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12_lecturedatareduction1

12_lecturedatareduction1 - Data Reduction Its Poetic 16.621...

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Data Reduction “It’s Poetic” 16.621 March 18, 2003
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Introduction A primary goal of your efforts in this course will be to gather empirical data so as to prove (or disprove) your hypothesis Typically the data that you gather will not directly satisfy this goal Rather, it will be necessary to “reduce” the data, to put it into an appropriate form, so that you can draw valid conclusions • In our discussion today we will examine some typical methods for processing empirical data Caution-garbage in/garbage out still applies
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Hiawatha Designs an Experiment by Maurice G. Kendall From The American Statistician Vol. 13, No. 5, 1959, pp 23-24 Verses 1 through 6
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Deyst’s 16.62X Project I have performed a very simple experiment The hypothesis was: my driving route distance, from West Garage to my driveway in Arlington, is eight miles On a number of trips I recorded the mileage, as indicated by the odometer of my automobile I now wish to reduce the data and draw some conclusions
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Experimental Project (cont.) My experimental procedure was: at the exit from West Garage I zeroed my trip odometer and when I reached my driveway at home I recorded the odometer reading On each of ten trips I took the same route home
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Error Sources Random errors Odometer readout resolution Odometer mechanical variations Route path variations Tire slippage Systematic errors Bias in the odometer readings Odometer scale factor error Tire diameter decreases due to wear
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Error Sources (cont.) The resolution I achieved in reading the odometer was within ± .025 miles The best knowledge I have about the other random errors is that they were all in the range of ± .10 miles I zeroed the odometer at the beginning of each trip so any bias in the measurements is small (i.e. about ± .005 miles)
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Error Sources (cont.) I did a scale factor calibration by driving 28 miles, according to mileage markers on Interstate 95, and in both directions I recorded 27.425 miles on my odometer Thus, the scale factor is 27.425 odometer indicted miles S . F . = = .980 28 actual miles And any error in the scale factor due to readout resolution is .025 ≅ ± .0006 e SF = ± 2 28
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Recorded Data Trip Number Mileage Reading S.F. Corrected Mileage reading 1 7.825 7.985 2 7.850 8.010 3 7.875 8.036 4 7.900 8.061 5 7.850 8.010 6 7.825 7.985 7 7.875 8.036 8 7.850 8.010 9 7.875 8.036 10 7.825 7.985
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Mileage Data Analysis My system model is that the route distance is constant To minimize the effect of random errors take the sample mean (average) of the data to obtain an estimate n d ˆ = 1 d i = 8.015 miles n i = 1
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Mileage Data Analysis (cont.) Variations of the individual measurements, about this estimate are e i = d i d ˆ The sample mean of these variations is n n 1 n e ˆ = e i = 1 ( d i d ˆ ) = 0 i = 1 n i = 1 So the estimate is unbiased
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Mileage Data Analysis (cont.)
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