Uncertainty Analysis

1. Uncertainty Analysis

2. Uncertainty Analysis – What it is • There is no such thing as a perfect measurements. All measurements of a variable contain inaccuracies. • The analysis of the uncertainties in experimental measurements and results is a powerful tool, particularly when it sis used in the planning and design of experiments • Although it may be possible to an uncertainty by improved experimental method or the careful use of statistical technique to reduce the uncertainty, it can never be eliminated

3. Issues of Analysis Systematic and Random Uncertainties

4. Issues of Analysis Systematic Uncertainties • Offset uncertainty Clearly there is a problem here: • the boiling point of water should be very close to 100.0 oC while the melting point should be very close to 0.0 oC • There is an offset uncertainty with the temperature measuring system of about 7.5 oC • Possible causes are inherent to measurement device (such as low battery, malfunctioning digital meter, incorrect type of thermocouple, etc)

5. Issues of Analysis Systematic Uncertainties • Gain uncertainty

6. Issues of Analysis Random Uncertainties • Random uncertainties produce scatter in observed values. • The cause : • limitation in the scale of the instrument  resolution uncertainty due to rounding up of measured value • reading uncertainty • random uncertainty due to environmental factor (electrical interference, vibration, power supply fluctuation, etc) • Use statistical technique to get an estimate of the probable uncertainty and to allow us to calculate the effect of combining uncertainties

7. Issues of Analysis True Value, Accuracy and Precision

8. Issues of Analysis Selection and Rejection of Data • A sensitive subject and one that can bring out strong feeling amongst experimenters: • One argue : All data are equal  no circumstances in which the rejection of data can be justified • Another argue : there as those that ‘know’ that a set of data is spurious and reject it without a second thought • Expert judgment  confidence level • Statistical test : • Chauvenet’s criterion  P = 1 – 1/(2N) •  - σ criterion   = 2, 3, …

9. Issues of Analysis Quoting the Uncertainty After making repeated measurement of a quantity, there are four important steps to take in quoting the value of the quantity: • Calculate the mean of the measured values • Calculate the uncertainty in the quantity, making clear the method used. Round the uncertainty to one significant figure (or two if the first figure is a ‘1’) • Quote the mean and uncertainty to the appropriate number of figures • State the units of the quantity

10. Issues of Analysis Uncertainty statement • Absolute uncertainty • With unit of the quantity • Fractional uncertainty • no unit • Percentage uncertainty • no unit

11. Determining Uncertainty – Single Quantity Simple Method Example:

12. Determining Uncertainty – Single Quantity Statistical Approach to variability in data

13. Determining Uncertainty – Single Quantity With 95 % level of confidence : Then :

14. Combining Uncertainty – Uncertainty Propagation • An experiment may require the determination of several quantities which are later to be inserted into an equation. • The uncertainties in the measured quantities combine to give an uncertainty of the calculated value • The combination of these uncertainties is sometimes called the propagation of uncertainty or error propagation measured quantity with uncertainty calculated quantity with propagation of uncertainty measured quantity with uncertainty

15. Combining Uncertainty – Simple Method • Most straightforward method and requires only simple arithmetic. • Each quantity in the formula is modified by an amount equal to the uncertainty in the quantity to produce the largest value and the smallest value Example : In an electrical experiment, the current through a resistor was found to be (2.5 ± 0.1) mA and the voltage across the resistor (5.5 ± 0.3) V. Determine the resistance of the resistor R and the uncertainty UR !

16. Combining Uncertainty – Partial Differentiation • Based on differentiation of function of several variables Properties: Sum: Difference: Product: Uncertainty propagation: Quotient:

17. Combining Uncertainty – Partial Differentiation Example : The temperature of (3.0 ±0.2) x 10-1 kg of water is raised by (5.5 ± 0.5) oC by heating element placed in the water. Calculate the amount of heat transferred to the water to cause this temperature rise ! The value of c = 4186 J kg-1oC-1 is assumed to be constant (neglecting its uncertainty)

18. Combining Uncertainty – Statistical Approach Taking uncertainty of the mean to relate to standard error of the mean and partial differential principle

19. General Uncertainty Analysis Consider a general case in which an experimental result, r, is a function of J measured variable Xi Then, the uncertainty in the result is given by : Note : all absolute uncertainties (UX) should be expressed with the same level of confidence

20. General Uncertainty Analysis Nondimensionalized forms: Note :

21. General Uncertainty Analysis Example: A pressurized air tank is nominally at ambient temperature (25 oC). Using ideal gas law, how accurately can the density be determined if the temperature is measured with an uncertainty of 2 oC and the tank pressure is measured with a relative uncertainty of 1%?

22. General Uncertainty Analysis Uncertainty analysis:

23. General Uncertainty Analysis Uncertainty analysis:

24. Detailed Uncertainty Analysis 1 2 j Elemental error sources …. Individual measurement system X1 B1,P1 X2 B2,P2 Xj Bj,Pj Measurement of individual variables …. r=r(X1,X2, …, Xj) Equation of result r Br,Pr Experimental result B = bias (systematic uncertainty) P = precision (random) uncertainty

25. Detailed Uncertainty Analysis The uncertainty in the result is: Systematic (bias) uncertainty: Correlated systematic uncertainty Precision (random) uncertainty:

26. Systematic Uncertainty • Systematic error can be determined and eliminated by calibration only to a certain degree (A certain bias will remain in the output of the instrument that is calibrated) • In the design phase of an experiment, estimate of systematic uncertainty may be based on manufacturer’s specifications, analytical estimates and previous experience • As the experiment progress, the estimate can be updated by considering the sources of elemental error: • Calibration error: some bias always remains as a result of calibration since no standard is perfect and no calibration process is perfect • Data acquisition error: there are potential biases due to environmental and installation effects on the transducer as well as the biases in the system that acquires, conditions and stores the output of the transducer • Data reduction errors: biases arise due to replacing data with a curve fit, computational resolution and so on

27. Random Uncertainty Analysis • Random uncertainty can be determined with various ways depending on particular experiment: • Previous experience of others using the same/similar type of apparatus/instrument • Previous measurement results using the same apparatus/instrument • Make repeated measurement • When making repeated measurement, care should be taken to the time frame that required to make the measurement: • Data sets should be acquired over a time period that is large relative to the time scale of the factors that have a significant influence on the data and that contribute to the random errors • Be careful of using a data acquisition system

28. Random Uncertainty Analysis Y ∆t Failure to determine random uncertainty due to inappropriate data acquisition Time, t

29. Some Detail Approach/Guidelines • Abernethy approach (1970-1980): • Adapted in SAE, ISA, JANNAF, NRC, USAF, NATO • Coleman and Steele approach (1989 renewed 1998): • Adapted in AIAA, AGARD, ANSI/ASME

30. Some Detail Approach/Guidelines • ISO Guide approach (1993): • Adapted by BIPM, IEC, IFCC, IUPAC, IUPAP, IOLM • Using a “standard uncertainty” • Instead of categorizing uncertainty as systematic and random, the “standard uncertainty”values are divided into type A standard uncertainty and type B standard uncertainty • Type A uncertainties are those evaluated “by the statistical analysis of series of observations” • Type B uncertainties are those evaluated “by means other than the statistical analysis of series of observations” • NIST Approach (1994): • Use “expanded uncertainty” U to report of all NIST measurement other than those for which Uchas traditionally been employed • The value of k = 2 should be used. The values of k other than 2 are only to be used for specific application