Analyzing Uncertainty and Errors

1. Analyzing Uncertainty and Errors ISAT 253 Spring 2005

2. Objectives • 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

3. Objectives • Define measurement errors • Differentiate between systematic and random errors Dr. Ken Lewis

4. Precision Accuracy Precision error Bias error Sensitivity Calibration Calibration standards Measurement standards Key Concepts Dr. Ken Lewis

5. 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. Dr. Ken Lewis

6. 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 Dr. Ken Lewis

7. Error summary Precision Error Random Error Bias Error Systematic Error Accuracy True Value Average Measure Dr. Ken Lewis

8. 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. Dr. Ken Lewis

9. Quantifying uncertainty • 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

10. Quantifying uncertainty The resolution uncertainty or resolution error. ±0.01 V Dr. Ken Lewis

11. Quantifying uncertainty 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

12. Quantifying uncertainty The precision Precision = Maximum deviation from the average Precision = ±|5.96 V - 6.03 V| = ±0.07 V ±0.07 V Dr. Ken Lewis

13. Quantifying uncertainty • Caution • Be sure that the precision statement isn’t based on one bad measurement Dr. Ken Lewis

14. 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 Dr. Ken Lewis

15. 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% Dr. Ken Lewis

16. 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. Dr. Ken Lewis

17. Span • The span is the difference between the high and low of the range Dr. Ken Lewis

18. Accuracy – example ±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 Dr. Ken Lewis

19. Sensitivity • The slope of the line relating input to output Dr. Ken Lewis

20. Sensitivity – Thermocouples Dr. Ken Lewis

21. Sensitivity -- Thermocouples

22. Sensitivity Calculation For example For Ideal devices Dr. Ken Lewis

23. Other Measurement Problems Dr. Ken Lewis

24. Other Measurement Problems • Hysteresis • Friction • Mechanical flexure of internal parts • Electrical capacitance • Will usually appear random Dr. Ken Lewis

25. 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? Accuracy uncertainty Hysteresis ±30 rpm Zero offset 200 rpm U = 250rpm + 30rpm + 200 rpm U = 580rpm

26. 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 measurementstandard. Dr. Ken Lewis

27. 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. Dr. Ken Lewis

28. 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. Dr. Ken Lewis

29. 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 Dr. Ken Lewis

30. 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. Dr. Ken Lewis

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

32. 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 Dr. Ken Lewis

33. Seven SI Base Units Dr. Ken Lewis

34. 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 Dr. Ken Lewis

35. Example -- Force • Recall Newton’s second law • Force = mass X acceleration • Acceleration = meter/second/second • =length/second2 Dr. Ken Lewis

36. Some SI Derived Units Dr. Ken Lewis

37. Some Standard SI Prefixes