Intermediate Lab PHYS 3870 Lecture 2 Defining Errors

Intermediate Lab PHYS 3870 Lecture 2 Defining Errors References: Taylor Ch. 1, 2, 3 Introduction Section 0 Lecture 1 Slide 1 Baird Problems (Ch 5 -See web site) Also refer to [HANDOUT] “Glossary of Important Terms in Error Analysis” INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 1

Problem Set #1 Due Next Week Introduction Section 0 Lecture 1 Slide 2 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 2

Introduction Section 0 Lecture 1 Slide 3 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 3

Useful Handout on Web Introduction Section 0 Lecture 1 Slide 4 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 4

What is Science? The scientific method goes further in: • Developing a description (model) of the system behavior based on observation • Generalizing this description (model) to other behavior and other systems • That is to say, the scientific method is experimentation and modeling intertwined Introduction Section 0 Lecture 1 Slide 5 • It is the scientific method that distinguishes science from other forms of endeavor INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 5

Scientific Method: Leads to new discoveries → how scientific progress is made! Careful measurements, Experiments Models, Empirical Introduction Section 0 Lecture 1 Slide 6 Laws, Generalization Hypothesis, Theory INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 6

Uncertainties in Observations Input Output SYSTEM The Universe • • Observations characterize the system to within the uncertainty of the measurements Uncertainties can arise from: Ø Limitations of instrumentation or measurement methods Ø Statistical fluctuations of the system Ø Inherent uncertainties of the system Introduction Section 0 Lecture 1 Slide 7 - Quantum fluctuations - Non-deterministic processes (e. g. , chaos): -INTRODUCTION TO Modern Physics PHYX 2710 There are systems where uncertainties dominate and preclude models Fall 2004 predicting the outcome - We will not (intentionally) deal with this type of system. Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 7

What is a Model? Introduction Section 0 Lecture 1 Slide 8 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 8

What is a Model? Introduction Section 0 Lecture 1 Slide 9 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 9

Intermediate Lab PHYS 3870 Designing “Good” Experiments What is “Good”? One that: Introduction Section 0 Lecture 1 Slide 10 • Gives “good” unambiguous results • Gets a “good” grade INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 10

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 11 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 11

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 12 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 12

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 13 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 13

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 14 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 14

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 15 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 15

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 16 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 16

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 17 Lecture 2 -Exp Design. xmcd INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 17

An Exercise in Experimental Design Is the linear fit the “best” model? Lecture 2 -Exp Design. xmcd Introduction Section 0 Lecture 1 Slide 18 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 18

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 19 INTRODUCTION TO Modern Physics PHYX 2710 Lecture 2 -Exp Design. xmcd Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 19

An Exercise in Experimental Design Introduction Section 0 Lecture 1 Slide 20 INTRODUCTION TO Modern Physics PHYX 2710 Lecture 2 -Exp Design. xmcd Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 20

Intermediate Lab PHYS 3870 Comparing Measurements to Models Qualitatively Introduction Section 0 Lecture 1 Slide 21 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 21

Intermediate Lab PHYS 3870 Dimensional Analysis Introduction Section 0 Lecture 1 Slide 22 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 22

Units and Dimensions Introduction Section 0 Lecture 1 Slide 23 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 23

Dimensional Analysis Introduction Section 0 Lecture 1 Slide 24 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 24

Units and Dimensions--A Whole New World! Introduction Section 0 Lecture 1 Slide 25 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 25

Introduction Section 0 Lecture 1 Slide 26 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 26

Units and Dimensions--A Whole New World! Introduction Section 0 Lecture 1 Slide 27 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 27

Units and Dimensions--A Whole New World! Introduction Section 0 Lecture 1 Slide 28 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 28

Intermediate Lab PHYS 3870 Graphical Analysis Introduction Section 0 Lecture 1 Slide 29 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 29

Graphical Analysis An “old School” approach to linear fits. • Rough plot of data • Estimate of uncertainties with error bars • A “best” linear fit with a straight edge • Estimates of uncertainties in slope and intercept from the error bars This is a great practice to get into as you are developing an experiment! Introduction Section 0 Lecture 1 Slide 30 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 30

Is it Linear? • A simple model is a linear model • You know it when you see it (qualitatively) • Tested with a straight edge • Error bar are a first step in gauging the “goodness of fit” Introduction Section 0 Lecture 1 Slide 31 Adding 2 D error bars is sometimes helpful. INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 31

Making It Linear or Linearization • A simple trick for many models is to linearize the model in the independent variable. Introduction Section 0 Lecture 1 Slide 32 • Refer to Baird Ch. 5 and the associated homework problems. INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 32

Linearizing Equations (1) Determine a value for mass. Introduction Section 0 Lecture 1 Slide 33 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 33

Linearizing Equations (2) Determine a value for the Rydberg constant, R. Introduction Section 0 Lecture 1 Slide 34 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 34

Special Graph Paper Semi-log paper tests for exponential models. Log-log paper tests for power law models. Both Semi-log and log -log paper are handy for displaying details of data spread over many orders of magnitude. Linear “Old School”Introduction Section 0 Lecture 1 Slide 35 graph paper is still a useful tool, especially for reality checking during the INTRODUCTION TO Modern Physics PHYX 2710 Semilog Fall 2004 experimental design process. Intermediate 3870 Fall 2013 DEFINING ERRORS Log-Log Lecture 2 Slide 35

Special Graph Paper Introduction Section 0 Lecture 1 Slide 36 INTRODUCTION TO Modern Physics PHYX 2710 Linear Fall 2004 Intermediate 3870 Fall 2013 Polar DEFINING ERRORS Lecture 2 Slide 36

Linearizing Equations and “Magic Graph Paper” Introduction Section 0 Lecture 1 Slide 37 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 37

Some Common Models Introduction Section 0 Lecture 1 Slide 38 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 38

Intermediate Lab PHYS 3870 Comparing Measurements to Models Errors as a Quantitative Tool Introduction Section 0 Lecture 1 Slide 39 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 39

Intermediate Lab PHYS 3870 Defining Errors Introduction Section 0 Lecture 1 Slide 40 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 40

What Is Error? • The term “error” does not mean mistake in science. • Rather, it means the inevitable uncertainty related to an observation or measurement of any physical quantity. • It is a best guess at the range of values of subsequent measurements. Example: x = 1. 0 0. 1 m This is shorthand for “the best estimate of x is 1. 0 m. Subsequent measurements of x will ‘almost certainly’ lie between 0. 9 m and 1. 1 m Introduction Section 0 Lecture 1 Slide 41 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 41

Why Are Errors Important? http: //www. math. nyu. edu/~crorres/Archime des/Crown. Intro. html Introduction Section 0 Lecture 1 Slide 42 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 42

A Timely Example of Errors in Measurements What time is it now? Introduction Section 0 Lecture 1 Slide 43 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 43

A Timely Example of Errors in Measurements Analysis with Mathcad sheet? Enter the data. Introduction Section 0 Lecture 1 Slide 44 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 44

A Timely Example of Errors in Measurements Analysis with Mathcad sheet? Introduction Section 0 Lecture 1 Slide 45 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 45

A Timely Example of Errors in Measurements Analysis with Mathcad sheet? What is the uncertainty? . Introduction Section 0 Lecture 1 Slide 46 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 46

A Timely Example of Errors in Measurements What time is it now? And the “best” answer is: ( 53 ± 9 ) sec [absolute error] or ( 53 sec ± 20% ) [relative (fractional) error] Introduction Section 0 Lecture 1 Slide 47 This is the best value and estimated uncertainty for a set INTRODUCTION TO Modern Physics PHYX 2710 of N measurements Fall 2004 of the time. Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 47

Summary of Stating Errors in Measurements Introduction Section 0 Lecture 1 Slide 48 Correct 1. 23± 0. 02 m (± 2%) INTRODUCTION TO Modern Physics PHYX 2710 Incorrect 1. 234567± 0. 024 m Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 48

Intermediate Lab PHYS 3870 Accuracy and Systematic Errors Introduction Section 0 Lecture 1 Slide 49 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 49

Random and Systematic Errors Precision is defined as a measure of the reproducibility of a measurement Such errors are called random (statistical) errors. If an experiment has small random error, it is said to have high precision. Accuracy is a measure of the validity of a measurement. If an experiment has small systematic error, it is said to have high accuracy. Introduction Section 0 Lecture 1 Slide 50 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 50

Accuracy and Systematic Error Consider 4 “dart experiments” Which experiments are precise (have good reproducibility or low random error)? Which experiments are accurate (are close to “true” result or have low systematic error)? Introduction Section 0 Lecture 1 Slide 51 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 51

Precision and Accuracy Introduction Section 0 Lecture 1 Slide 52 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 52

Accuracy and Systematic Error Introduction Section 0 Lecture 1 Slide 53 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 53

A Timely Example of Errors in Measurements What time is it? Log on to the National Institute of Standards and Technology (NIST) web site. GOOD: A low resolution time stamp to 0. 3 s http: //nist. time. gov/ BETTER: “The NIST servers listen for a NTP request on port 123, and respond by sending a udp/ip data packet in the NTP format. The data packet includes a 64 -bit (1 part in 2 1019) timestamp containing the time in UTC seconds since Jan. 1, 1900 with a resolution of 200 ps. http: //www. boulder. nist. gov/timefreq/service/its. htm Introduction Section 0 Lecture 1 Slide 54 BEST: If that isn’t good enough, try these details on current research, with proposed precision of ± 1 x 10 -18 s (± 1 as!!!) (also see Sc. American, 2002 special issue on time keeping). http: //www. boulder. nist. gov/timefreq/ INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 54

A Timely Example of Errors in Measurements What time is it? EVEN BETTER: NIST's Second "Quantum Logic Clock" is World's Most Precise Clock NIST scientists have built a second "quantum logic clock, " using quantum information processing techniques on a single ion of aluminum to make a clock that would not gain or lose more than one second in Introduction Section 0 Lecture 1 Slide 55 about 3. 7 billion years. For more information, please see http: //www. nist. gov/pml/div 688/logicclock_0 20410. cfm. INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 55

Intermediate Lab PHYS 3870 Precision and Random (Statistical) Errors Introduction Section 0 Lecture 1 Slide 56 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 56

Precision of Measurements Introduction Section 0 Lecture 1 Slide 57 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 57

Precision and Random (Statistical) Error 1 Introduction Section 0 Lecture 1 Slide 58 2 3 … 123. 45 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 58

Reading a Vernier Scale A vernier scale provides a way to gain added precision from an instrument scale Introduction Section 0 Lecture 1 Slide 59 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 59

Intermediate Lab PHYS 3870 The Ruler Exercise Introduction Section 0 Lecture 1 Slide 60 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 60

The Ruler Exercise Introduction Section 0 Lecture 1 Slide 61 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 61

The Ruler Exercise • Measure the width of the conference table. • Record you measured value and associated error on the white board. Name Ruler Type Measured Value Uncertantity. Introduction Section 0 Lecture 1 Slide 62 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 62

The Ruler Exercise Introduction Section 0 Lecture 1 Slide 63 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 63

The Ruler Exercise Introduction Section 0 Lecture 1 Slide 64 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 64

Precision, Accuracy and Systematic Errors Precision is defined as a measure of the reproducibility of a measurement Such errors are called random (statistical) errors. If an experiment has small random error, it is said to have high precision. Accuracy is a measure of the validity of a measurement. Introduction Section 0 Lecture 1 Slide 65 If an experiment has small systematic error, it is said to have high accuracy. INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 65

Intermediate Lab PHYS 3870 Comparing Measurements to Models Quantitatively Introduction Section 0 Lecture 1 Slide 66 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 66

Comparison with Other Data Introduction Section 0 Lecture 1 Slide 67 Is there agreement? (With what? ) Are these comparisons of precision or accuracy? INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 67

Direct Comparison with Standard Introduction Section 0 Lecture 1 Slide 68 Is there agreement? (With what? ) Are these comparisons of precision or accuracy? INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 68

Intermediate Lab PHYS 3870 Using Errors to Quantitatively Test Models Basic Approach [Baird, Ch 4. 1]. a. Know data and uncertainties (presumably) b. Use this to identify system, inputs and outputs Introduction Section 0 Lecture 1 Slide 69 c. Now develop a model d. Then test model by comparison with data (first qualitatively, then quantitatively) INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 69

Testing a Model? Introduction Section 0 Lecture 1 Slide 70 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 70

Summary of Stating Errors in Measurements Introduction Section 0 Lecture 1 Slide 71 Correct 1. 23± 0. 02 m (± 2%) INTRODUCTION TO Modern Physics PHYX 2710 Incorrect 1. 234567± 0. 024 m Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 71

Quantifying Precision and Random (Statistical) Errors Introduction Section 0 Lecture 1 Slide 72 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 72

Standard Deviation Introduction Section 0 Lecture 1 Slide 73 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 73

Standard Deviation of the Mean Introduction Section 0 Lecture 1 Slide 74 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 74

Errors in Models—Error Propagation Introduction Section 0 Lecture 1 Slide 75 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 75

Specific Rules for Error Propogation Introduction Section 0 Lecture 1 Slide 76 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 76

Independent Uncertainties Introduction Section 0 Lecture 1 Slide 77 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 77

Specific Rules for Independent Error Propagation Introduction Section 0 Lecture 1 Slide 78 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 78

Independent (Random) Uncertaities and Gaussian Distributions Introduction Section 0 Lecture 1 Slide 79 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 79

General Formula for Error Propagation Introduction Section 0 Lecture 1 Slide 80 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 80

General Formula for Error Propagation Introduction Section 0 Lecture 1 Slide 81 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 81

General Formula for Multiple Variables Introduction Section 0 Lecture 1 Slide 82 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 82

A Complex Example for Multiple Variables Introduction Section 0 Lecture 1 Slide 83 INTRODUCTION TO Modern Physics PHYX 2710 Fall 2004 Intermediate 3870 Fall 2013 DEFINING ERRORS Lecture 2 Slide 83