1. University of Florida Mechanical and Aerospace Engineering 1 Useful Tips for Presenting Data and Measurement Uncertainty Analysis Ben Smarslok

2. University of Florida Mechanical and Aerospace Engineering 2 Outline Why is presenting data properly important? Explain important terminology and definitions –NIST vs. ISO vs. ASME/ASTM –Oberkampf definitions of model uncertainty (not included) Experimental scenarios and corresponding methods –Uncertainty propagation –Crossed vs. nested factors (ANOVA vs. VCA) –p-values –Interlaboratory Studies (not included)

3. University of Florida Mechanical and Aerospace Engineering 3 Best guess by experimenter Half the smallest division of measurement Standard deviation:  Standard error:  m =  /  n Expanded uncertainty of ± 2  or ± 3  (95% or 99% confidence interval) Standard uncertainty: u Combined standard uncertainty: u c *(Courtesy of Duane Deardorff presentation from UNC) m = 75 ± 5 g What is the meaning of ± 5 ?

4. University of Florida Mechanical and Aerospace Engineering 4 What does x ± u mean? Engineers think in terms of ±2  (95%) Physicists generally report ±1  (68% CI) Chemists report ±2  or ±3  (95% or 99% CI) Survey/poll margin of error is 95% CI Accuracy tolerances are often 95% or 99% NIST Calibration certificate is usually 99% Conclusion: The interpretation of ± u is not consistent within a field, let alone between fields – It is very important to explain the statistical relevance of the uncertainty bounds!!!

5. University of Florida Mechanical and Aerospace Engineering 5 Presenting Uncertainty Precisely Choose a standard for presenting uncertainty (I prefer NIST), and reference the standard Explain the source of the uncertainty –Type A – calculated by statistical methods (it is useful to explain the design of experiments and the number of samples involved) –Type B – determined by other means, such as estimate from experience or manufacturers specifications Use terms carefully! –Error vs. Uncertainty: Error is the deviation from the true value and measured value (never known), which is estimated as uncertainty –Bias vs. variability (will explain later) Avoid use of ambiguous ± notation without explanation Pet peeve: –COV = covariance

6. University of Florida Mechanical and Aerospace Engineering 6 Uncertainty classification: –Random uncertainty / variability – scatter in the measurements (v) –Systematic uncertainty / bias – systematic departure from the true value (b) NIST Classification of Measurement Uncertainties Type of evaluation: –Type A – calculated by statistical methods –Type B – determined by other means, such as estimate from experience x t = true value of specimen  = experimental population average x = experimental sample average v x = random error of sample  x = systematic error of sample Range is at 95% (2  ) level of a normal distribution

7. University of Florida Mechanical and Aerospace Engineering 7 Uncertainty Analysis Example Consider our transverse modulus work (E 2 ) Hooke’s Law: We will work through this problem backwards  P = Load  A = Area   = transverse strain 1 2 Total Uncertainty Bias & Variability ComponentsContributors

8. University of Florida Mechanical and Aerospace Engineering 8 Level 1: Total Uncertainty In general, –where, v X and b X were propagated from component uncertainties = Student’s t distribution at 95% confidence level (depends on # of DOF) Total uncertainty of E 2 at 1  (68%) confidence for comparison to experimental results Or, at the commonly accepted 95% level

9. University of Florida Mechanical and Aerospace Engineering 9 Level 2: Uncertainty Propagation Law of Propagation of Uncertainties (LPU): –where, p are the inputs (components) and q is the output E 2 Example: –Uncertainty contributors were analyzed for each of the components of E 2 –Random and systematic effects propagated separately –Only systematic uncertainties can have correlated effects Thickness and width are correlated

10. University of Florida Mechanical and Aerospace Engineering 10 Level 3: NIST Component Measurement Uncertainty Table

11. University of Florida Mechanical and Aerospace Engineering 11 Level 4: Contributors of Component Uncertainty (Further Analysis) Numerous different methods to analysis the significance of uncertainty contributors It is important to use the appropriate analysis method depending on the design of experiments (DOE) –Either design the experiments properly or match the corresponding method to the data you already have Most DOEs fall into one of these two categories: Crossed Same patients in each hospital. Patients unique to each hospital. Nested

12. University of Florida Mechanical and Aerospace Engineering 12 Crossed Design: ANOVA Crossed (or factorial) DOEs correspond to analysis of variance (ANOVA) Consider thickness in the E2 example –Since the SAME specimens were measured in the SAME positions with the SAME users, then the factors were crossed –3-way ANOVA with crossed, random variables was conducted Nominal: 0.09 x 1 in. Uncertainty contributors: Specimen – variability from specimen to specimen Position – variation across measurement surface User – error from user technique Measurement repeatability

13. University of Florida Mechanical and Aerospace Engineering 13 Thickness ANOVA 3-way ANOVA of crossed, random variables –Statistical software available for ease of use: Excel for 2 factors or SAS for 3 or more Factors: –A = specimena = 4 –B = positionb = 3 –C = userc = 4 –Repetitions:n = 3 ANOVA model: ANOVA results were not directly used in uncertainty analysis, but were used to identify significant contributors and validate uncertainty estimates Hypothesis Test for A:

14. University of Florida Mechanical and Aerospace Engineering 14 Results: Thickness ANOVA Use ANOVA to deterimine the significance of the contributors of uncertainty in thickness Position is most significant factor with p-value = 0.006 Not as interested in interactions in this study Used to validate estimated range of uncertainties of thickness and width ~~~

15. University of Florida Mechanical and Aerospace Engineering 15 Nested Design: VCA Nested DOEs correspond to variance component analysis (VCA) Consider a two-stage nested design of one specimen for thickness –Relevant if positions and users were unique each time –Specimens considered individually since the thickness does not have to be the same from one specimen to the next –Data was organized according to position –y1, y2, and y3 refer to the repeated basic measurements

16. University of Florida Mechanical and Aerospace Engineering 16 Variance Component Analysis of Thickness Goal: Develop a nested design to determine the contribution of each factor in the overall variance Variance of the measurement process for one specimen –Position – the three locations on the specimen where the thickness was measured (unique to each specimen) –User – four different users per position performed the measurements –Basic Measurement – three repeated measurements by each user at each position Compare the weight of each contributor to determine significance where, i is a component in the process

17. University of Florida Mechanical and Aerospace Engineering 17 Concluding Remarks Using proper statistical terminology and representation is necessary to have meaningful results You can say your results are “pretty good”, but give what your definition of “pretty good” is! Depending on the project, more or less uncertainty analysis may be required It is important to design your experiments with the statistical analysis in mind Age-old question: How many measurements do I need? –Obviously depends on the circumstances, so there is no straight forward answer –Best recommendation: Feel comfortable enough with your results that you can predict the next measurement within a desired range