Introduction to Data Analysis Analysis of Experimental Errors How to Report and Use Experimental Errors Statistical Analysis of Data Simple statistics of data Plotting and displaying the data Summary November 2001 10 001 Introduction to Computer Methods Errors and Uncertainties experimental error mistake experimental error blunder experimental error inevitable uncertainty of the measurement The measured value alone is not enough We also need the experimental error November 2001 10 001 Introduction to Computer Methods Testing the Theories and Models Experimental data should be consistent with you theory model and inconsistent with alternative ones to prove them wrong Example Bending of the light near the Sun 1 simplest classical theory 0 2 careful classical analysis 0 9 3 Einstein s general relativity 1 8 Solar eclipse needed to check Dyson Eddington Davidson year 1919 a 2 95 confidence 1 7 2 3 Consistent with 1 8 and inconsistent with 0 9 November 2001 10 001 Introduction to Computer Methods Types of Experimental Errors Random errors Systematic errors rotating table stopwatch voltage measurements measuring glass car performance calibration errors i revealed by repeating the measurements in general are invisible ii may be estimated statistically November 2001 detected by comparison with results of alternative method or equipment 10 001 Introduction to Computer Methods How to Report Use the Errors Experimental result measured value of x xbest x the best estimate for x uncertainty error margin of error xbest x x xbest x Usually November 2001 95 confidence is suggested 95 sure x inside the limits 5 chance x is outside the limits 10 001 Introduction to Computer Methods Some Basic Rules Experimental errors should be always rounded to one significant digit g 9 82 0 02385 wrong g 9 82 0 02 correct Thus error calculations become simple estimates Exception if the leading significant digit of the error is 1 keep one digit more Everest is November 2001 8848 1 5 m high 10 001 Introduction to Computer Methods Some Basic Rules The last significant figure in the answer should be the same order of magnitude as the uncertainty 92 8 0 3 92 3 90 30 During the calculation retain one more digit than is finally justified to decrease the error November 2001 10 001 Introduction to Computer Methods Error in a Sum or a Difference Two or more independently measured variables x xbest dx y ybest dy x x y y z x y x y z If x y z are large but x y z is small trouble Overall error is large November 2001 10 001 Introduction to Computer Methods Calculating Relative Errors relative error fractional uncertainty x x x x 50 1 cm 0 02 x x x 100000 1 cm 0 00001 x November 2001 10 001 Introduction to Computer Methods Relative errors of product and ratio of two variables x xbest x x x y ybest y y y xy xy xy xbest x ybest y xbest ybest xbest y ybest x xy xbest y ybest x xy x y xy ybest xbest For product xy the relative error is the sum of relative errors of x and y November 2001 10 001 Introduction to Computer Methods Relative errors of product and ratio of two variables x y x y xbest x xbest 1 x xbest x y ybest y ybest 1 y ybest xbest 1 x xbest xbest max ybest 1 y ybest ybest x y 1 xbest ybest xbest 1 x xbest xbest min ybest 1 y ybest ybest x y 1 xbest ybest x y x y x y ybest xbest November 2001 10 001 Introduction to Computer Methods 2 Simple Rules When the measured quantities are added or subtracted the errors add When the measured quantities are multiplied or divided the relative errors add November 2001 10 001 Introduction to Computer Methods Propagation of Errors We have upper bounds on errors for sum difference and product quotient of 2 measurables Can we do any better If errors are independent and random the errors are added in quadrature q x y q x y 2 2 x y q x z u w q x 2 z u w 2 2 x z u w November 2001 10 001 Introduction to Computer Methods 2 Propagation of Errors l1 5 3 0 2cm l2 7 2 0 2cm l l1 l2 l l1 l2 2 2 0 2 0 2 2 2 3mm l1 l2 2mm 2mm 4mm For 2 measurables there is no great difference but for n measurables the difference is 1 n November 2001 10 001 Introduction to Computer Methods Propagation of Errors Relative error of product quotient x z q u w q x z u w q x z u w x z u w x z u w 2 2 2 2 If the relative errors for n measurables are the same we gain 1 n in relative error November 2001 10 001 Introduction to Computer Methods Propagation of Errors D C electric motor V voltage I current work done by motor mgh efficiency e energy delivered to motor VIt relative error for m h V I 1 relative error for t 5 q m h V I t q m h V I t 2 2 2 2 2 1 1 1 1 5 29 5 2 2 2 2 2 m h V I t 1 1 1 1 5 9 I t m h V November 2001 10 001 Introduction to Computer Methods Propagation of Errors If many measurables the error is dominated by the one from the worse measurable e t e t What if measure x but need an error for f x Suggest that the error is small Do Taylor expansion of f about xbest df x x f x f xbest dx November 2001 10 001 Introduction to Computer Methods Propagation of Errors Relative error of a power q xn q x n q x General case function of several variables q x z q q q x z x z 2 q q q x z x z November 2001 10 001 Introduction to Computer Methods 2 Propagation of Errors Example measuring g with a simple pendulum L length T oscillation period T 2 L g 1 2 T T 2 T2 T 2 g 4 2 L T2 g L T 2 g L T 2 2 L 92 95 0 1 cm T 1 936 0 004 sec 4 2 92 95 cm 2 g best 979 cm sec 1 936 sec 2 g 2 2 0 1 2 0 2 0 4 L L 0 1 T T 0 2 g g 0 004 x 979 cm sec2 4 cm sec2 November 2001 10 001 Introduction to Computer Methods Statistical Analysis of Random Errors Random small Systematic small Random large Systematic small November 2001 Random small Systematic large Random large Systematic large 10 001 Introduction to Computer Methods Statistical Analysis of Random Errors Random small Systematic Random small Systematic Random large Systematic November 2001 Random large Systematic 10 001 Introduction to Computer Methods Statistical Analysis of Random Errors 1st case we knew the value of the measured …