1 Quantifying & Propagation of Uncertainty Module 2 Lecture THREE (4-4) 3/14/05

2 What have you learned so far? Determine Random Uncertainty in the Measurement of the Measurand Using Single measurement Using ONE sample Using M samples Determine Overall Random uncertainty caused by Elemental Errors Determine Total Uncertainty caused by Bias and Random uncertainties Determine Total Uncertainty caused by more than ONE variable

3 Random & Bias Errors Single Measurement B x =0

4 Random & Bias Errors in Multiple Measurements

5 Determination of Total (Systematic and Random ) Uncertainties Total Systematic and Random Uncertainty W x (RSS) B i ’s are the systematic uncertainties caused by k elemental error sources and Pi’s are the random uncertainties caused by m elemental error sources

6 Examples Lecture Slides (check values?) Readings (course web)

7 Mathematical Approach for Determining Uncertainties o It allows us to study the impact of uncertainties caused by MORE THAN ONE INDEPENDENT variable on the TOTAL uncertainty on the DEPENDENT variable o The mathematical mechanism to do this is “Partial Derivative”.

8 Calculate Total uncertainty dR is the Total uncertainty in the measurement of R (result-dependent variable) caused by Elemental uncertainties dx i in the variables x i (independent variables) Using the RSS method

9 Partial Derivative - Notation The “total” change in the area is represented by the derivative dA as Total Change in area Partial Change in area due to d W Partial Change in area due to d L

10 In Terms of Uncertainty in Measurements Total Uncertainty Sensitivity of A with respect to L Uncertainty in L Uncertainty in W Sensitivity of A with respect to W

11 TOTAL Uncertainty of Dependent Variable in Terms of Random and Bias Uncertainties of Independent Variables Uncertainty in L Uncertainty in W

12 Determine Total Uncertainty Using the RSS Formula Assume the dimensions of the rectangular (LxW= 60x50) and uncertainties in the measurements of L and W are mm and mm, respectively. Meaning? We are 95% Confident that True Value of Area = m 2 Assuming that THERE is No Bias Errors Other wise We are 95% Confident that Mean Value of Area Measurements (population) = m 2

13 How to perform PD? Transform the Multi-variable function to ONE variable function replace all variables with constants, except the ONE variable that is differentiated Perform ordinary differentiation Replace back the constants with the equivalent variables

14 Example: Area A = L x W

15 Example: Area A = L x W

16 Example: Area A = L x W

17 Example: Resistance

Another Differentiation Rule?

21 Example: Resistance

22 Example: Resistance Determine the Total uncertainty in measuring the resistance R, for the nominal values of L = 2m, A= 1 mm 2, and resistivity = 0.025x10 -6 Ω.m. The uncertainties in the measurement of L, A, and resistivity are m, + 0.1mm 2, and x10 -6 Ω.m, respectively?

23 Example: Electrical Resistance General Formula

24 Example: Electrical Resistance The RSS formula

26 Determine Total Fractional Uncertainty Using Fractional Uncertainties of Variables If the dependent variable R is a product of the measured variables, i.e Then, the fractional uncertainty in R is directly related to the fractional uncertainty of the variables

27 The Two Forms are Equivalent Which is equivalent to

28 Back to Area Example Formula for Area A = L x W

31 Any Question????? Any Question???? Danke Schon Thank you Good Luck with Exam