1. ME 322: Instrumentation Lecture 2 January 23, 2015 Professor Miles Greiner Quad measurement calculations and results for Lab 2, Probability Distribution Functions, Examples (symmetric, one-sided)

2. Announcements Lab 1 work sheet due now HW 1 due Monday – Use ME 322 ID number (from WebCampus), not name If you have requested to move to a new section – we will send an email to let you know if we were able to fulfil you request Extra-Credit Opportunity – Help out at Science OlympiadScience Olympiad Saturday, February 7, 2015, 9:00am to 3:00pm ~1% course grade extra credit – ~One homework or lab assignment – ~4 points on a test Sign up by Wednesday, January 28, 2015

3. Results of the Quad Measurements Data is on Lab 2 website How to process that data, – You will repeat and present this in Lab 2, Analysis of Quad Measurement Spreadsheet has Measured Data and room for Calculations – H, D C, N Ci, N CA, F = D C /N CA – N Si, N SA, S=F*N SA – N Li, N LA, L=F*N LA – A = LS – C = A*($3.49/200 ft 2 ) How to Plot – F versus H; L vs S (scatter plots) – Cost Estimate Histogram (install analysis toolpack; file, options, add-in) Questions – Is stride length F highly correlated with height, H? – What are the distributions, sample mean and standard deviation of the cost estimates? Are any of them “out of place?” – Are the measured values of L and S correlated? Should they be? – If you budget the amount of your cost estimate, you are only 50% sure to have enough to cover quad (be above the average value, which we assume is the most accurate estimate) How much money should you budget to be 90% sure to have enough

4. Randomly Varying Processes The output of a measurement instrument is affected by the measurand (the quantity being measured) and many uncontrolled (undesired) factors Consider a process (such as a measurement) that has a very large number of factors that can, independently, increase or decrease the value of the outcome Its not likely that all or a large majority of the factors will push the outcome in the same direction. Its more likely that “most” of the factors will cancel each other, and push the outcome only “slightly” in one direction or the other. This describes how uncontrolled factors affect the output of measurement systems (instruments)

5. Gaussian (Normal) Probability Distribution Function

6. How can we use this? If a sample is very large, and if the process variations are “normally” distributed, – Then expect sample histogram to take a bell shape, – And, if we know  and , the probability that the next measurement x will be in the range x 1 < x < x 2 is Note, for any  and  :

7. Non-Dimensionalization Define – Number of standard deviations x is above the mean We can show that the probability that the next measurement is between z 1 and z 2 is: – – Where – This integral is tabulated on page 146, for z > 0

8. Graphical Representation Area from center (z = 0) to z For z > 0 : Note: This integral is tabulated on page 146, for z > 0

10. For negative values of z For z 1 < 0 :

11. Symmetric Example Find the Probability a measurement is within one standard deviation (  ) of the mean (  ).

12. Page 146

13. Next measurement is within 2  and 3  of the mean

15. One-sided example

17. Lab 2 If you make a measurement, there is a 50% likelihood it is below the mean (best) value. How much should you add to your best estimate to be 90% you are above the mean? Answer: 1.282 standard deviations

18. Extra Slides

19. Area of UNR Quad Find Short Side (S) – N Si – N SA Find Long Side (L) – N Li – N LA