Professor Joseph Kroll

Dr. Jose Vithayathil

University of Pennsylvania

19 January 2005

Physics 414/521 Lecture 1

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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)

SI = Système Internationale = International System of Units

see http://physics.nist.gov/cuu/Units/units.html

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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:1. 3 m/s2. 2.96 m/s3. 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 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?

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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 systematicor 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 with√ 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 § 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!

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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:

a and b fully (100%) correlated:Error islarger!

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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 resultas 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 <a>

Definition: Variance s

Here is the true value of quantity a

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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