Professor Joseph Kroll Dr. Jose Vithayathil University of Pennsylvania 19 January 2005 Physics...
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Transcript of Professor Joseph Kroll Dr. Jose Vithayathil University of Pennsylvania 19 January 2005 Physics...
Professor Joseph Kroll
Dr. Jose Vithayathil
University of Pennsylvania
19 January 2005
Physics 414/521 Lecture 1
19 January 2005 Physics 414/521 - Lecture 1 2
Outline
• Standard units
• Discussion of errors– statistical
– systematic
– reminder about error propagation
• Mean & Variance
19 January 2005 Physics 414/521 - Lecture 1 3
Standard Units (SI)
SI = Système Internationale = International System of Units
see http://physics.nist.gov/cuu/Units/units.html
19 January 2005 Physics 414/521 - Lecture 1 4
Examples of Definitions of Standard Units
• Length– 1 meter = length of path travelled by light in vacuum in
1/299,792,458 seconds
– speed of light in vacuum = c = 299,792,458 m/s exactly
• Time– 1 second = 9,192,631,710 periods of radiation corresponding to
transition between two hyperfine levels of ground state of Cs-133
– hyperfine level due to interaction of electron spin and nuclear spin Cesium-133: 55 electrons, 54 in stable shells, 55th in outer shell not disturbed by inner electrons
– see http://tycho.usno.navy.mil/cesium.html (Cesium clocks)
• Mass– 1 kilogram = mass of standard Platinum-Iridium cylinder
19 January 2005 Physics 414/521 - Lecture 1 5
Measurements & Errors
Consider 3 measurements of speed of light c:1. 3 m/s2. 2.96 m/s3. 2.9013 m/s
Which measurement is the best measurement?
19 January 2005 Physics 414/521 - Lecture 1 6
Measurements & Errors (cont.)
Depends on what we mean by best
Accuracy: how close we are to true value
Precision: how exactly is the result measured – this quantityis usually what we are trying to estimate with our “error.”
3 m/s is the most accurate
but significant figures implies 2.9013 is the most precise
Without an error you can not evaluate a measurementaside: is this a measurement in vacuum?
19 January 2005 Physics 414/521 - Lecture 1 7
Errors
Report measurement of “a” as a § a
a represents estimate of uncertainty on measurement – also use a & a as notation for uncertainty
Classify errors as one of two types:1. Statistical (Random)2. Systematic
Reported error may include both statistical and systematicor they may be reported separately: a § astat § asyst
19 January 2005 Physics 414/521 - Lecture 1 8
Statistical Errors
Statistical: often called “random” error– improves (gets smaller) with additional measurement
Example: determination of the half-life of a radioactive substance
Count number of disintegrations N in a fixed amount of time– this single experiment provides an estimate of the half-life– repeat several times: improve the measurement statistically– in this type of example error scales with√ N– we will examine quantitatively later
19 January 2005 Physics 414/521 - Lecture 1 9
Systematic Errors
• Come from a variety of sources– measurement instrument
• e.g., improperly calibrated measurement device
– procedure • e.g., may need model to interpret data – what happens if you try a
different model? (will see an example later)
– mistakes
• Often difficult to estimate– if you can estimate them – may find a way to eliminate them
• May not scale (get smaller) with more statistics– but sometimes do have a statistical component
• e.g., calibration of measurement instrument may be based on limited statistics data sample – more calibration data – more precise calib.
19 January 2005 Physics 414/521 - Lecture 1 10
Error Propagation
If we have two measurements: a § a & b § bWhat is the error on quantity f = f(a,b)?
The error on f (fa) from a:
The error on f (fb) from b:
The total error on f (f) from a & b:
n.b., assumes errors are uncorrelated!
19 January 2005 Physics 414/521 - Lecture 1 11
Error Propagation (cont.)
This is called “adding errors in quadrature”
Some examples:
19 January 2005 Physics 414/521 - Lecture 1 12
Error Propagation (cont.)
Again: previous formulas assumed no correlations,that is, a and b are independent (uncorrelated)
This might not be true
Example: measuring an area of rectangle: A = ab
a and b independent:
a and b fully (100%) correlated:Error islarger!
19 January 2005 Physics 414/521 - Lecture 1 13
Error Progagation (cont.)
What about a ratio r = b/a?
If a & b fully correlated: r increases or decreases ?
With unknown systematics it is often better to report resultas a ratio
19 January 2005 Physics 414/521 - Lecture 1 14
Mean and Variance
How to combine i = 1, …, n measurements ai of the same quantity?
Definition: Average or Mean <a>
Definition: Variance s
Here is the true value of quantity a
19 January 2005 Physics 414/521 - Lecture 1 15
More on Variance
Usually you don’t know the true value :Use your best estimate: the mean <a>
note with a little algebriac manipulation:
N-1 for unbiasedestimate