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Basic Error Analysis Considerations

Basic Error Analysis Considerations. Phys 403 Spring 2011. Error analysis could fill a whole course. The few thoughts presented here are meant as a starting point for you to learn more on your own.

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Basic Error Analysis Considerations

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  1. Basic Error Analysis Considerations Phys 403 Spring 2011 Error analysis could fill a whole course. The few thoughts presented here are meant as a starting point for you to learn more on your own

  2. Sofar:we have not trained you well for error analysis, typically preferring that you “get” the main physics idea first • In “real life”: a result without error has no interpretation and therefore is no result! • Every lab report (and paper ) should have a section on uncertainties • Two types you should identify • Systematic • Instrumentation measurement precision; alignment uncertainty, temperature stability, sample purity, lens properties, … • To reduce uncertainty, identify largest source(s) and make dedicated fixes and improvements • Statistical • When the error is based on the number of events (N) or entries; error usually scales as 1/sqrt(N) • To reduce uncertainty, collect more data

  3. Accuracy versus Precision • The accuracy of an experiment is a measure of how close the result of the experiment comes to the true value. • It is a measure of the “correctness” of the result • The precision of an experiment is a measure of how exactly the result is determined (without reference to what the result means) • Absolute precision – same units as value • Relative precision – fractional units of value Precision vs Accuracy Source: P. Bevington, Data Reduction and Error Analysis for the Physical Sciences

  4. Example for Statistical Errors + Error Propagation  Life time Measurement ! Measurement: o observe radioactive decay o measure counts/∆t vs t o exponential fit to determine decay constant (or life time)

  5. The Poisson Distribution I r: decay rate [counts/s] t: time inteval [s]  Pn(rt) : Probability to have n decays in time interval t! r A statistical process is described through a Poisson Distribution if: o random process for a given nucleus probablility 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.

  6. The Poisson Distribution II r: decay rate [counts/s] t: time inteval [s]  Pn(rt) : Probability to have n decays in time interval t! Is nuclear decay a random process? Yes, follows Poisson Distribution! (Rutherford and Geiger, 1910) r

  7. The Poisson Distribution III r = decay rate = 10 counts/s t: time inteval = 1 s  Pn(rt) : Probability to have n decays in time interval t r n Pn(rt=10) 0 4.5x10-5 5 0.038 10 0.125 15 0.035 20 0.002

  8. The Poisson Distribution at Large rt Poisson distribution: discrete Gaussian distribution: continous

  9. Measured Count Rate and Errors <n> true average count rate with σ Single measurement Nm=rmt=24 And σm=√Nm=√24=4.5 Experimental result <n>=24 +/- 4.5

  10. Propagation of Errors Error propagation for one variable: f(x) Error propagation for two variables: x

  11. Example I, Error on Half-Life Propagate error in decay constant λ into half life:

  12. Example II, Rates for γγ Correlations Measured coincidence rate: S’ = S + B, Δ S’=√S’ Measured background rate: B, ΔB= √B Signal: S = S’ – B Error :

  13. Interpreting fitting results … • Some of you will be comparing a slightly low lifetime with the real one of 2.2 ms. Yours will usually be lower due to negative muon capture. Here, 0.73 ± 0.18; a bit “low” but okay. Too low means errors are underestimated Too high means fit is bad 2.08 ± 0.05 ms

  14. Let’s consider systematics … A Table is helpful; Consider how you might develop and fill in a table for the Quantum Eraser (or other optics) experiment

  15. From the γγ Experiment: • Systematic test of system count-rate stability • Large drifts , temperature effects from power dissipation in discriminators Coincidence Count Rates time in hours

  16. More systematicsIn Ferroelectric analysis … • Calibration of temperature as sample is being evaluated. How accurate is it? How well are phase transitions identified? • Quality of particular sample. Would 2nd , 3rd “identical” sample produce same results? • General reproducibility of traces for multiple paths. • Do results depend on speed of change in T? • How about the direction of the T change (heating vs cooling?) • How do you tell the difference between physics that depends on heating versus cooling compared to just instrumentation effects, like time lag ? • Temp lag between thermometer and sample…leads to ? • How do external factors like quality of the lead connections affect results? Are connections being evaluated too?

  17. Conclusions… • Uncertainties exist in all experiments • Learn to identify them and begin to believe that it is essential to understand them • Their origin • Their magnitude • Don’t fall in love with your first idea. • Test it; reject it. That’s okay. Move on to next one • Use the right number of sig. figs. and “look” at your numbers closely; they matter; the rest is words

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