analyzing uncertainty and errors isat 253 spring 2005
TRANSCRIPT
Spring 2005 Dr. Ken Lewis 2
Objectives Understand the concept of uncertainty Define measurement uncertainties and errors For a set of measurements, learn to calculate
specific Sensitivities Accuracies Precisions
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Objectives Define measurement errors Differentiate between systematic and random
errors
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Key Concepts Precision Accuracy Precision error Bias error Sensitivity Calibration Calibration standards
Measurement standards
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Recall Resolution
The smallest increment of a unit of measure that an instrument can detect or measure.
Accuracy How close the measurement is to the “true value”
Precision The consistent repeatability of a measurement.
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Recall -- Types of Error Bias error ( average of the measurements – true)
Non random Systematic Destroys accuracy
Precision error (measurement readings – average) Random Hard to control without changing the measurement
system
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Error summary
TrueValue
AverageMeasure
Precision ErrorRandom Error
Bias ErrorSystematic Error
Accuracy
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Which type of error is it? A carpenter bought a piece of lumber at Lowe’s and
measured it in the store as 8’-2”. At home, when she measured it again to cut it, she measured it as 8’-1” using the same tape measure.
We measure the pH of a solution of 0.1 mol acetic acid and 0.2 mol ammonium acetate as 4.8 at 25°C, but standards show it should be 4.78.
A statistical process control (SPC) gauge is 6 microns high every time it is used.
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Quantifying uncertaintyVolts
5.98
6.05
6.10
6.06
5.99
5.96
6.02
6.09
6.03
5.99
Ten measurements were made on a battery
The true voltage is known to be 6.11 volts.
The average of the measurements is 6.03 volts
Find
The resolution error
The systematic error or accuracy
The precision
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Quantifying uncertaintyVolts
5.98
6.05
6.10
6.06
2.99
5.96
6.02
6.09
6.03
5.99
The resolution uncertainty or resolution error.
±0.01 V
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Quantifying uncertaintyVolts
5.98
6.05
6.10
6.06
2.99
5.96
6.02
6.09
6.03
5.99
The systematic error or accuracy
-0.08 V
Accuracy = average value - true value
Accuracy = 6.03 V - 6.11 V = -0.08 V
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Quantifying uncertaintyVolts
5.98
6.05
6.10
6.06
2.99
5.96
6.02
6.09
6.03
5.99
The precision
±0.07 V
Precision = Maximum deviation from the average
Precision = ±|5.96 V - 6.03 V| = ±0.07 V
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Quantifying uncertainty
Caution Be sure that the precision
statement isn’t based on one bad measurement
Volts
5.98
6.05
6.10
6.06
2.99
5.96
6.02
6.09
6.03
5.99
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Reporting Uncertainties Uncertainties can be reported as:
a number in the measurement units a percentage of the instrument’s full scale a percentage of the measurement itself
Resolution error can be reported as: ± 1 of the least significant digit ± ½ of the least significant digit
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Numerical Example
Consider our battery/voltmeter problem: Average measured value = 6.03 V Assume meter range = 0-10 V
Precision can be reported in three ways: Measurement units: 6.03 V ± 0.07 V Percent of full scale: 6.03 V ± 0.7% FS Percent of the measure: 6.03 V ± 1%
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Range A measuring system is designed to operate over a
finite range 0 – 500°C 20 – 200 psig 0 – 300 lbs
The range given describes the limits of proper response
What happens outside the range is no gauranteed.
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Span The span is the
difference between the high and low of the range
Range Span
0 – 500°C 500°C
20 – 200 psig 180 psig
0 – 300 lbs 300 lbs
±3 volts 6 volts
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Accuracy – example ±5% full scale
Input % of full scale
Ou
tpu
t % o
f fu
ll sc
ale
0 100
0
120
ideal instrument
Accuracy: ±5% full scale
Problem: below the full scale reading the error will be greater than ±5%.
0 – 200°C ± 10°C So at 30°C, the reading
will be somewhere between 20°C and 40°C
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Sensitivity The slope of the line relating input to output
Input range °C
ou
tpu
t ra
ng
e m
V
0 100
0
120
ideal instrument
Sensitivity
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Sensitivity – ThermocouplesOutput of Common Thermocouples
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200 1400 1600 1800
Temperature (°C)
T
E
J
K
R
S
Ou
tpu
t (m
V)
E
J
T
K
R
S
Sensitivity -- ThermocouplesType Material Range °C Sensitivity
mV/°C
T Copper/constantan -250 – 400 0.052
E Chromel/constantan -270 -- 1000 0.076
J Iron/constantan -210 – 760 0.050
K Chromel/alumel -270 – 1372 0.039
R Pt/Pt – 13% Rh -50 – 1768 0.011
S Pt/Pt – 10% Rh -50 to 1768 0.012
C W, 5% Re/W, 26% Re 0 -- 2320 0.020
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Sensitivity Calculation
( ) ( ) =
( ) ( )
d output outputsensitivity K
d input input
For example
(10 0)0.010
(100 0)
V VK
C C
For Ideal
devices
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Other Measurement Problems
Input Range °C
Out
put R
ange
mV
0 100
0
25
Ideal
Real
ZeroOffsetError
Non-linearity
Sensitivityerror
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Other Measurement Problems
Input Range °C
Ou
tpu
t R
an
ge
mV
0 100
0
25
DecreasingTemperature
IncreasingTemperature
Hysteresis
Hysteresis Friction Mechanical flexure of
internal parts Electrical capacitance
Will usually appear random
Example – A tachometer Tachometer measures
shaft rotation speeds in the range of 0 – 5000rpm Accuracy: ±5% FS Hysteresis: 30 rpm Zero offset: 200 rpm
What is the maximum error you expect in a shaft speed reading of 3500 rpm?
0.05 5000 250rpm rpm
Accuracy uncertainty
Hysteresis ±30 rpm
Zero offset 200 rpm
U = 250rpm + 30rpm + 200 rpm
U = 580rpm
580or 16.6%
3500
rpm
rpm
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Calibration A process wherein a set of measurements are
made of measurand values that can determined independently
Readings are compared to the known ‘true’ values and errors determined
Implies that the measurements are referenced against a measurement standard.
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What is a meter? Measurement Standards
1793 Govt. of France decrees the unit of length to be 10-7 of the earth’s quadrant passing through Paris and called the meter.
1889 Treaty of the Meter (Conférence Général des Poids et Mésures, CGPM) established a platinum-iridium bar.
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What is a meter? Measurement Standards
1960 definition based on the krypton86 radiation from an electrical discharge lamp.
1983: The meter is the SI unit of length and is defined as the length of the path traveled by light in a vacuum during the time interval of 1/299,792,458 of a second.
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What is a second?Measurement standards The second is the duration of 9,192,631,770
periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
Other base units – great site http://physics.nist.gov/cuu/Units/current.html
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What is the POINT?Measurement Standards There are many basic standards. Before the French Revolution every different
duchy had their own version of standards of weight, length, etc. Need to standardize to allow transfers of
knowledge Need to have standards to allow calibration of
instruments to make results reliable and interchangeable.
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What’s the Point? Here $200,000,000Mars Polar Lander
December 3, 1999 EDWARD EULER, Lockheed
Martin: The mistake was that we had to give the Jet Propulsion Lab some data that is used to compute very, very small little thrust pulses onboard the spacecraft. And we did give them the data in the wrong units... and in English units, and it should have been in metric. And they used the data as if it were metric, and underestimated the magnitude of these small, little pulses that come out of the jets of the Orbiter by about a factor of five. And that in turn made it very difficult to get the proper navigation, or determine the position and velocity of the spacecraft, which eventually led to the failure.
http://www.pbs.org/newshour/bb/science/july-dec99/mars_lander_12-2.html
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Measurement Standards Many standards;
1 foot is 12 inches 2.54 centimeters is exactly 1 inch There are 28 grams in 1 ounce A CD is 12 centimeters in diameter All video players (VHS) can interpret correctly
any VHS tape The electric voltage and current in California is
the same as it is in New Hampshire
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Seven SI Base Units
Quantity Dimension SI Unit Symbol
Time [t] second s
Length [L] meter m
Mass [m] kilogram kg
Current [i] ampere A
Temperature [T] Kelvin K
Luminosity --- candela cd
Amount --- mole mol
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Standards All primary standards except mass “can” be
reproduced in a good well equipped laboratory.
The standard for mass ‘International Prototype Kilogram’ is a Pt—Ir cylinder kept in Paris France
Standards for all other physical variables are; Derived from the base standards Physical laws
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Example -- Force Recall Newton’s second law
Force = mass X acceleration Acceleration = meter/second/second
=length/second2
2 Force =
ml
t
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Some SI Derived UnitsQuantity Dimension SI Unit Symbol
Area [L2] meter2 m2
Volume [L3] meter3 m3
Velocity [L/t] meter/second m/s
Acceleration [L/t2] meter/second2 m/s2
Force [mL/t2] Newton N or (kg-m/s2)
Energy [mL2/t2] joule J or (N-m)
Power [mL2/t3] watt W or (J/s)
Voltage [mL2/(t3i)] volt V or (W/A)
Pressure [m/(Lt2)] Pascal Pa or (N/m2)
Viscosity [m/(Lt)] Pascal-second Pa-s