PHY 335 Data Analysis for Physicists

Transcription

1 PHY 335 Data Analysis or Physicists Instructor: Kok Wai Ng Oice CP 171 Telephone e mail: kng@uky.edu Oice hour: Thursday 11:00 12:00 a.m. or by appointment Time: Tuesday and Thursday 9:30 10:45 Place: CP 297 Course eb page: pa edu/~kng/all2010 Text book: An Introduction to Error Analysis 2 nd edition by John R. Taylor University Science Books Make sure you have received a course syllabus.

2 PHY520: Introduction to Quantum Mechanics 1. Grading policy Homeork 50% 2 Tests 30% Final examination 20% Total 100% 2. Final examination ill be on Dec 14 (Tuesday) 10:30 to 12:30 pm in CP 297

3 Goal In this course e ill learn the standard techniques commonly used by scientists in dierent areas to handle the errors generated in a measurement. This ill allo us to make legitimate argument rom a not so perect set o data. Also, useul inormation is oten hidden behind the uncertainty. It can only be recovered by statistical methods.

4 Topics Part 1 Single measurement 1. Basic stu (Chapter 1 and 2) 2. Propagation o uncertainties (Chapter 3) Part 2 Multiple measurements as independent results 1. Mean and standard deviation (Chapter 4) 2. Basic on probability distribution unction (not in text explicitly) 3. The Binomial distribution (Chapter 10) 4. The Poisson distribution (Chapter 11) 5. Normal distribution (irst hal o Chapter 5) 6. χ 2 test ho ell does the data it the distribution model? (Chapter 12) Part 3 Multiple measurements as one sample 1. Central limit theorem (not in text explicitly) 2. Normal distribution (second hal o Chapter 5) 3. Propagation o error (Chapter 3) 4. Rejection o data (Chapter 6) 5. Merging to sets o data together (Chapter 7) Part 4 Dependent variables 1. Curve itting (Chapter 8) 2. Covariance and correlation (Chapter 9)

5 Errors There are to types o error: 1. Systematic error 2. Random error

6 Systematic Errors Systematic error is a ixed oset that is reproduced in every independent measurement: 1. It is diicult to realize systematic errors. 2. Methods to realize systematic errors: a. Measure a standard source. b. Compare data ith ihothers doing the same measurement. c. Observe and think. 3. Systematic error is easy to deal ith once it is realized.

7 Examples o systematic error

8 Random Errors Random error causes luctuation (random) in data. It cannot be reproduced in every independent measurement: 1. Random error can be easily realized, though e may not kno its source. 2. Methods to reduce random errors: a. Make the environment quieter. b. Improve equipment or use another method. 3. Random errors cannot be indeinitely reduced to zero because there are limits set by natural processes like quantum mechanics. 4 To deal ithrandom errors require special knoledge hich e 4. To deal ith random errors require special knoledge, hich e ill learn rom this class.

9 Example o Random Errors

10 Uncertainties 1. Uncertainties are caused by a. Errors. b. Resolution o equipment. 2. A measured value can never be exact, because o uncertainties. 3 We use a range to represent the most likely occurrence o the 3. We use a range to represent the most likely occurrence o the true value. Example: L = 413 ±3 m means the true value is most likely beteen 410m and 416m. We ill understand its exact meaning as e learn more.

11 Signiicant Figures 1. Signiicant igures is a compact ay to express the uncertainty in a measurement that is commonly accepted by the scientiic community. 2. In this convention, the uncertainty is ±50% o a unit o the least signiicant digit. For example, i you rite L=413m, you actually means L= 413 ± 0.5 m, or L lies beteen 412.5m and 413.5m. 3. In most cases, the number o signiicant igures is obvious. For example, 1234s has 4 signiicant igure s has 7 signiicant igures. 4. Conusion may arise hen there are sequence o zeros in ront or ater the number.

12 Rules in Counting Signiicant Figures Case 1. When there is a decimal a. All zeros at the let, even ith the decimal among them, have no signiicance. For example: s has 3 signiicant igures m m has5 signiicant igures b. All zeros at the right, even ith the decimal among them, have signiicance. For example: s has 6 signiicant igures m has 5 signiicant igures g has 8 signiicant igures Case 2. When there is no decimal a. Allzeros at the let have no signiicance (there is no need or these zeros). For example: s=456s has 3 signiicant igures b. When the zeros are at the right, you cannot tell! For example: kg can have 3, 4, 5, 6, or 7 signiicant igures. Some people put a mark at the zero o least signiicance. ii For example: kg has 5 signiicant igures. This ambiguity ill not be there i e use scientiic notation, because there is alays a decimal in scientiic notation (case 1 above).

13 Scientiic Notation In scientiic notation, e rite a number in the orm o k 10 n, here k is a decimal number beteen 0 and 10 (0 k < 10) and the exponent n is an integer. Example: s = s has 3 signiicant igures m = m has 5 signiicant igures s = m has 6 signiicant igures m = m has 5 signiicant igures g = g has 8 signiicant igures s= s has 3 signiicant igures kg = kg i it has 3 signiicant igures = kg i it has 4 signiicant igures = kg i it has 5 signiicant igures = kg i it has 6 signiicant igures = kg i it has 7 signiicant igures

14 Precision o error Rule There is not such thing as precision o uncertainty or error, so e should limit the uncertainty to 1 signiicant igure. For this reason, scientiic notation is enough or most situation. Examples: s = s = (4.56 ± 0.005) 10 4 s m = m = ( ± ) 10 5 m g = g = ( ± ) ) 10 4 g I you do not ant to express your uncertainty as a 5, you need to use the ± to express it explicitly, but make sure it has only one (at most 2) signiicant igure. Examples: ( ± ) 10 5 m ( ± ) 10 4 g

15 Percentage error I you are measuring the length o a road o several miles long, an error o one inch is a very accurate measurement. Hoever, the same error is not acceptable in the measurement o an inch long string. So it makes sense to compare the error to the actual value: Fraction uncertainty = x x or x Percentage error = 100% x In terms o signiicant igures: Given a value o n signiicant igures, depending on its actual value, the raction error is beteen n and (n+1).

16 Propagation o error or single variable unction I variable x has a measured value o x 0 and uncertainty x (i.e. x = x 0 ± x), ho ill this aect the uncertainty o a unction (x) that depends on the value o x? d = x d x d = (x0) ± x d x Do not orget to round o your calculation o to one signiicant igure, and then round o to the appropriate p unit.

17 Example The height hihthh o a building is measured by the angle θ. θ is measured at 35 o but tit has an uncertainty o ±1 o. Assuming there is no uncertainty in measuring the horizontal distance. What is h and its uncertainty? h 200m θ =1 1 o = rad h = 200 tan θ h = 200 tan 35 h = 140 ± 50 m o 35 o h = 200 d tan θ dθ dθ 2 = 200 sec θ d θ = (200 sec = 52.16m 2 35 = m 140m o 50m ) (0.175)

18 Propagation o error or multivariable unction No consider a multivariable unction (u, v,, ). I measurements o u, v,,. All have uncertainty u, v,,., ho ill this aect the uncertainty o the unction? = L text) o (Equation (3.48) v u u ± = L v u u ),..., v, (u This orks or cases like systematic errors, hen the errors o most o the variables have the same sign. For cases like random errors, this overestimate and give an upper bound o the actual error: bound o the actual error: W ill t d th d l t i th v u u L We ill study the case o random error later in the course.