1. Statistics Presentation Ch En 475 Unit Operations

2. Quantifying variables (i.e. answering a question with a number) 1. Directly measure the variable. - referred to as “measured” variable ex. Temperature measured with thermocouple 2. Calculate variable from “measured” or “tabulated” variables - referred to as “calculated” variable ex. Flow rate m =  A v (measured or tabulated) Each has some error or uncertainty

3. Some definitions: x = sample mean s = sample standard deviation  = exact mean  = exact standard deviation As the sampling becomes larger: x   s    t chart z chart  not valid if bias exists (i.e. calibration is off) A. Error of Measured Variable Several measurements are obtained for a single variable (i.e. T). What is the true value? How confident are you? Is the value different on different days? Questions

4. Let’s assume “normal” Gaussian distribution For small sampling: s is known For large sampling:  is assumed How do you determine the error? small large (n>30) we’ll pursue this approach Use z tables for this approach Use t tables for this approach Don’t often have this much data

5. Example nTemp 140.1 239.2 343.2 447.2 538.6 640.4 737.7

6. Standard Deviation Summary (normal distribution) 40.9 ± (3.27) 1s: 68.3% of data are within this range 40.9 ± (3.27x2) 2s: 95.4% of data are within this range 40.9 ± (3.27x3) 3s: 99.7% of data are within this range If normal distribution is questionable, use Chebyshev's inequality:Chebyshev's inequality At least 50% of the data are within 1.4 s from the mean. At least 75% of the data are within 2 s from the mean. At least 89% of the data are within 3 s from the mean. The above ranges don’t state how accurate the mean is - only the % of data within the given range

7. Student t-test (gives confidence of where  (not data) is located)  =1- probability r = n-1 = 6Prob.t+-90%.051.9432.40 95%.0252.4473.02 99%.0053.7074.58 5% t true mean measured mean 2-tail

8. T-test Summary 41 ± 2 90% confident  is somewhere in this range 41 ± 3 95% confident  is somewhere in this range 41 ± 5 99% confident  is somewhere in this range  = exact mean 40.9 is sample mean

9. Comparing averages of measured variables Day 1: Day 2: What is your confidence that  x ≠  y (i.e. they are different)  n x +n y -2 1-  confident different  confident same Larger t: More likely different 1-tail

10. Problem 1 1. Calculate average and s for both sets of data 2. Find range in which 95.4% of the data falls (for each set). 3. Determine range for  for each set at 95% probability 4. At what confidence are pressures different each day? Data points Pressure Day 1 Pressure Day 2 1750730 2760750 3752762 4747749 5754737

11. Example: You take measurements of , A, v to determine m =  Av. What is the range of m and its associated uncertainty? Calculate variable from multiple input (measured, tabulated, …) variables (i.e. m =  Av) What is the uncertainty of your “calculated” value? Each input variable has its own error B. Uncertainty of Calculated Variable Details provided in Applied Engineering Statistics, Chapters 8 and 14, R.M. Bethea and R.R. Rhinehart, 1991).

12. To obtain uncertainty of “calculated” variable DO NOT just calculate variable for each set of data and then average and take standard deviation DO calculate uncertainty using error from input variables: use uncertainty for “calculated” variables and error for input variables Plan: obtain max error (  ) for each input variable then obtain uncertainty of calculated variable Method 1: Propagation of max error - brute force Method 2: Propagation of max error - analytical Method 3: Propagation of variance - analytical Method 4: Propagation of variance - brute force - Monte Carlo simulation

13. Value and Uncertainty Value used to make decisions - need to know uncertainty of value Potential ethical and societal impact How do you determine the uncertainty of the value? Sources of uncertainty ( from Rhinehart, Applied Engineering Statistics, 1991 ): 1. Estimation - we guess! 2. Discrimination - device accuracy (single data point) 3. Calibration - may not be exact (error of curve fit) 4. Technique - i.e. measure ID rather than OD 5. Constants and data - not always exact! 6. Noise - which reading do we take? 7. Model and equations - i.e. ideal gas law vs. real gas 8. Humans - transposing, …

14. Estimates of Error (  ) for input variables (  ’s are propagated to  find uncertainty) 1. Measured: measure multiple times; obtain s;  ≈ 2.5s Reason: 99% of data is within ± 2.5s Example: s = 2.3 ºC for thermocouple,  = 5.8 ºC 2. Tabulated :  ≈ 2.5 times last reported significant digit (with 1) Reason: Assumes last digit is ± 2.5 (± 0 assumes perfect, ± 5 assumes next left digit is fuzzy) Example:  = 1.3 g/ml at 0º C,  = 0.25 g/ml Example: People = 127,000  = 2500 people

15. Estimates of Error (  ) for input variables 3. Manufacturer spec or calibration accuracy: use given spec or accuracy data Example: Pump spec is ± 1 ml/min,  = 1 ml/min 4.Variable from regression (i.e. calibration curve):  ≈ 2.5*standard error (std error is stdev of residual) Example: Velocity is slope with std error = 2 m/s 5. Judgment for a variable: use judgment for  Example: Read pressure to ± 1 psi,  = 1 psi

16. Estimates of Error (  ) for input variables If none of the above rules apply, give your best guess Example: Data from a computer show that the flow rate is 562 ml/min ± 3 ml/min (stdev of computer noise). Your calibration shows 510 ml/min ± 8 ml/min (stdev). What flow rate do you use and what is  ? In the following propagation methods, it’s assumed that there is no bias in the values used - let’s assume this for all lab projects.

17. Method 4: Monte Carlo Simulation (propagation of variance – brute force) Choose N (N is very large, e.g. 100,000) random ±δ i from a normal distribution of standard deviation σ i for each variable and add to the mean to obtain N values with errors: Choose N (N is very large, e.g. 100,000) random ±δ i from a normal distribution of standard deviation σ i for each variable and add to the mean to obtain N values with errors: rnorm(N,μ,σ) in Mathcad generates N random numbers from a normal distribution with mean μ and std dev σrnorm(N,μ,σ) in Mathcad generates N random numbers from a normal distribution with mean μ and std dev σ Find N values of the calculated variable using the generated x’ i values. Find N values of the calculated variable using the generated x’ i values. Determine mean and standard deviation of the N calculated variables. Determine mean and standard deviation of the N calculated variables. y = y avg ± 1.96 SQRT(  y ) 95%y = y avg ± 1.96 SQRT(  y ) 95% y = y avg ± 2.57 SQRT(  y ) 99%y = y avg ± 2.57 SQRT(  y ) 99%

18. Monte Carlo Simulation Example Estimate the uncertainty in the critical compressibility factor of a fluid if Tc = 514 ± 2 K, Pc = 61.37 ± 0.6 bar, and Vc = 0.168 ± 0.002 m 3 /kmol? Estimate the uncertainty in the critical compressibility factor of a fluid if Tc = 514 ± 2 K, Pc = 61.37 ± 0.6 bar, and Vc = 0.168 ± 0.002 m 3 /kmol?

19. Example: Propagation of variance Calculate  and its 95% probable error All independent variables were measured multiple times (Rule 1); averages and s are given M = 5.0 kg s = 0.05 kg L = 0.75 m s = 0.01 m D = 0.14 m s = 0.005 m

20. Monte Carlo

21. Problem 2 The DIPPR Database reports the following measured property values with associated uncertainties for methyl propionate: The DIPPR Database reports the following measured property values with associated uncertainties for methyl propionate: TC = 530.6 K ± 1%TC = 530.6 K ± 1% PC = 40.04 bar ± 3%PC = 40.04 bar ± 3% VC = 0.282 m 3 /kmol ± 5%VC = 0.282 m 3 /kmol ± 5% Uncertainties are essentially 95% confidence intervals or 1.96s. Uncertainties are essentially 95% confidence intervals or 1.96s. Determine ZC and use a Monte Carlo calculation to determine a 95% confidence interval. Determine ZC and use a Monte Carlo calculation to determine a 95% confidence interval.

22. Overall Summary measured variables: use average, std dev (data range), and student t-test (mean range and mean comparison) calculated variable: determine uncertainty -- Max error: propagating error with brute force -- Max error: propagating error analytically -- Probable error: propagating variance analytically -- Probable error: propagating variance with brute force (Monte Carlo)

23. Data and Statistical Expectations 1. Summary of raw data (table format) 2. Sample calculations– including statistical calculations 3. Summary of all calculations- table format is helpful 4. If measured variable: average and standard deviation, confidence of mean 5. If calculated variable: Uncertainty using Monte Carlo (if possible)