1 CM4110 Unit Operations Lab Measurement Basics Fundamentals of Measurement and Data Analysis D. Caspary September, 2008

2 CM4110 Unit Operations Lab Measurement Basics Outline: Principles of measurement Error Analysis “Propagation of error”

3 CM4110 Unit Operations Lab Measurement Basics Principles of Measurement Nothing can be measured exactly Measurements are approximations of true value of a characteristic or property Associated with every measurement is “uncertainty” or “error”

4 CM4110 Unit Operations Lab Measurement Basics Principles of Measurement Uncertainty is introduced thru Instrument Error (or Reading /Measurement Error) Uncertainty is observed as fluctuations in replicated experimental data called Experimental Error

5 CM4110 Unit Operations Lab Measurement Basics Reporting Measured Values Engineering and scientific reporting must be ethical and honest – always report appropriate estimate of uncertainty with the results Learn/ Use appropriate statistical tools Use common sense

6 CM4110 Unit Operations Lab Measurement Basics Example problem statement : “Calculate the overall heat transfer coefficient for a shell and tube heat exchanger.” How will Instrument and Experimental Error affect the calculated results?

7 CM4110 Unit Operations Lab Measurement Basics Planning your experimental strategy: What is known? from mfg. data, tables, etc. What do I need to measure? What instruments are available? What is the precision of each instrument? And, what about accuracy in measurements? How will precision and accuracy of these instruments affect the calculated results?

8 CM4110 Unit Operations Lab Measurement Basics Instrument Error can show up as:  Systematic error – determinate (or fixed) error – defines accuracy  Random error – indeterminate error associated with the instrument – defines precision

9 CM4110 Unit Operations Lab Measurement Basics Accuracy and Precision are independent Accurate measurement – small systematic error Precise measurement – small random variation (random error)

10 CM4110 Unit Operations Lab Measurement Basics Poor Accuracy Poor Precision Poor Accuracy Good Precision Good Accuracy Poor Precision Good Accuracy Good Precision

11 CM4110 Unit Operations Lab Measurement Basics Reporting Instrument Error For analog scales Typically plus or minus ½ the smallest increment For digital readouts Report the value as displayed, then look up accuracy and precision spec’s in manufacturer’s data

12 CM4110 Unit Operations Lab Error Analysis Estimating Experimental Error – again, the Experimental Strategy Usually you will perform a set of experiments: How many replicates of each test should you perform? How will variation in each replicate affect the result?

13 CM4110 Unit Operations Lab Error Analysis Three Types of Experimental Error  Gross error – mistakes  Systematic error – determinate (or fixed) error. Correct this first!  Random error – indeterminate error. Use statistics to extimate.

14 CM4110 Unit Operations Lab Error Analysis … liars, damned liars, and statisticians… “Your goal is to present the Location and Dispersion of your results.” Wheeler and Chambers, Understanding Statistical Process Control, SPC Press, 1992

15 CM4110 Unit Operations Lab Error Analysis Location of Data With three or more replicates typically report the Average With a single value (or 2 values), report the value(s).

16 CM4110 Unit Operations Lab Error Analysis Dispersion of Data Range Lowest value and highest value Often used for small data sets Easy to report Not used for our purposes as it hides data – says nothing about the dispersion of the “middle values”

17 CM4110 Unit Operations Lab Error Analysis Dispersion of Data RMS Deviation (aka Standard Deviation) calculate the average for the sample set calculate the deviation from the average for each value square the individual deviations sum all the squares of the deviations find the average squared deviation take the square root of the average squared deviation

18 CM4110 Unit Operations Lab Error Analysis Dispersion of Data Standard Deviation (aka Average Std. Dev.) Calculate like RMS deviation except use (n-1) in the denominator when calculating the average squared deviation As data set gets large, Std. Dev. approaches the value for RMS Dev.

19 CM4110 Unit Operations Lab Error Analysis Rules of Thumb Be realistic (honest) in reporting the measurement error or uncertainty. Normally report Average, Error, and sample size for UO Lab measurements Do not hide data. Do not allow yourself to adjust the results to match some “expected value”.

20 CM4110 Unit Operations Lab Error Analysis Discarding Data Bad Data caused by obvious blunders can be discarded if it has “assignable cause” “Unexplained” Data cannot be discarded because it doesn’t meet our expectations no assignable cause (is random) Any data filtering must be consistent and unbiased.

21 CM4110 Unit Operations Lab Propagation of Error Estimating the error in your calculated results: The Error in measured quantities that are arithmetically combined must also be combined. Use standard practice for “propagating” error through calculations. Error can be reported in EU’s or %.

22 CM4110 Unit Operations Lab Propagation of Error Text References Understanding Statistical Process Control, 2 nd edition, D.J. Wheeler, D.S. Chambers, SPC Press, Experimental Methods for Engineers, 3 rd edition, J.P. Holman, McGraw-Hill, Data Reduction and Error Analysis for the Physical Sciences, P.R. Bevington, Mcgraw-Hill, 1969.

23 CM4110 Unit Operations Lab Propagation of Error Web References – shows how to arithmetically combine individual errors to get error in calculated result. ysis/Propagation.html – propagation of error and error analysis for all situations Dr. Pintar’s Error Analysis Handout – link on course web page for definitions and a worked out example from actual UO Lab data