PHYS2150 Lecture 3
First labs due Friday the 11th at 4pm.
•Comparing values
•Rejecting data
Comparing values
Rejecting
Data
•Weighted averages
Weighted averages
COMPARING WITH ‘KNOWN’
VALUE:
• Measure: g = [997 ± 32] cm/s2 = x± δx
Usually negligible
• Known value: 980.665 cm/s2 = x0± δx0
• Discrepancy is (x-x0)± δ(x-x0), where
• [δ(x-x0)]2 = (δx)2 + (δx0)2
• Significance of Discrepancy (in sigmas)
• Use erf table to determine agreement confidence level:
62% agreement: good!
• If the significance is less than 1.0 (measurement was
within one standard deviation of the correct one), then
the measured value is near enough to the known value
to be acceptable.
Comparing with Known Values
• Percent Error = | Measured – Accepted| x 100%
Accepted
•
This is more of a ball park estimate. In general, this quantity
should be less than 10% in this lab. This gives you another
way to look at how far apart the two quantities are.
ANOTHER EXAMPLE: e/m
ANOTHER EXAMPLE: e/m
artifact
ANOTHER EXAMPLE: e/m
ANOTHER EXAMPLE: e/m
Note: Systematic error comes out to be 0.064 which is much closer to
the std. dev. than the uncertainty on the mean. For only 14 data
points, std. dev. should be used, not uncertainty on the mean.
ANOTHER EXAMPLE: e/m
ANOTHER EXAMPLE: e/m
ANOTHER EXAMPLE: e/m
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•
•
•
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Compare measured value
[1.946 ± 0.019 (stat.) ± 0.064 (sys.)]x10-11 C/kg
to accepted value 1.75882 x 10-11 C/kg
Discrepancy from the accepted value is
|1.946x10-11 - 1.75882 x 10-11| C/kg = 0.187x10-11 C/kg
Significance of discrepancy = Discrepancy= 0.187 C/kg
σtotal
(0.019) 2 + (0.064) 2 C/kg
= 2.8 (often stated as “off by 2.8 sigma”)
If uncertainties calculated correctly, should get within 1 σ 68% of
time. Otherwise, something else is wrong.
ANOTHER EXAMPLE: e/m
•
What is the probability of value being further than 2.8σ from
the mean for a gaussian distribution?
•
Probability of being inside 2.8σ is P(2.8σ) = 99.49% from table
p.287 Taylor. (Check this in the table!)
•
Probability of being outside 2.8σ is (1- 0.9949) = 0.51%
•
It is very unlikely that a reasonably measured quantity would
be that far from the mean. Either a mistake or underestimated
systematic uncertainties.
•
The discrepancy between the measured and accepted value
was significant. (bad)
Note: If they had used the std. dev instead of the uncert. on mean, the
significance would be 1.97σ. Probably more correct. (still bad)
Note: If this happens to you, you will need to talk more in your report about what
sources of error might have caused this. Ask questions if you are not sure.
REJECTING DATA
•
BE VERY CAREFUL HERE! You are treading in the footsteps
of a long line of practitioners of pathological science.
•
Generally, you should have external reasons for cutting data
•
In reality, even that’s not enough: do you only go searching
for problems when you get a result you didn’t expect? Your
analysis is biased!
•
All prescriptions for deciding on the validity of data points are
controversial!
•
I’ll describe one that’s common in textbooks (but not in real
life): Chauvenet’s criterion. First, a cautionary tale.
HOW TO LOOK FOR A
PARTICLE
•
Those of you doing the K meson experiment have already
seen this
•
Look in high-energy collisions for events with multiple output
particles that could be decay products (displaced from
primary interaction, if particle is long-lived as with the K0)
•
Reconstruct a relativistic invariant mass from the momenta
of the decay products
•
Make a histogram of the masses from candidate events
•
Look for a peak, indicating a state of well-defined mass
ONE PEAK OR TWO?
•
CERN experiment in late 1960s
observed A2 mesons
•
Particle appeared to be a doublet
•
Statistical significance of split is very
high
•
There is really only one particle.
HOW DID IT HAPPEN?
• In an early run, a dip showed up. It was a statistical fluctuation,
but people noticed it and suspected it might be real.
• Subsequent runs were looked at as they came in. If no dip
showed up, the run was investigated for problems. There’s
usually a minor problem somewhere in a complicated
experiment, so most of these runs were cut from the sample.
• When a dip appeared, they didn’t look as carefully for a
problem.
• So an insignificant fluctuation was boosted into a completely
wrong “discovery.”
• Lesson: Don’t let result influence which data sets you use.
CHAUVENET’S CRITERION
• Assume data (xi; i=1,N) come from a gaussian distribution.
•Determine mean <x> and standard deviation σ.
•
Assuming gaussian distribution, calculate how many data points (fraction
times N) should be outside a particular number of standard deviations.
•
Pick a threshold
• Chauvenet says 0.5
• In real life you would probably want to use a smaller number)
• reject data points farther from the mean
•
So if you use 0.5 as your threshold, you would throw out data points beyond
a number of sigma where you expect only 0.5 data points.
CHAUVENET’S CRITERION
•
•
Example:
•
Have 10 data points.
•
Know that 5% of data points should be outside 2 σ, 0.5 data points.
•
So, throw out an event if it’s more than 2σ from the mean.
Can’t do this iteratively (recaculate mean and sigma with remaining data,
then throw out some more points)!
•
It would make your measurement meaningless.
A FINAL COMMENT ON
REJECTING DATA
• Rejecting data is a last resort. If you are suspicious of
a data point, you should first:
•
•
If you can, take more data!
•
Ask what could have gone wrong (but be careful that you
aren’t going on a witch hunt)
•
Is there a problem with the way you recorded/calibrated
that could cause this data point to be different?
•
Whatever you decide, lay it all out in the report, including the
result including the rejected data. Be honest about it.
NEVER reject data points because you don’t like the answer!
WEIGHTED AVERAGES
• If you measure the same thing twice, and the errors are
different, how do you combine the results?
• A proper averaging gives more weight to measurements
with smaller uncertainties (they are more “right”).
• The reported error on the average must reflect this in
some way.
• Weighted averages are the most basic form of fitting.
WEIGHTED AVERAGES
•
We showed that the mean of measurements gives the best
measurement of the true value
•
This assumed that each measurement had the same
uncertainty! How to combine when they are different?
•
Assume two measurements of g:
g1=(980.2±3.7) cm/s2
g2=(977.4±1.9) cm/s2
•
If unweighted, mean = 978.8 cm/s2, but this ignores the fact
that measurement 2 has smaller uncertainty so the true value
is likely to be closer to it than to measurement 1.
TURNING MAXIMUM LIKELIHOOD
ARGUMENT INTO A WEIGHTED
AVERAGE
Weights go as 1/σ2, so more precise measurements count much more
than less precise ones.
USING WEIGHTED
AVERAGES
Before averaging, it’s a good idea to check compatibility of the results:
• Same idea as discrepancy in Lecture 2.
• Two measured values: (xA ± σA) and (xB ± σB)
• Difference is |xA – xB|
⎛ |x −x |
A
B
• Significance of difference (in σ’s) is δ = ⎜⎜
2
2
σ
σ
+
A
B
⎝
⎞
⎟σ
⎟
⎠
• Can convert significance of difference in σ to probability using erf
(calculated or table)
Typically less than 1% probability implies measurements inconsistent
ARE OUR MEASUREMENTS
CONSISTENT?
The values are within their combined error bars.
TAKE THE AVERAGE:
• Straight average is 9.788 m/s2
• Weighted average 9.780±0.017
• Weighted average is much closer
to the more precise measurement
• Uncertainty is lower than on the
individual measurements: we have
more information in the weighted
average.
WEIGHTED AVERAGE:
WHEN?
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•
•
•
•
•
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NO, if:
You don’t know the intrinsic uncertainty, or
The intrinsic uncertainty is the same for all measurements
You need to calculate the standard deviation to determine
the uncertainty
YES, if:
You know the intrinsic uncertainties, AND
They are not the same for all measurements