1. Professor Joseph Kroll Dr. Jose Vithayathil University of Pennsylvania 19 January 2005 Physics 414/521 Lecture 1

2. 19 January 2005Physics 414/521 - Lecture 12 Outline Standard units Discussion of errors –statistical –systematic –reminder about error propagation Mean & Variance

3. 19 January 2005Physics 414/521 - Lecture 13 Standard Units (SI) SI = Système Internationale = International System of Units see http://physics.nist.gov/cuu/Units/units.html

4. 19 January 2005Physics 414/521 - Lecture 14 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, 55 th in outer shell not disturbed by inner electrons –see http://tycho.usno.navy.mil/cesium.html (Cesium clocks)http://tycho.usno.navy.mil/cesium.html Mass –1 kilogram = mass of standard Platinum-Iridium cylinder

5. 19 January 2005Physics 414/521 - Lecture 15 Measurements & Errors Consider 3 measurements of speed of light c: 1. 3 m/s 2. 2.96 m/s 3. 2.9013 m/s Which measurement is the best measurement?

6. 19 January 2005Physics 414/521 - Lecture 16 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?

7. 19 January 2005Physics 414/521 - Lecture 17 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 §  a stat §  a syst

8. 19 January 2005Physics 414/521 - Lecture 18 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

9. 19 January 2005Physics 414/521 - Lecture 19 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.

10. 19 January 2005Physics 414/521 - Lecture 110 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 (  f a ) from  a: The error on f (  f b ) from  b: The total error on f (  f) from  a &  b: n.b., assumes errors are uncorrelated!

11. 19 January 2005Physics 414/521 - Lecture 111 Error Propagation (cont.) This is called “adding errors in quadrature” Some examples:

12. 19 January 2005Physics 414/521 - Lecture 112 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 is larger!

13. 19 January 2005Physics 414/521 - Lecture 113 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

14. 19 January 2005Physics 414/521 - Lecture 114 Mean and Variance How to combine i = 1, …, n measurements a i of the same quantity? Definition: Average or Mean Definition: Variance s Here  is the true value of quantity a

15. 19 January 2005Physics 414/521 - Lecture 115 More on Variance Usually you don’t know the true value  : Use your best estimate: the mean note with a little algebriac manipulation: N-1 for unbiased estimate