Basic Error Analysis. Physics 401 Spring 2015 Eugene V Colla

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1 Basic Error Analysis Physics 401 Spring 2015 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 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

of accuracy can be due to length uncertainty because of

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 401 R in 11

17K 15 LHe Resonator U 2 (m/s) 10 P403 results T l =2.

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

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 0.30 0.25 e s =0 0.30 0.25 0.

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

) 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 Siméon Denis Poisson (1781-1840) P 0.3 0.2 0.1 0.

14 ) 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 Siméon Denis Poisson ( ) P rt=1 rt=4 rt= number of counts A statistical process is described through a Poisson Distribution if: o 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 to decay. physics

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

) rt ) rt Pn t e n n! n 0,1,2,... Poisson and Gaussian distributions 0.1 probability of occurence 0.08 0.06 0.04 0.

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 σ N physics

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

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 y = f(x1, x2... xn) 2 n f 2 i i i i1 xi f ( x, x ) x 1.15 f(x i ) f±fx 1.10 x i ± x i x i physics

29 Derive resonance frequency f from measured inductance L± L and capacitance C± C L 1 1 f ( L, C) 2 LC 10 1mH, C 10 2μF f f f ( L, C, L, C) L C L C 2 2 f L f C 1 C L 1 L C ; Results: f(l 1,C 1 )= Hz f= hz f(l 1,C 1 )=503.3±56.3Hz physics

30 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 ±V, d=d 0 ±d 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

31 d 2 x Q F S T 32 fc V g t t g g t rise ) ) ) Q S T F F T S F S T T t t g g t rise T t t t t t t t g rise g g rise g rise physics

$Step 1. Collect your data + parameters of the experiment in: \\Phyaplportal\PHYCS401\Common\Origin templates\oil drop experiment\section L1.opj Use different columns for each student or team.$

32 Step 1. Collect your data + parameters of the experiment in: \\Phyaplportal\PHYCS401\Common\Origin templates\oil drop experiment\section L1.opj Use different columns for each student or team. This Origin project is for data collecting only but not for data analysis. For data analysis you have to copy these data and experiment parameters obtained by different students/team and paste it in one in your personal Origin project. Setup and environmental parameters physics 401 Raw data 32

folder and open it Prepare equations calculations of data in next columns (Set column

33 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

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

Step 4. Histogram. Bin size Millikan oil drop experiment Origin will automatically but not optimally adjust the bin size h. In tis page figure h=0.5.

35 Step 4. Histogram. Bin size Millikan oil drop experiment 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 physics

36 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

37 Step 4. Multipeak Gaussian fitting Millikan oil drop experiment To do this you have to add an extra plot to the graph Counts vs. Bin Center physics

38 Step 4. Multipeak Gaussian fitting Millikan oil drop experiment This plot can be used for peak fitting. physics

39 Step 4. Multipeak Gaussian fitting Millikan oil drop experiment This plot can be used for peak fitting. Final result for first two peaks: Q/e=0.93±0.01 Q/e=1.87±0.02 This pretty close to e and 2e Here w = 2 and error of the mean = σ N physics

Basic Error Analysis. Physics 401 Fall 2017 Eugene V Colla

Basic Error Analysis. Physics 401 Fall 2017 Eugene V Colla Basic Error Analysis Physics 401 Fall 2017 Eugene V Colla Errors and uncertainties The Reading Error Accuracy and precession Systematic and statistical errors Fitting errors Appendix. Working with oil