Professor Joseph Kroll

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Physics 414/521 Lecture 1

• Professor Joseph Kroll
• Dr. Jose Vithayathil
• University of Pennsylvania
• 19 January 2005

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Outline

• Standard units
• Discussion of errors
• statistical
• systematic
• reminder about error propagation
• Mean Variance

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Standard Units (SI)
see http//physics.nist.gov/cuu/Units/units.html
SI Système Internationale International
System of Units
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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

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Measurements Errors

• Consider 3 measurements of speed of light c
• 3 m/s
• 2.96 m/s
• 2.9013 m/s

Which measurement is the best measurement?
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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 quantity is 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
measurement
aside is this a measurement in vacuum?
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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
systematic or they may be reported separately a
? astat ? asyst
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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 withv N we
will examine quantitatively later
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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.

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Error Propagation
If we have two measurements a ?a b ?b What
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!
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Error Propagation (cont.)
This is called adding errors in quadrature
Some examples
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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
Error is larger!
?a and ?b fully (100) correlated
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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 result as a ratio
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Mean and Variance
How to combine i 1, , n measurements ai of the
same quantity?
Definition Average or Mean ltagt
Definition Variance s
Here ? is the true value of quantity a
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More on Variance
Usually you dont know the true value ? Use your
best estimate the mean ltagt
N-1 for unbiased estimate
note with a little algebriac manipulation