1. 1 Review from previous class  Error VS Uncertainty  Definitions of Measurement Errors  Measurement Statement as An Interval Estimate  How to find bias and random uncertainty?

2. 2 Error VS Uncertainty ERROR Known for certainty. Correct it. A fixed number. Not a statistical variable. The difference between the ‘true value’ ( X True ) and the measured value X i. UNCERTAINTY Not known for certainty. No correction scheme is possible. An uncertainty is a possible value that an error may have. A statistical variable. Uncertainty can only be estimated.

3. 3 Definitions of Measurement Errors Assume that we know and that we neglect the variable-but-deterministic error, the followings are the definitions of errors associated with the i th measurement. i be the subscript for the i th measurement/realization be the measured value of X at reading/sample i. be the true value of X. be the total measurement error for reading i. be the systematic/bias error, which is fixed/constant over realizations. be the random/precision error for reading i, which randomly varying over realizations. X Statistical Experiment: Frequency of occurrences the population distribution Measurement i Assumption: Note that throughout, we shall neglect the variable- but-deterministic error.

4. 4 Measurement Statement as An Interval Estimate  The above statement is interpreted as an interval estimate: Measurement statement as an interval estimate. XiXi X Best Estimate X Best Estimate - U X X Best Estimate + U X Specified range with some confidence X True (not known)

5. 5 Bias VS (Band Width of) Scatter in Data For repeated measurements, roughly:  Bias:results in the deviation of the (population) mean from the true value.  Random:results in the scatter in data in a set of repeated measurements. It is viewed and quantified as (the width of) the band (of scatter) not absolute position. i

6. 6  Uncertainty Propagation Equation (UPE)  Design Uncertainty Analysis  Detailed Uncertainty Analysis Today Contents

7. 7 Questions: What is the uncertainty of derived quantities? Given all ‘s, how can we find ? Measured quantity 1 Measured quantity 2 Measured quantity j Derived quantity

8. 8 UPE : Uncertainty Propagation Equation In our experiments, we have Note: all errors are assumed to be independent. Propagated uncertainty can be found from Physical quantity of uncertainties

9. 9 UPE: Derived Equation Interested students can find more details in the references given below. UPE as a relation between variances. STEP 1:Square of the  th realization [Neglect H.O.T.] [1] Dunn, P. F., 2005, “Measurement and data analysis for engineering and science,” McGraw-Hill, New York. [2] Coleman, H. W., Glenn, W., and Steels, Jr., 1998, Experimentation and uncertainty analysis for engineers, 2nd Edition, Wiley, New York. x (unit) r (unit) Measured uncertainty band Resultant uncertainty band For r= f(x) Expand uncertainties as a Taylor series, and focus only uncertainty terms, error

10. 10 STEP 2:Decompose the “errors” we have The square-equation becomes

11. 11 STEP 3:Sum over , divide by N, take limit N  infinity, we have the relation between variances where are the corresponding variances, and are the corresponding covariances. And,and are the corresponding cross correlation coefficients. Neglect correlations between two errors

12. 12 STEP 4:Estimate the population properties with the sample properties. Since we never know the population properties, we estimate them with the sample properties. Replace the above population variances with sample variances where are the corresponding sample standard deviation, and are the corresponding sample covariances.  So far, no assumption regarding the types of error distribution has been made.

13. 13 STEP 5:Multiply by the coverage factor K. Assume Normal distribution for r. Convert to U r.  To obtain the expanded uncertainty U r [ISO terminology] at the specified confidence limit C, we multiply S r with the coverage factor K.  It is in choosing K that assumptions regarding the type(s) of the error distributions must be made.  Because of the central limit theorem, we assume that the error distribution for r is normal. [The same argument can be made for all X i.]  Hence, K corresponds to t value from t distribution at C. We thus have

14. 14 STEP 6:Further simplification for our purpose. For simplicity, we shall assume the followings for this class.  All errors are independent. Hence, all cross covariances vanishes.  We shall consider only the case where all variables have the same degree of freedom.  Hence, the above relation reduces to the equation introduced earlier: where is the sample standard deviation for bias error of X i, is the sample standard deviation for random error of X i, Note: The cross covariances between bias errors can be significant when the two have common error sources. For example, X i and X j may be calibrated from the same standards.

15. 15 Summary: UPE 1.The fraction uncertainty ( FU ) or per cent uncertainty (x 100) is defined as It is often convenient to think of uncertainty in terms of fraction or %. 2.The uncertainty magnification factor for a variable X i, UMF Xi,, is defined as It is clear from the UPE that variables X i ‘s that have larger UMF relatively (for equal fraction uncertainty) contribute more to the final uncertainty in r (U r ) than those with smaller UMF.

16. 16 Important Notes Regarding The Applications of DRE and UPE 1. Current measurement process: As mentioned earlier, the DRE must reflect the current measurement process. Hence, e.g., even though the definition of density is  In order to apply the UPE suitably, we must write – depending upon our current measurement process, e.g., or as the case may be.

17. 17 Important Notes Regarding The Applications of DRE and UPE 2.All X i ’s variables that may affect the uncertainties: Note that the writing of the DRE must include all X i ’s variables that may affect the final uncertainty in r, and not only the measurands (measure variables). For example, a) if it is determined that the uncertainty in the gas constant R will have little/acceptable (if neglected) effect on the determination of the density from measured pressure p and measured temperature T, we may write the DRE as (assume that the perfect gas law is acceptable): b) However, if this is not the case, we must take the uncertainty in the ‘constant’ R into account and write:

18. 18 Important Notes Regarding The Applications of DRE and UPE 3. The derived quantities must be alone on the LHS of the DRE. In writing the DRE, if r is to be determined/derived from other measured quantities (not measured directly), r must be alone on the LHS of the DRE. For example, even though a) if pressure p is to be determined/derived from the measured density  and the measured temperature T, the DRE must be written as b) on the other hand, if the density  is to be determined/derived from the measured pressure p and the measured temperature T, the DRE must be written as As one can see, the uncertainties are accumulated from measurement results and the two UPE equations are not the same.

19. 19 Some Useful Notes on UPE 1. UPE of common DRE’s 1.1 NOTE: The variables with higher power relatively (for equal per cent uncertainty) contribute more to the final uncertainty in r, U r.

20. 20 Some Useful Notes on UPE 1.2 NOTE: Because r is the difference ( aX 1 - bX 2 ) and U r depends upon the division by the difference, care must be taken when attempting to find ‘small difference ( X 1 - X 2 ) between large quantities.’ For example, the difference in heights is desired by measurements of X 1 and X 2 separately: X 1 = 100 + 1 cm and X 2 = 99 + 1 cm. As one can see, although the uncertainties in X 1 and X 2 are each of only ~ 1%, the final uncertainty in r = (X 1 - X 2 ) can be as large as ~ 200%. This is one of the reasons why sometimes differential type instruments (measure the difference directly) are desired in such situation.

21. 21 General VS Detailed Uncertainty Analysis Elemental error sources Measurement system i (or any variables that can affect U of r.) 1 1 Measurement statement i (or any variables that can affect U of r.) DRE Statement for r General Uncertainty AnalysisDetailed Uncertainty Analysis ii No distinction is made between bias and random uncertainties, U Decompose uncertainty into bias and random uncertainties, U  B and P UPE

22. 22 Design-Stage Uncertainty  Design-stage uncertainty analysis refers to an initial analysis performed prior to the measurement.  It is used to assist in the selection of equipment and procedures based on their relative performance and cost.  In the design stage, we consider only sources of uncertainty in general. Only uncertainty caused by a measurement system’s resolution and estimated intrinsic errors are considered.  The analysis can tell us the minimum uncertainty in the measured value that would result from the measurement. Calibration (Measurement) Process StandardInstrument Current Measurement Process Manufacturer or in-house calibrator Experimenter / User Bias uncertainty of current measurement process

23. 23 Why Detailed Uncertainty Analysis? Detailed Uncertainty Analysis Detailed uncertainty analysis helps us in the interpretation of measured data. Detailed uncertainty analysis can point out to the potential point of improvement in our experiment, for example, number of sampling, etc. Decompose uncertainty into bias and random uncertainties in order to analyze each component effectively. where = the overall uncertainty =the bias uncertainty =the random uncertainty Bias uncertainty can be reduced by calibration, not repeated measurements. Random uncertainty can be reduced by repeated measurements.

24. 24 Scheme for Detailed Uncertainty Analysis (for Single Test) Elemental error sources Measurement system i (or any variables that can affect U of r.) 1 Measurement statement i (or any variables that can affect U of r.) DRE Statement for r i

25. 25 STEP 1. Estimate Bias Uncertainty STEP 1: Identify elemental error sources Standard one is instrument uncertainty but there are more. Manufacturer data are interpreted as @ 95%. 1 i STEP 2: Combine elemental errors using RSS. Estimate it @ C %. (if given as standard deviation) STEP 3: Write down the UPE for bias uncertainty STEP 4: Find B r using UPE.

26. 26 STEP 2. Estimate Random Uncertainty (with available measured data) STEP 1: Determine the random uncertainty from the statistics of measured data. Single sample ( S X from auxiliary test.) Multiple N sample ( S X from current test.) 1 i STEP 2: Write down the UPE for random uncertainty STEP 4: Find P r using UPE. In case measured data are already available.

27. 27 STEP 3. Combine Bias and Random Uncertainties 1 Measurement statement i (or any variables that can affect U of r.) Statement for r i

28. 28 Scheme for Detailed Uncertainty Analysis (for Multiple Test, M tests)  Similar to that of single test.  However, since there are multiple r,  it is recommended to use statistics of the tests, and not UPE, to estimate the random uncertainty P r :  For bias uncertainty, the same method as for single test can be used.

29. 29  Interested students can find further information in the references given.  Or, the NIST website http://physics.nist.gov/cuu/Uncertainty/index.htmlhttp://physics.nist.gov/cuu/Uncertainty/index.html  The relevant standard is ISO: Guide to the Expression of Uncertainty in Measurement often called ISO GUM.  NIST also has technical notes that are free http://physics.nist.gov/Pubs/guidelines/TN1297/tn1297s.pdf http://physics.nist.gov/Document/tn1297.pdf Reference