Say y f x1 x2 x3 Then y f x1 x2 x3 y 2 y 2 y 2 And u y x u x1 x u x 2 x u x 3 1 2 3 or n 2 u y i ui i 1 1 2 1 2 Manometers are pressure measuring devices that determine a pressure by measuring the height of a column of fluid We would like to achieve an accuracy of 1 of the maximum reading 10kPa This is to be done using a well manometer which has a 0 2 mm resolution Estimate the uncertainty that can be tolerated in the density of the gage fluid which has a nominal value of 2500 kg m3 The relationship between the pressure and column height is P rgh Assume we know g to a higher degree of accuracy than the other variables If we are going to meet the design specifications we need to have a maximum uncertainty in P of uP 0 01 10 000 100 Pa 95 What is the instrument resolution uncertainty u0 0 0001 m 95 Ok to make things easier let s write out the full expression for the uncertainty 1 0 P 2 P 2 P 2 2 uP uh ur ug 100 Pa 95 h r g Let s fill in P kg rg 2500 9 81 24525 0 2 2 h ms P gh 9 81 h r Looking for this What is the expected value of h 2 P m P 10 000 gh 9 81 0 408 4 00 2 h 0 408 m r s rg 2500 9 81 Now we have all of the information that we need to solve P 2 P uP uh ur h r 1 2 2 100 Pa uP 24525 0 0 0001 4ur 1 2 2 2 95 100 Pa 95 We can report our answer as ur 10 kg m3 10 19 0 4 2500 95 Maximum allowable uncertainty in the fluid density Check the specs on the fluid density Carefully Zero order analysis Scatter in the data due to resolution First Order Uncertainty time dependent processes producing scatter in the measured data Nth order Overall measurement uncertainty instrument calibration information Systematic error will be denoted by B Bias does not vary during repeated measurements and is independent of sample size Random error will be denoted by P Precision We will use the statistical analysis developed in earlier classes to deal with these errors Systematic and Random Uncertainties must be evaluated with the same confidence level 95 to be combined In advanced stage uncertainty analysis it is desirable to keep systematic and bias errors separated Consider a measurement system with several elemental errors all contributing to the overall error in the system We can divide these elemental errors into Pk Elemental Standard Random Uncertainty Where P is given by P Sx S P x N Single measurement Measurement of the mean We propagate the standard random error through P P12 P22 P32 1 2 Bk Elemental Systematic Errors We propagate the systematic error through B B12 B22 B32 1 2 RSS method So at this point we have P and B P is the measurement Standard Random Uncertainty B is the measurement Systematic Uncertainty We have assumed Bias Errors reported at 95 We combine the measurement standard random uncertainty and the measurement systematic uncertainty through 1 2 2 u x B 2 tv P P What do we use for n each elemental error may have different sample size Find n through Welch Satterthwaite formula K 2 Pk n k K1 4 Pk k 1 vk 2 vk Nk 1 Example In the control of a chemical process temperature is measured by a sensor that according to the manufacturer has a calibration of 0 50C In 20 measurements in a test of the temperature sensor s repeatability an average value of 1500C was determined with a standard deviation of 1 50C It is estimated that there can be a spatial variation of 20C and an installation effect of 10C In a separate test of the data transmission system it was determined that for 10 measurements the standard deviation was 0 50C The control program uses a linear relationship between temperature and voltage which can introduce an uncertainty of up to 10C in the applicable range What are the elemental errors classify as B and P calibration B repeatability P spatial variation B installation B data acquisition P linearity B If we put numbers to these we have BIAS calibration 0 50C spatial variation 20C installation 10C linearity 10C Precision repeatability P 1 50C 20 measurements data acquisition P 0 50C 10 Measurements Combine elemental Bias Errors B 0 52 22 12 12 2 5 Combine elemental precision Errors USE SD NOT SDOM P 52 1 52 1 58 C ux B 2 tv P P What is n 1 2 2 ux 2 52 tv P 1 58 repeatability P 1 50C 20 measurements data acquisition P 0 50C 10 Measurements What is tn P t23 95 2 07 P 95 2 K 2 Pk n k K1 4 Pk k 1 vk 1 5 n 2 So we have 1 2 2 4 C 1 2 2 u x 2 52 2 07 1 58 0 Total uncertainty of a single T measurement 4 0 5 vk Nk 1 2 4 1 5 0 5 19 9 2 22 9 23 In the previous example we looked at the uncertainty in a single measurement of temperature based on available data We would use SDOM if we were interested in the best estimate of the true temperature R f x1 x2 x3 We know that R f x1 x2 x3 We need to figure out uR f Px1 Px2 Px3 Bx1 Bx2 Bx3 Again we can divide these elemental errors into Pk Elemental Standard Random Uncertainty Where P is given by P Sx Sx P N We propagate the systematic Single measurement error through L 2 B i Bi i 1 Measurement of the mean We propagate the standard random error through L 2 P i Pi i 1 Bk Elemental Systematic Errors 1 2 1 2 So at this point we have P and B P is the measurement Standard Random Uncertainty B is the measurement Systematic Uncertainty We have assumed Bias Errors reported at 95 We combine the measurement standard random uncertainty and the measurement systematic uncertainty through 2 K 2 k Pk k 1 n 1 4 K k Pk 2 2 2 u x B tv P P k 1 vk Finally the degrees of freedom in this case is calculated from