limits of +/-1.96 mean that
the 2.5% of parts in each tail are out of spec. Thus
Z usl  1.96 
2
 For  m
X usl   act
 act
X usl  1.96   act   act
2 / 74 the observed standard deviation is
  act
2   2 / 74  
 obs   act
act
act 1  1 / 74  1.0067  act
and
Z usl ,Observed
(1.96 act   act )   act
1.96


 1.947
1.0067  act
1.0067
Then the acceptable parts now rejected are
21.960  1.947  20.9750 0.9742  0.0016
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
18
Summarizing how it all fits together…..

When a set of measurements are made, the results are always
observed values, Yobs  Yact   bias  Ym
2
2  0  2
 obs
  act
m
 obs   act   bias  0

 m ea su rm en t
%GRR  5.15Tolerance
100%
If the actual mean and standard deviation are known then the
measurement system bias and variance can be calculated
 bias   obs   act


2
2
 m2   obs
  act
If the item being measured is a standard reference
2
 m2   obs
0
If the measurement system bias and variance are known then the
actual mean and actual variance can be calculated
 act   obs   bias
L Berkley Davis
Copyright 2009
2   2  2
 act
obs
m
MER301: Engineering Reliability
Lecture 16
19
Sources of Measurement
System Error
Inputs
Process
Inputs
Outputs
Measurement
Process
Outputs
• Observations
• Measurements
Observed Process Variation
Measurement Variation
Actual Process Variation
Long-term
Process
Variation
Short-term
Process
Variation
within sample
variation
Variation due
to operator
Variation due
to gauge
Measurement
System
Repeatability
Reproducibility
Resolution
Accuracy
(Bias)
Precision (Pure
Error)
Stability (time
dependent)
Linearity (value
dependent)
Repeatability
Total Variation made up of Actual Process Variation and Measurement
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
System Variation
20
Measurement System Errors
True
Accuracy
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
True
Average
Operator A
Observed
Average
(Low End)
Accuracy
(Low End)
Stability
True
Average
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
L Berkley Davis
Copyright 2009
Linearity
Engineering Reliability
Lecture 16
21
Elements that contribute to Accuracy
and Precision Errors
 Instrument Capability



Resolution
Gage Repeatability
Linearity
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
Stability
Observed
Average
True (Low End)
Average
True
Average
Accuracy
(Low End)
Operator A
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Union College
Mechanical Engineering
Linearity
Engineering Reliability
Lecture 16
21
 Measurement System - Short Term (ST)
 Instrument Capability
 Equipment Calibration(Bias)
 Test/Re-Test Study(Repeatability)
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
Stability
Observed
Average
True (Low End)
Average
True
Average
Accuracy
(Low End)
Operator A
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Linearity
 Measurement System - Long Term (LT) Use
Union College
Mechanical Engineering
 Measurement System - Short Term Use
 Reproducibility
 Stability
L Berkley Davis
Copyright 2009
Engineering Reliability
Lecture 16
21
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
Observed
Average
True (Low End)
Average
Operator A
Accuracy
(Low End)
Stability
True
Average
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Union College
Mechanical Engineering
Engineering Reliability
First Two are MER301:
Entitlement….Third
is Reality
Lecture 16
Linearity
Engineering Reliability
Lecture 16
21
22
Elements that contribute to Precision or
Variation Errors
Gage Performance
Characteristics
Accuracy
 Instrument Capability
True
(Bias)
Resolution
2
Gage Repeatability
instrument
Linearity



Time 2
Time 1
Repeatability
(precision)
Observed Average

True
Average
Observed
Average
(Low End)
Accuracy
(Low End)
Operator A
Stability
True
Average
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Linearity
 Measurement System- Short Term (ST) Use
Union College
Mechanical Engineering
Engineering Reliability
Lecture 16
21
 Instrument Capability
2
2
2
 Equipment
Calibration(Bias)


measuremen t , ST
instrument
repeatibility
 Test/Re-Test Study(Repeatability)



 Measurement System - Long Term (LT) Use
 Measurement System - Short Term Use (ST)
2
2
2
2 
2



 Reproducibility(Gage
R&R)
m
LT
instrument
repeatability
reproducibility
 Stability(Gage R&R)

L Berkley Davis
Copyright 2009




First Two are Entitlement….Third
is Reality
MER301: Engineering Reliability
Lecture 16
23
Measurement System Analysis
Yobserved  Yactual  bias  Ymeasuremen t
observed   actual  bias  0
2
2
2
 observed
  actual
 0   measuremen
t
Ymeasuremen t  Yinstrument  Yrepeatability  Yreproducibility
Y
measurement
0
From pages
119-120…
2
2
2
2
 measuremen






t
instrument
repeatability
reproducib ility
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
24
Updating how variances all fit together

When a set of measurements are made, the results are always
observed values, Yobs  Yact   bias  Ym
2
2
2
2  0  2   2  2
 obs
  act




m
act
instrument
repeatability
reproducibility

If the actual mean and standard deviation are known then the
measurement system bias and variance can be calculated

observed  actual  bias
2
2
2
2
2
 m2   obs
  act
  instrument
  repeatabil


ity
reproducibility
If the item being measured is a standard reference
2
2
2
2
 m2   obs
 0   instrument
  repeatabil


ity
reproducibility

If the measurement system bias and variance are known then the
actual mean and actual variance can be calculated
2   2  2
 act
act  obs  bias
obs
m
2
2
2   2  ( 2
 act
obs
instrument   repeatability   reproducibility )
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
25
Emissions Sampling
Heated Sampling Line
Cal/Zero
Gases
Calibration Gas
NOx
Instrument
Sample
Conditioning
Yactual-
NOx from
Gas turbine
 obs   act   bias
2
2  2
 obs
  act
measuremen t
Yobs- NOx Reading
Yobserved  Yactual   bias  Ymeasuremen t
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
26
Elements that contribute to Accuracy
and Precision Errors
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
True
Average
Operator A
Observed
Average
(Low End)
Stability
True
Average
Accuracy
(Low End)
Observed
Average
(High End)
Accuracy
(High End)
Operator B
 Instrument Capability
Resolution
Gage Repeatability
Linearity
Reproducibility
Union College
Mechanical Engineering
Linearity
Engineering Reliability
Lecture 16
21
Emissions Sampling
Measurement SystemShort Term(ST) Use
Instrument Capability
Equipment Calibration
Test/Re-Test Study
Heated Sampling Line
Cal/Zero
Gases
Calibration Gas
NOx
Instrument
Sample
Conditioning
Yactual-
NOx from
Gas turbine
Measurement SystemLong term (LT) Use
Measurement System -Short
Term(ST) Use
Reproducibility
Stability
 obs   act   bias
2
2  2
 obs
  act
measuremen t
Yobs- NOx Reading
Yobserved  Yactual   bias  Ymeasuremen t
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
29
2
2
2
2
 measuremen






t
instrument
repeatability
reproducibility
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
27
How Can we Address Accuracy
and Precision Errors?
 Establish magnitude and sources of
measurement system error due to bias
and precision errors
 Tools
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
True
Average
Observed
Average
(Low End)
Stability
True
Average
Observed
Average
(High End)
Instrument Capability Analysis
Test/Re-test – system precision/repeatability
Calibration - bias
“Continuous Variable” Gage R&R (Gage
Reproducibility and Repeatability)
 Attribute Variable Gage R&R
 Destructive Gage R&R




L Berkley Davis
Copyright 2009
Operator A
Accuracy
(Low End)
Accuracy
(High End)
Operator B
Reproducibility
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
Linearity
Engineering Reliability
Lecture 16
21
28
Measurement System Analysis
 Instrument Capability Analysis…..



Resolution-smallest increment that the gage can resolve in the
measurement process. Gage should be able to resolve tolerance
band into ten or more parts. Resolution Uncertainty = u   4  
0
0
Instrument Accuracy- measure of instrument repeatability or
instrument “noise”.. Found by repeated measurements of the
same test item. Uncertainty = u   4  
r
r
Linearity- consistency of the measurement system across the
Gage Performance
entire range of the measurement system.
Characteristics
Linearity Uncertainty = ul  4   l
The variations are combined as follows
True

Repeatability
(precision)
Observed Average
u
2
instrument

L Berkley Davis
Copyright 2009
Accuracy
(Bias)
2
instrument
 u o  u r  ul
2
2
   
2
o
MER301: Engineering Reliability
Lecture 16
2
r
True
Average
2
2
l
Operator A
Observed
Average
(Low End)
Accuracy
(Low End)
True
Average
Time 2
Time 1
Stability
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Union College
Mechanical Engineering
Linearity
Engineering Reliability
Lecture 16
21
29
Instrument Capability Analysis…..

Variation for any one instrument equals the sum of the
resolution, repeatability and linearity terms
Yinstrument  Yresolution  Yrepeatability  Ylinearity
2
2
2
2
 instrument
  resolution
  repeatabil


ity
linearity

The Variation for “n” instruments equals the sum of the
variations for each individual instrument
Yinstrument  Yinstrument  Yinstrument      Yinstrument
1
2
2
2
2
2
 instrument
  instrument
  instrument
      instrument
1

L Berkley Davis
Copyright 2009
2
n
n
Each of the “n” instruments has resolution, repeatability, and
linearity terms that must be taken into account
MER301: Engineering Reliability
Lecture 16
30
Instrument Capability Analysis
- Resolution of Instruments/Sensors


The measurement uncertainty u 0 due to resolution is generally
taken as a specified fraction of the smallest increment an
instrument can resolve, ie as a fraction of the smallest scale
division
General Rule: assign a numerical value for the mean value of
u0 equal to one half of the instrument resolution. This
means that half of the smallest scale division is assumed to
equal a 95% Confidence Interval ( a 4 o
wide band) for
variation due to resolution
uo  4   0  12 resolution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
31
Instrument Capability Analysis
Repeatability and Linearity
 The manufacturer of an instrument will provide
information on the capability of the instrument in the
specification sheets provided with the instrument
 The numerical values given for Instrument or Sensor
Accuracy and Linearity are almost always uncertainties
 Let
= uncertainty due to the equipment
r
accuracy/repeatability error where u r  4   r
 Let ul = uncertainty due to linearity error
where ul  4   l
 The inherent capability/uncertainty of the
instrument/sensor is then estimated as:
u

uinstrument
 uo 2  u r 2  ul 2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
32
Example 16.3-Instrument Capability
 The Capability of a force measuring instrument
is described by catalogue data. Calculate an
estimate of the variation attributable to this
instrument. Express the result both in
dimensional terms (N) and in dimensionless
terms for a reading R=50N
Resolution
Range
Linearity
Repeatability
L Berkley Davis
Copyright 2009
0.25N
0 to 100N
within 0.20N over range
within 0.30N over range
MER301: Engineering Reliability
Lecture 16
33
Example 16.3(con’t)
 An estimate of the instrument uncertainty
depends on the combined uncertainties due
to resolution, repeatability and linearity
uo  4   o  0.25/ 2  0.125 N
u r  4   r  0.30N
ul  4   l  0.20N
The instrument uncertainty is then

u instrument
 (0.2) 2  (0.3) 2  (0.125) 2  0.38N
u instrument
L Berkley Davis
Copyright 2009
u d 0.38N


 0.0076
R
50N
MER301: Engineering Reliability
Lecture 16
34
Measurement System Analysis
 Instrument Capability Analysis Summary…..




Resolution-smallest increment that the gage can resolve in the
measurement process. Gage should be able to resolve tolerance
band into ten or more parts. Resolution Uncertainty = u   4  
0
0
Instrument Accuracy- measure of gage repeatability or gage
“noise”.. Found by repeated measurements of the same test item.
Uncertainty = u   4  
r
r
Linearity- consistency of the measurement system across the
entire range of the measurement system.
Linearity Uncertainty = ul  4   l
The variations are combined as follows
2
 instrument
  o2   r2   l2
2
uinstrument
 u o 2  u r 2  ul 2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
35
Measurement System Analysis
 Measurement System Short Term Use



Includes Instrument Capability
Repeatability - variation when one operator repeatedly
makes the same measurement with the same
measuring equipment Test/Re-test Study
Calibration/Bias
 Measurement System-Long Term Use



L Berkley Davis
Copyright 2009
Includes Measurement System –Short Term Use
Reproducibility- variation when two or more operators
make same measurement with the same measuring
equipment
Stability-variation when the same operator makes the
same measurement with the same equipment over an
extended period of time
MER301: Engineering Reliability
Lecture 16
36
Test/Retest Example 16.4
 Test/Retest (Repeatability) Study on a
Measurement System. Thirty repeat
measurements were taken on a Standard
Reference Item with a thickness of 50mils
The tolerance band for the application is
20mils(+/-10).
 Data, in mils
 53,45,52,47,54,52,52,55,52,48,48,53,55,51,47,52,47,
35,45,54,48,51,53,44,52,52,55,59,53,53
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
37
Example 16.4(con’t)
 Objective is to establish the precision and
accuracy of the measurement system
 Precision-Repeatability
In a good Measurement System, 99% of the
measurements of a given item should fall
within a band less than 1/10 of tolerance band
5.15   measuremen t
GRR 
 1 / 10
tolerance
 Accuracy/Bias
LSL
USL
Process
T
Failed Good
units
Gauge
Passed Bad
Units
Gauge
Bias = sample mean- true value
 bias  Yobserved  Yactual
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
38
Example 16.4 Run Chart and Histogram
Run Chart for C1
60
50
Frequency
C1
10
40
10
20
5
30
Observation
Number of runs about median:
Expected number of runs:
Longest run about median:
Approx P-Value for Clustering:
Approx P-Value for Mixtures:
13.0000
14.9333
6.0000
0.2190
0.7810
Number of runs up or dow n:
Expected number of runs:
Longest run up or dow n:
Approx P-Value for Trends:
Approx P-Value for Oscillation:
19.0000
19.6667
3.0000
0.3829
0.6171
0
35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0
C1
These results look bad to the eye…
there are outliers and mean is high
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
39
Test/Retest Study Example 16.4
Summary

Descriptive Statistics
Variable
N
C1
30
Variable
Minimum
C1
35.000
Mean
50.567
Maximum
59.000
 Conclusions
Median
52.000
Q1
47.750
StDev
4.561
Q3
53.000
 bias 
SE Mean
0.833
Example 16.3- Precision
Not Accurate, Not Precise
Ac
 Given the tolerance band of 20 mils,there is
an unacceptable level of device precision
Not Accurate, Precise
GRR  5.15  measuremen t / 20  23.48/ 20  1.174  (1/ 10)
Union College
Mechanical Engineering
 Given the Reference Test item had a known
thickness of 50mils, the bias(inaccuracy) is:
 bias  bias = 50.57 – 50.0 = 0.57mils
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
40
A
MER301: Engineering R
Lecture 16
Measurement System Analysis
 Measurement System-Short Term Use
 Repeatability-variation when one operator repeatedly
makes the same measurement with the same
measuring equipment Test/Re-test Study
 Measurement System - Long Term Use
 Reproducibility- variation when two or more operators
make same measurement with the same measuring
equipment
 Stability-variation when the same operator makes the
same measurement with the same equipment over an
extended period of time
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
41
Elements that contribute to Accuracy
and Precision Errors
 Instrument Capability



Resolution
Gage Repeatability
Linearity
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
Stability
Observed
Average
True (Low End)
Average
True
Average
Accuracy
(Low End)
Operator A
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Union College
Mechanical Engineering
Linearity
Engineering Reliability
Lecture 16
21
 Measurement System - Short Term (ST)
 Instrument Capability
 Equipment Calibration(Bias)
 Test/Re-Test Study(Repeatability)
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
Stability
Observed
Average
True (Low End)
Average
True
Average
Accuracy
(Low End)
Operator A
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Linearity
 Measurement System - Long Term (LT) Use
Union College
Mechanical Engineering
 Measurement System - Short Term Use
 Reproducibility
 Stability
L Berkley Davis
Copyright 2009
Engineering Reliability
Lecture 16
21
Gage Performance
Characteristics
Accuracy
True
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
Observed
Average
True (Low End)
Average
Operator A
Accuracy
(Low End)
Stability
True
Average
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
Union College
Mechanical Engineering
Engineering Reliability
First Two are MER301:
Entitlement….Third
is Reality
Lecture 16
Linearity
Engineering Reliability
Lecture 16
21
42
Measurement System Analysis:
A Summary of the Basic Equations
Yobserved  Yactual   bias  Ymeasuremen t
 observed   actual   bias  0
2
2
2
 0   measuremen
  actual
 observed
t
2
2
2
2






 measuremen
reproducib ility
repeatability
instrument
t
2
2
2
  repeatabil
  instrument
 ST
ity
2
2
2
2




  measuremen
 LT
reproducib ility
ST
t
2
2
2
2
2


  repeatabil
  instrument
  actual
 observed
reproducib ility
ity
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
43