Basic Error Analysis. Physics 401 Fall 2018 Eugene V Colla

Transcription

1 Basic Error Analysis Physics 401 Fall 2018 Eugene V Colla

2 Errors and uncertainties The Reading Error Accuracy and precession Systematic and statistical errors Fitting errors Appendix. Working with oil drop data Nonlinear fitting physics 401 2

3 T = 63 ±? Best guess T~0. 5 Wind speed 4mph±? Best guess ±0. 5mph physics 401 3

4 Clearance fit physics 401 4

5 Measurement of the speed of the light 1675 Ole Roemer: 220,000 Km/sec Does it make sense? What is missing? Ole Christensen Rømer NIST Bolder Colorado c = 299,792,456.2±1.1 m/s. physics 401 5

6 L 0. 5mm L=53mm±ΔL(?) Acrylic rod L 0. 03mm How far we have to go in reducing the reading error? We do not care about accuracy better than 1mm If ruler is not okay, we need to use digital caliper Probably the natural limit of accuracy can be due to length uncertainty because of temperature expansion. For 53mm L mm/K Reading Error = ± 1 2 (least count or minimum gradation). physics 401 6

7 Fluke 8845A multimeter Example Vdc (reading)=0.85v V = = 22μV physics 401 7

8 The accuracy of an experiment is a measure of how close the result of the experiment comes to the true value Precision refers to how closely individual measurements agree with each other physics 401 8

9 physics 401 9

10 Systematic Error: reproducible inaccuracy introduced by faulty equipment, calibration or technique. Random errors: Indefiniteness of results due to finite precision of experiment. Measure of fluctuation in result after repeatable experimentation. Philip R. Bevington Data Reduction and Error Analysis for the Physical sciences, McGraw-Hill, 1969 physics

11 Sources of systematic errors: poor calibration of the equipment, changes of environmental conditions, imperfect method of observation, drift and some offset in readings etc. Example #1: measuring of the DC voltage R in Current source I R U E off E off =f(time,temperature) expectation U=R*I U = actual result R I R R in 1 + R E off physics 401R in 11

12 Example #3: poor calibration Measuring of the speed of the second sound in superfluid He4 20 Published data T l =2.17K 15 LHe Resonator U 2 (m/s) 10 P403 results T l =2.1K 10mA 5 HP34401A DMM T (K) Temperature sensor physics

13 Result of measurement Systematic error X meas = X true + e s + e r Correct value Random error 0.35 B 0.35 e s B e s = P 0.15 P X true X i X X i true physics

14 Pn t ) rt ) n! n e rt n 0,1,2,... r: decay rate [counts/s] t: time interval [s] Pn(rt) : Probability to have n decays in time interval t Siméon Denis Poisson ( ) A statistical process is described through a Poisson Distribution if: rt=1 0.3 rt=4 0.2 P rt= random process for a given nucleus probability for a decay to occur is the same in each time interval. o universal probability the probability to decay in a given time interval is same for all nuclei. o no correlation between two instances (the decay of on nucleus does not change the probability for a second nucleus tophysics decay o number of counts 20

15 ) rt ) rt Pn t e n n! n 0,1,2,... r: decay rate [counts/s] t: time interval [s] P n (rt) : Probability to have n decays in time interval t Properties of the Poisson distribution: rt=10 < n >= rt σ = rt n n0 P ( rt) 1, probabilities sum to 1 P 0.1 n n P ( rt) rt, the mean n0 n number of counts ( ) 2 n n P ( ), n0 n rt rt standard deviation physics

16 ) rt ) rt Pn t e n n! n 0,1,2,... Poisson and Gaussian distributions 0.1 probability of occurence "Poisson distribution" "Gaussian distribution" Carl Friedrich Gauss ( ) number of counts 1 Pn ( x) e 2 ( xx) Gaussian distribution: continuous physics

17 2 x 1 Pn ( x) e 2 ( xx) Error in the mean is given as N number od events σ N, physics

18 Source of noisy signal Expected value 5V Actual measured values physics

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20 Result U x c N - standard deviation N number of samples For N=10 6 U=4.999± % accuracy

21 Ag b decay Model 108 Ag t 1/2 =157s 110 Ag t 1/2 =24.6s ExpDec2 Equation y = A1*exp(-x/t1) + A2*exp(-x/t2) + y0 Residuals 0 Count Reduced Chi-Sqr Adj. R-Square Value Standard Error C y C A C t C A C t time (s) time (s) Count 20 Model Gauss Equation y=y0 + (A/(w*sq rt(pi/2)))*exp(-2 *((x-xc)/w)^2) Reduced Chi-S qr Adj. R-Square Value Standard Error Counts y Counts xc Counts w Counts A Counts sigma Counts FWHM Counts Height t t y A1 exp A2 exp y t t Residuals physics

22 40 Ag b decay Test 1. Fourier analysis 20 Residuals time (s) Count 20 Model Gauss Equation y=y0 + (A/(w*sq rt(pi/2)))*exp(-2 *((x-xc)/w)^2) Reduced Chi-S qr Adj. R-Square Value Standard Error Counts y Counts xc Counts w Counts A Counts sigma Counts FWHM Counts Height No pronounced frequencies found Residuals physics

23 40 20 Ag b decay Test 1. Autocorrelation function Residuals time (s) Count 20 Model Gauss Equation y=y0 + (A/(w*sq rt(pi/2)))*exp(-2 *((x-xc)/w)^2) Reduced Chi-S qr Adj. R-Square Value Standard Error Counts y Counts xc Counts w Counts A Counts sigma Counts FWHM Counts Height Correlation function M 1 y( m) f ( n) g( n m) n Residuals autocorrelation function M 1 y( m) f ( n) f ( n m) n0 physics

24 Ag b decay Clear experiment Data + noise t 1 (s) t 2 (s) physics

25 Ag b decay Histogram does not follow the normal distribution and there is frequency of is present in spectrum physics

26 Ag b decay Autocorrelation function Conclusion: fitting function should be modified by adding an additional term: t t y( t) y A exp A exp A3 sin( t ) t t physics

27 FFT autocorrelation Clear experiment Data + noise Modified fitting t 1 (s) t 2 (s) physics

28 time In general we could expect both components of errors Q meas = Q true + e s + e r e s - systematic error comes from uncertainties of plates separation distance, applied DC voltage, ambient temperature etc. V =V DC ±DV, d=d 0 ±Dd e r - random errors are related to uncertainty of the knowledge of the actual t g and t rise. Uncertainty of time of crossing the marker line. It is random. physics

29 Q F S T 1 9d Systematic error 3 2 x 3 2 fc V g t t g g t rise F 1 t g 1 32 f c g 1 2 S 3 3 9d 2 x T V g t t g g t rise dq dq dq dq dq DQ F S T S T df D D D D D ds dt ds dt ) ) ) ) ) DS DT FT ) DS ) FS ) DT ) Q S T 10/1/2018 Physics

30 Systematic error 2 2 DS DT DQ Q S T DS Dd DV 3 Dx 3 D 1 D 1 Dg Dd 3 Dx S d V 2 x g d 2 x DT Dt Dt t t t t t g rise g g rise g rise 10/1/2018 Physics

31 Step 1. Origin Project For Raw Data : \\engr-file-03\phyinst\apl Courses\PHYCS401\Students\2. Millikan Raw Data All these projects with raw data should be stored in: \\engr-file-03\phyinst\apl Courses\PHYCS401\Students\ 2. Millikan Raw Data Here should only the files with raw data but not other files which you using for calculations. All other files you can save in your personal folder physics

32 Step 2. Working on personal Origin project Make a copy of the Millikan1 project to your personal folder and open it Prepare equations calculations of data in next columns (Set column values ). Switch Recalculate in Auto mode Paste these 5 parameters and raw data from Section L1-L4.opj projects Calculate manually the actual air viscosity physics

33 Millikan oil drop experiment Step 3. Histogram graph First use the data from the column with drop charges and plot the histogram physics

34 Step 4. Histogram. Bin size Origin will automatically but not optimally adjust the bin size h. In tis page figure h=0.5. There are several theoretical approaches how to find the optimal bin size. 3. 5σ h = n 1/3 Is the sample standard deviation and n is total number of observation. For presented in Fig.1 results good value of h ~0.1 Millikan oil drop experiment h=5 physics

35 Step 4. Histogram. Bin size Millikan oil drop experiment To change the bin size click on graph and unplug the Automatic Binning option Bin size in this histogram is 0.1 physics

36 Step 4. Find the bin Worksheet Millikan oil drop experiment Right click on histogram and choose Go to Bin Worksheet. physics

37 Step 5. Add Counts vs bin plot physics

38 Step 5. Multipeak Gaussian fitting Millikan oil drop experiment This plot can be used for peak fitting. physics

39 Step 5. Multipeak Gaussian Fitting For 1 st peak x c ~0.882±0.007 physics

40 (x i,y i ) is an experimental data array. x i is an independent variable and y i - dependent f(x,b) is a model function and b is the vector of fitting (adjustable) parameters The goal of the fitting procedure is to find the set of parameters which will generate the function f closest to the experimental points. To reach this goal we will try to minimize the sum of squared deviation function (S): m i1 b 2 S( b) y f ( x, ) i i physics

41 The goal of fitting is not only to find the curve best matching the experimental data but also to find the corresponding parameters which in majority cases are the important physical parameters There are several known mathematical algorithms for optimizing these parameters but in general the fitting procedure could have not only unique solution and the choice of initial parameters is very important issue m i1 b 2 S( b) y f ( x, ) i i physics

42 b i 1 40 b i 2 S(bi) 20 0 Local minimum Main minimum bi Let we have the S function dependent on parameter b i as shown on this graph physics