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 !
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)
x10-5

8. The Poisson Distribution at Large rt
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 f(x) x Error propagation for one variable:
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 …
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
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.

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
Error
Value
Method to determine
Synchronization of the reconvergence of the split beams
Instability of interference pattern due to air fluctuations
xx % in reading high/low
Stability of reading vs time
Beam splitter imperfections
xx % on visibility

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 systematics In 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