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Analyzing Uncertainty and Errors

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Analyzing Uncertainty and Errors

ISAT 253

Spring 2005

- Understand the concept of uncertainty
- Define measurement uncertainties and errors
- For a set of measurements, learn to calculate specific
- Sensitivities
- Accuracies
- Precisions

Dr. Ken Lewis

- Define measurement errors
- Differentiate between systematic and random errors

Dr. Ken Lewis

Precision

Accuracy

Precision error

Bias error

Sensitivity

Calibration

Calibration standards

Measurement standards

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

- 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

Dr. Ken Lewis

Precision Error

Random Error

Bias Error

Systematic Error

Accuracy

True

Value

Average

Measure

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

- 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

Dr. Ken Lewis

The resolution uncertainty or resolution error.

±0.01 V

Dr. Ken Lewis

The systematic error or accuracy

Accuracy = average value - true value

Accuracy = 6.03 V - 6.11 V = -0.08 V

-0.08 V

Dr. Ken Lewis

The precision

Precision = Maximum deviation from the average

Precision = ±|5.96 V - 6.03 V| = ±0.07 V

±0.07 V

Dr. Ken Lewis

- Caution
- Be sure that the precision statement isn’t based on one bad measurement

Dr. Ken Lewis

- 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

Dr. Ken Lewis

- 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%

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

- The span is the difference between the high and low of the range

Dr. Ken Lewis

- 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

Dr. Ken Lewis

- The slope of the line relating input to output

Dr. Ken Lewis

Dr. Ken Lewis

For example

For

Ideal

devices

Dr. Ken Lewis

Dr. Ken Lewis

- Hysteresis
- Friction
- Mechanical flexure of internal parts
- Electrical capacitance

- Will usually appear random

Dr. Ken Lewis

- 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?

Accuracy uncertainty

Hysteresis

±30 rpm

Zero offset 200 rpm

U = 250rpm + 30rpm + 200 rpm

U = 580rpm

- 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 measurementstandard.

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

- 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

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

- 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.

Dr. Ken Lewis

http://www.pbs.org/newshour/bb/science/july-dec99/mars_lander_12-2.html

- 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

Dr. Ken Lewis

Dr. Ken Lewis

- 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

Dr. Ken Lewis

- Recall Newton’s second law
- Force = mass X acceleration

- Acceleration = meter/second/second
- =length/second2

Dr. Ken Lewis

Dr. Ken Lewis