Analyzing Uncertainty and Errors ISAT 253 Spring 2005.

Analyzing Uncertainty and Errors

ISAT 253

Spring 2005

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


Objectives Define measurement errors Differentiate between systematic and random

errors


Key Concepts Precision Accuracy Precision error Bias error Sensitivity Calibration Calibration standards

Measurement standards


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.


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


Error summary

TrueValue

AverageMeasure

Precision ErrorRandom Error

Bias ErrorSystematic Error

Accuracy


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.


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



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



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



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


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


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


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%


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.


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


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


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


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


Sensitivity Calculation

( ) ( ) =

( ) ( )

d output outputsensitivity K

d input input

For example

(10 0)0.010

(100 0)

V VK

C C

For Ideal

devices


Other Measurement Problems

Input Range °C

Out

put R

ange

mV

0 100

0

25

Ideal

Real

ZeroOffsetError

Non-linearity

Sensitivityerror


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


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.


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.


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.


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


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.


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


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


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


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


Example -- Force Recall Newton’s second law

Force = mass X acceleration Acceleration = meter/second/second

=length/second2

2 Force =

ml

t


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

Some Standard SI Prefixes

Multiple Prefix Symbol

10-12 pico p

10-9 nano n

10-6 micro

10-3 milli m

10-2 centi c

10-1 deci d

10+3 kilo k

10+6 mega M

10+9 giga G

10+12 tera T

Analyzing Uncertainty and Errors ISAT 253 Spring 2005.

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