The Treatment of Numerical Experimental Results
|
|
- Ashlie Stokes
- 5 years ago
- Views:
Transcription
1 Memorial University of Newfoundl Department of Physics Physical Oceanography The Treatment of Numerical Experimental Results The purpose of these notes is to introduce you to some techniques of error analysis which you will use in the laboratory The notes are in no sense complete; other techniques such as the χ 2 test t-test are widely used, but are not discussed here Introduction An experimental result has no physical meaning unless an uncertainty (or error [1] )is assigned to it Getting the wrong answer when multiplying or dividing numbers is not an error, but a mistake is something which you should recognize be able to correct The error is not found by comparing your answer to some number in the textbook: uncertainties cannot be avoided in experimental physics Multiple measurements of the same quantity using the same measuring instrument, may not give the same result each time due to rom errors If a systematic error is present, a set of very precise measurements may miss the true value completely For example, a stopwatch is able to measure times to within 001 s, but there is often an associated error due to reaction time of about 03 s Absolute Relative Uncertainty Suppose a ruler is used to measure the length of a rod It is difficult to determine length exactly, but we decide on a value somewhere between 101 cm 103 cm, written as L =102 cm± 01 cm where 102 cm is the most likely for the length the value ±01 is called the absolute uncertainty in the measurement The uncertainty defines a range of possible values for the length, so that 101 cm L 103 cm We often use relative uncertainty, where relative uncertainty = absolute uncertainty, measured value [1] The terms experimental error experimental uncertainty are assumed to have the same meaning in these notes The Treatment of Numerical Experimental Results (1)
2 which, for the rod measurement is ± = ±0009 (no units) The relative uncertainty (or precision of the measurement) is often quoted as a percentage so that the percent uncertainty above is 09% In the laboratory, the size of the error is usually estimated according to the equipment being used You need to select a large enough uncertainty such that the true value lies within the range of uncertainty most of the time This is not always straightforward but becomes easier with experience Combining Errors in Laboratory Results In experiments where the desired result is calculated from two or more quantities (eg speed = distance/time) the errors in each quantity must be combined to give an uncertainty in the final answer To calculate z ±δz, where z = f(x, y) ±δx ±δy are the uncertainties in x y, we could, in principle, calculate z for all values of x ± δx, y ± δy A less tedious method is to calculate the maximum value of δz from the equation: δz = f f δx + δy, (1) x y which for simple mathematical operations reduces to the following forms: z = x + y δz = δx + δy (2a) z = x y δz = δx + δy (2b) z = x y z = x y δz z = δx x + δy y δz z = δx x + δy y (2c) (2d) (2) The Treatment of Numerical Experimental Results
3 Examples 1) When the desired result depends on more than two quantities, the error may be calculated by breaking down the algebraic expression term by term Thus if a = bc/d, the maximum error in a is given by: δa a = δb b + δc c + δd d (3) 2) If z = x n,wheren can be positive or negative, we differentiate to obtain δz = nx n 1 δx or, as a relative uncertainty, δz z = n δx (4) x 3) The density of a metal cylinder may be calculated from ρ = 4m πd 2 l where the symbols have their usual meaning The corresponding error equation is δρ ρ = δm m +2δd d + δl l Note that 4 π do not contribute to the relative error since they are constants 4) y = a sin θ; or δy y = δa cos θδθ + a sin θ or δy y = δa a +cotθδθ provided θ δθ are measured in radians 5) y = a cos θ; δy = δa sin θ + a cos θδθ (using Eq (1)) or δy = δa cos θ + a sin θδθ δy y = δa a +tanθδθ 6) y =lnx; δy = 1 x δx The Treatment of Numerical Experimental Results (3)
4 Mean Stard Error The arithmetic mean x of a set of N readings is defined by N i=1 x = (5) N where x i is the ith reading, means add up all the individual values of x i from i =1to N The mean is the best estimate of the true value Repeated measurements generally follow a normal or Gaussian probability distribution; the probability of occurrence of an individual value x i may be calculated from { 1 (xi x) 2 } P (x i )= exp σ x 2π where x is the mean σ x is the stard deviation, defined by (xi x) σ x = 2 (6) N 1 The stard deviation is a measure of the deviation of a typical reading from the mean value It may be shown that 68% of the measurements lie within one stard deviation of x nearly all measurements (95%) are expected to lie within 2σ x of the mean The quantity σx 2 is called the variance It is generally more useful to consider the stard error of the mean, σ x It may be shown that for N measurements, each subject to an error δx i, the error in the mean is Experimental results are usually expressed in the form x i 2σ 2 x σ x = σ x (7) N x ± σ x In the special case of radioactive decay, the mean square deviation (from Poisson statistics) is given by σ = N where N is the number of counts Data are recorded in the form N ± N Hence it is necessary to count for at least 10,000 decays to obtain an accuracy of 1% Foran accuracy of 1 in 10 3, 10 6 decay events are required (4) The Treatment of Numerical Experimental Results
5 Significant Figures Suppose you use a calculator to obtain a stard deviation of If this is larger than the reading error in your measurements, then this will be the error in each of the datapoints For a sample of N datapoints, the expected uncertainty in the stard deviation (ie, the error in the error) is: δσ est = σ est 2N 2 Suppose δσ est = This means that the actual value of the stard deviation lies between = , = A moment s thought about this should convince you that many of these digits have no significance The value of the estimated stard deviation is more like 099 ± 023 or maybe even 10 ± 02 Writing 0988 ± 0233 has more digits than are actually significant Even if you repeat a measurement 50 times, the estimated stard deviation has at most only two digits that have any meaning Imagine that one of the data points has a numerical value of If we estimate the stard deviation to be 099, then the point value is 1235 ± 099 It would be wrong to say ± 099, since the 5 has no meaning In the laboratory, the reading error will be the error in each individual measurement This will be little more than a guess made by the experimenter, it is doubtful that you can guess to more than one significant figure Thus a reading error almost by definition has only one significant figure, that number determines the significant figures in the value itself You need to be particularly careful when writing down computer-generated results A slope of ± is meaningless: the error should written as ±0002, which means that the slope should be written as 0078 to keep the same number of figures after the decimal point Similarly, if the voltage across a resistor is 154 ± 01 volts the current is 17 ± 01 amps, the resistance is not ohms because any additional figures beyond the first decimal place are meaningless The final value for R should be written as (91 ± 06) Ω The Treatment of Numerical Experimental Results (5)
6 The Errors in a Straight Line Graph Often the result of an experiment depends on the slope of a straight line graph Errors associated with the data are illustrated by error bars, the size of which defines the range of uncertainty in one (or both) axes A straight line is represented by the equation where m is the slope b is the y-intercept y = mx + b y v a l u e s δx δy maximum best minimum b xvalues The uncertainty in the slope is estimated by considering extremes of maximum minimum slope which might concievably fit the data, as illustrated in the diagram Denoting the slopes of these lines by m max m min respectively, δm may be calculated from δm = m max m min 2 (8) the error in the intercept: δb = b max b min (9) 2 (6) The Treatment of Numerical Experimental Results
7 Method of Least Squares Linear regression by the Method of Least Squares finds the equation of the best fit line by minimizing the sum of the squares of deviations of the individual y values from the straight line δx 5 y v a l u e s δx 2 δx 4 δx 3 δx 1 xvalues The slope intercept are given by m = (xy) 1/ N ( x)( y) (x2 ) 1 / N ( x) 2 (10) b = 1 / N ( y m x ) (11) with uncertainties (y2 ) b y m (xy) δb = N 2 δm = (12) δb (x2 ) 1 / N ( x) 2 (13) The Treatment of Numerical Experimental Results (7)
8 Worked Example Linear regression is easily performed by computer, however, a detailed calculation is shown here for completeness [2] x =10, 20, 30, 40, 50, 60 y =716, 725, 743, 761, 770, 779, Example of Least Squares Fit y values x values i x i y i x 2 i yi 2 x i y i sum [2] Data taken from L Hmurcik et al Linear regression analysis in a first physics lab, Am J Phys, 57, (1989) (8) The Treatment of Numerical Experimental Results
9 hence x =210; y =4494; x 2 =910; y 2 = xy = Finally we obtain The corresponding uncertainties are given by m = / / 6 (210) 2 =0134 b = 1 / 6 ( ) = δb = 4 δm = / 6 (210) 2 =0009 =004 The relative uncertainty in the slope, therefore, is 0009 =0067 or 67% 0134 the relative uncertainty in the intercept is =0005 or 05% When the data points do not follow a straight line, the best curve through the points may also be obtained by linear regression provided the function is a polynomial of the form y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + Other mathematical functions may be fitted using non-linear regression In most of the lab experiments that you will do, the uncertainty in the slope of the straight line will be greater than the uncertainties in other measured quantities This means that you can usually ignore the errors in everything but the slope The Treatment of Numerical Experimental Results (9)
APPENDIX A: DEALING WITH UNCERTAINTY
APPENDIX A: DEALING WITH UNCERTAINTY 1. OVERVIEW An uncertainty is always a positive number δx > 0. If the uncertainty of x is 5%, then δx =.05x. If the uncertainty in x is δx, then the fractional uncertainty
More informationDealing with uncertainty
Appendix A Dealing with uncertainty A.1 Overview An uncertainty is always a positive number δx > 0. If you measure x with a device that has a precision of u, thenδx is at least as large as u. Fractional
More informationDealing with uncertainty
Appendix A Dealing with uncertainty A.1 Overview An uncertainty is always a positive number δx > 0. If you measure x with a device that has a precision of u, thenδx is at least as large as u. Fractional
More information1 Random and systematic errors
1 ESTIMATION OF RELIABILITY OF RESULTS Such a thing as an exact measurement has never been made. Every value read from the scale of an instrument has a possible error; the best that can be done is to say
More informationMeasurements and Data Analysis
Measurements and Data Analysis 1 Introduction The central point in experimental physical science is the measurement of physical quantities. Experience has shown that all measurements, no matter how carefully
More informationBRIDGE CIRCUITS EXPERIMENT 5: DC AND AC BRIDGE CIRCUITS 10/2/13
EXPERIMENT 5: DC AND AC BRIDGE CIRCUITS 0//3 This experiment demonstrates the use of the Wheatstone Bridge for precise resistance measurements and the use of error propagation to determine the uncertainty
More informationError Analysis in Experimental Physical Science Mini-Version
Error Analysis in Experimental Physical Science Mini-Version by David Harrison and Jason Harlow Last updated July 13, 2012 by Jason Harlow. Original version written by David M. Harrison, Department of
More informationAppendix II Calculation of Uncertainties
Part 1: Sources of Uncertainties Appendix II Calculation of Uncertainties In any experiment or calculation, uncertainties can be introduced from errors in accuracy or errors in precision. A. Errors in
More informationElectromagnetism lab project
Electromagnetism lab project Contents 1. Overview of the course 2. How to analyse errors in measurements 3. How to make graphical representations (plots) Overview Four lab experiments Biot Savart s law
More informationMeasurement of Electrical Resistance and Ohm s Law
Measurement of Electrical Resistance and Ohm s Law Objectives In this experiment, measurements of the voltage across a wire coil and the current in the wire coil will be used to accomplish the following
More informationUncertainty, Error, and Precision in Quantitative Measurements an Introduction 4.4 cm Experimental error
Uncertainty, Error, and Precision in Quantitative Measurements an Introduction Much of the work in any chemistry laboratory involves the measurement of numerical quantities. A quantitative measurement
More informationError analysis for IPhO contestants
Error analysis for IPhO contestants Heikki Mäntysaari University of Jyväskylä, Department of Physics Abstract In the experimental part of IPhO (and generally when performing measurements) you have to estimate
More informationMeasurement and Uncertainty
Physics 1020 Laboratory #1 Measurement and Uncertainty 1 Measurement and Uncertainty Any experimental measurement or result has an uncertainty associated with it. In todays lab you will perform a set of
More informationPhysics: Uncertainties - Student Material (AH) 1
UNCERTAINTIES Summary of the Basic Theory associated with Uncertainty It is important to realise that whenever a physical quantity is being measured there will always be a degree of inaccuracy associated
More informationError analysis for the physical sciences A course reader for phys 1140 Scott Pinegar and Markus Raschke Department of Physics, University of Colorado
Error analysis for the physical sciences A course reader for phys 1140 Scott Pinegar and Markus Raschke Department of Physics, University of Colorado Version 1.0 (September 9, 2012) 1 Part 1 (chapter 1
More informationLecture 2: Reporting, Using, and Calculating Uncertainties 2. v = 6050 ± 30 m/s. v = 6047 ± 3 m/s
1 CHAPTER 2: Reporting and Using Uncertainties Quoting a result as: Best Estimate ± Uncertainty In the Archimedes experiment result, we had a table which read Measurement of Crown Density by Two Experts
More informationFundamentals of data, graphical, and error analysis
Fundamentals of data, graphical, and error analysis. Data measurement and Significant Figures UTC - Physics 030L/040L Whenever we take a measurement, there are limitations to the data and how well we can
More informationIntermediate Lab PHYS 3870
Intermediate Lab PHYS 3870 Lecture 4 Comparing Data and Models Quantitatively Linear Regression Introduction Section 0 Lecture 1 Slide 1 References: Taylor Ch. 8 and 9 Also refer to Glossary of Important
More informationUncertainty and Graphical Analysis
Uncertainty and Graphical Analysis Introduction Two measures of the quality of an experimental result are its accuracy and its precision. An accurate result is consistent with some ideal, true value, perhaps
More informationErrors: What they are, and how to deal with them
Errors: What they are, and how to deal with them A series of three lectures plus exercises, by Alan Usher Room 111, a.usher@ex.ac.uk Synopsis 1) Introduction ) Rules for quoting errors 3) Combining errors
More informationIntroduction to Data Analysis
Introduction to Data Analysis Analysis of Experimental Errors How to Report and Use Experimental Errors Statistical Analysis of Data Simple statistics of data Plotting and displaying the data Summary Errors
More informationError Analysis and Graph Drawing
Error Analysis and Graph Drawing I. Introduction: I.1 It is impossible to do an experimental measurement with perfect accuracy. There is always an uncertainty associated with any measured quantity in an
More informationExperiment 4 Free Fall
PHY9 Experiment 4: Free Fall 8/0/007 Page Experiment 4 Free Fall Suggested Reading for this Lab Bauer&Westfall Ch (as needed) Taylor, Section.6, and standard deviation rule ( t < ) rule in the uncertainty
More information1 Measurement Uncertainties
1 Measurement Uncertainties (Adapted stolen, really from work by Amin Jaziri) 1.1 Introduction No measurement can be perfectly certain. No measuring device is infinitely sensitive or infinitely precise.
More informationDescribing the Relationship between Two Variables
1 Describing the Relationship between Two Variables Key Definitions Scatter : A graph made to show the relationship between two different variables (each pair of x s and y s) measured from the same equation.
More informationERROR AND GRAPHICAL ANALYSIS WORKSHEET
Student Names: Course: Section: Instructor: ERROR AND GRAPHICAL ANALYSIS WORKSHEET Instructions: For each section of this assignment, first read the relevant section in the Yellow Pages of your Lab Manual.
More informationRegression and Nonlinear Axes
Introduction to Chemical Engineering Calculations Lecture 2. What is regression analysis? A technique for modeling and analyzing the relationship between 2 or more variables. Usually, 1 variable is designated
More informationPhysics 115 Experiment 1. Introduction to Measurement and Error Analysis (PHY 115 and 117)
Physics 115 Experiment 1 Introduction to Measurement and Error Analysis (PHY 115 and 117) Introduction In the sciences, measurement plays an important role. The accuracy of the measurement, as well as
More informationPARADISE VALLEY COMMUNITY COLLEGE PHYSICS COLLEGE PHYSICS I LABORATORY
PARADISE VALLEY COMMUNITY COLLEGE PHYSICS 111 - COLLEGE PHYSICS I LABORATORY PURPOSE OF THE LABORATORY: The laboratory exercises are designed to accomplish two objectives. First, the exercises will illustrate
More informationName: Lab Partner: Section: In this experiment error analysis and propagation will be explored.
Chapter 2 Error Analysis Name: Lab Partner: Section: 2.1 Purpose In this experiment error analysis and propagation will be explored. 2.2 Introduction Experimental physics is the foundation upon which the
More informationUncertainty in Measurements
Uncertainty in Measurements Joshua Russell January 4, 010 1 Introduction Error analysis is an important part of laboratory work and research in general. We will be using probability density functions PDF)
More informationExperimental Uncertainty (Error) and Data Analysis
Experimental Uncertainty (Error) and Data Analysis Advance Study Assignment Please contact Dr. Reuven at yreuven@mhrd.org if you have any questions Read the Theory part of the experiment (pages 2-14) and
More informationLab 1: Measurement, Uncertainty, and Uncertainty Propagation
Lab 1: Measurement, Uncertainty, and Uncertainty Propagation 17 ame Date Partners TA Section Lab 1: Measurement, Uncertainty, and Uncertainty Propagation The first principle is that you must not fool yourself
More informationAPPENDIX A - MEASUREMENT AND UNCERTAINTY: APPENDIX B - COMPUTERS AND SOFTWARE: APPENDIX C - EQUIPMENT: APPENDIX D MATH REVIEW:
APPENDIX A - MEASUREMENT AND UNCERTAINTY: General Information Experimental Error Error Analysis Uncertainty and Propagation of Error APPENDIX B - COMPUTERS AND SOFTWARE: imac & Logger Pro Graphing Graphical
More informationExperimental Uncertainty (Error) and Data Analysis
E X P E R I M E N T 1 Experimental Uncertainty (Error) and Data Analysis INTRODUCTION AND OBJECTIVES Laboratory investigations involve taking measurements of physical quantities, and the process of taking
More informationAccuracy and precision in measurements
Accuracy and precision in measurements Scientists aim towards designing experiments that can give a true value from their measurements, but due to the limited precision in measuring devices, they often
More informationSPH3U1 Lesson 03 Introduction. 6.1 Expressing Error in Measurement
SIGNIFICANT DIGITS AND SCIENTIFIC NOTATION LEARNING GOALS Students will: 6 ERROR Describe the difference between precision and accuracy Be able to compare values quantitatively Understand and describe
More informationExperiment 2 Random Error and Basic Statistics
PHY191 Experiment 2: Random Error and Basic Statistics 7/12/2011 Page 1 Experiment 2 Random Error and Basic Statistics Homework 2: turn in the second week of the experiment. This is a difficult homework
More informationA Quick Introduction to Data Analysis (for Physics)
A Quick Introduction to Data Analysis for Physics Dr. Jeff A. Winger What is data analysis? Data analysis is the process by which experimental data is used to obtain a valid and quantifiable result. Part
More informationx y x y ax bx c x Algebra I Course Standards Gap 1 Gap 2 Comments a. Set up and solve problems following the correct order of operations (including proportions, percent, and absolute value) with rational
More informationPHY 123 Lab 1 - Error and Uncertainty and the Simple Pendulum
To print higher-resolution math symbols, click the Hi-Res Fonts for Printing button on the jsmath control panel. PHY 13 Lab 1 - Error and Uncertainty and the Simple Pendulum Important: You need to print
More informationMethods and Tools of Physics
Methods and Tools of Physics Order of Magnitude Estimation: Essential idea: Scientists aim towards designing experiments that can give a true value from their measurements, but due to the limited precision
More informationUncertainty in Physical Measurements: Module 5 Data with Two Variables
: Often data have two variables, such as the magnitude of the force F exerted on an object and the object s acceleration a. In this Module we will examine some ways to determine how one of the variables,
More informationYear 12 Physics INDUCTION WORK XKCD. Student. Class 12 A / B / C / D / E Form
Year 12 Physics 2018-19 INDUCTION WORK XKCD Student Class 12 A / B / C / D / E Form DYP 2018 1. Physical Quantities Maths and Physics have an important but overlooked distinction by students. Numbers in
More informationPre-Lab: Primer on Experimental Errors
IUPUI PHYS 15 Laboratory Page 1 of 5 Pre-Lab: Primer on Eperimental Errors There are no points assigned for this Pre-Lab. n essential skill in the repertoire of an eperimental physicist is his/her ability
More informationCourse Project. Physics I with Lab
COURSE OBJECTIVES 1. Explain the fundamental laws of physics in both written and equation form 2. Describe the principles of motion, force, and energy 3. Predict the motion and behavior of objects based
More informationPropagation of Error Notes
Propagation of Error Notes From http://facultyfiles.deanza.edu/gems/lunaeduardo/errorpropagation2a.pdf The analysis of uncertainties (errors) in measurements and calculations is essential in the physics
More informationUncertainties in AH Physics
Advanced Higher Physics Contents This booklet is one of a number that have been written to support investigative work in Higher and Advanced Higher Physics. It develops the skills associated with handling
More informationMeasurements and Data Analysis An Introduction
Measurements and Data Analysis An Introduction Introduction 1. Significant Figures 2. Types of Errors 3. Deviation from the Mean 4. Accuracy & Precision 5. Expressing Measurement Errors and Uncertainty
More informationIB Physics Internal Assessment Report Sample. How does the height from which a ball is dropped affect the time it takes to reach the ground?
IB Physics Internal Assessment Report Sample Design How does the height from which a ball is dropped affect the time it takes to reach the ground? In this experiment, the release height of the ball is
More informationPHY 101L - Experiments in Mechanics
PHY 101L - Experiments in Mechanics introduction to error analysis What is Error? In everyday usage, the word error usually refers to a mistake of some kind. However, within the laboratory, error takes
More informationPHYS 2211L - Principles of Physics Laboratory I Propagation of Errors Supplement
PHYS 2211L - Principles of Physics Laboratory I Propagation of Errors Supplement 1. Introduction. Whenever two or more quantities are measured directly in order to indirectly determine the value of another,
More informationPractical 1 RC Circuits
Objectives Practical 1 Circuits 1) Observe and qualitatively describe the charging and discharging (decay) of the voltage on a capacitor. 2) Graphically determine the time constant for the decay, τ =.
More information1. Data based question. This question is about change of electrical resistance with temperature.
1. Data based question. This question is about change of electrical resistance with temperature. The table below gives values of the resistance R of an electrical component for different values of its
More informationUncertainty in Physical Measurements: Module 5 Data with Two Variables
: Module 5 Data with Two Variables Often data have two variables, such as the magnitude of the force F exerted on an object and the object s acceleration a. In this Module we will examine some ways to
More informationmeas (1) calc calc I meas 100% (2) Diff I meas
Lab Experiment No. Ohm s Law I. Introduction In this lab exercise, you will learn how to connect the to network elements, how to generate a VI plot, the verification of Ohm s law, and the calculation of
More informationToday s lecture. WEST VIRGINIA UNIVERSITY Physics
Today s lecture Units, Estimations, Graphs, Trigonometry: Units - Standards of Length, Mass, and Time Dimensional Analysis Uncertainty and significant digits Order of magnitude estimations Coordinate Systems
More informationNotes Errors and Noise PHYS 3600, Northeastern University, Don Heiman, 6/9/ Accuracy versus Precision. 2. Errors
Notes Errors and Noise PHYS 3600, Northeastern University, Don Heiman, 6/9/2011 1. Accuracy versus Precision 1.1 Precision how exact is a measurement, or how fine is the scale (# of significant figures).
More informationExperiment 0 ~ Introduction to Statistics and Excel Tutorial. Introduction to Statistics, Error and Measurement
Experiment 0 ~ Introduction to Statistics and Excel Tutorial Many of you already went through the introduction to laboratory practice and excel tutorial in Physics 1011. For that reason, we aren t going
More informationChapter 1. A Physics Toolkit
Chapter 1 A Physics Toolkit Chapter 1 A Physics Toolkit In this chapter you will: Use mathematical tools to measure and predict. Apply accuracy and precision when measuring. Display and evaluate data graphically.
More informationError Analysis How Do We Deal With Uncertainty In Science.
How Do We Deal With Uncertainty In Science. 1 Error Analysis - the study and evaluation of uncertainty in measurement. 2 The word error does not mean mistake or blunder in science. 3 Experience shows no
More informationExperimental Uncertainties
Experimental Uncertainties 1 Measurements of any physical quantity can never be exact. One can only know its value with a range of uncertainty. If an experimenter measures some quantity X, the measurement
More informationYoung s Modulus of Elasticity. Table 1. Length of the wire a /m b /m ±
Joe Chow Partner: Sam Elliott 15 section # 3 Desk # 7 3 July 010 Young s Modulus of Elasticity Purpose To determine Young s Modulus for a metal wire. Apparatus micrometer # 7, metric measuring tape Properly
More informationBRIEF SURVEY OF UNCERTAINITY IN PHYSICS LABS
BRIEF SURVEY OF UNCERTAINITY IN PHYSICS LABS THREE CASES OF UNCERTAINTY CALCULATION There are two main situations when dealing with uncertainty calculation of a given parameter; or it is measured or it
More informationDr. Julie J. Nazareth
Name: Dr. Julie J. Nazareth Lab Partner(s): Physics: 133L Date lab performed: Section: Capacitors Parts A & B: Measurement of capacitance single, series, and parallel combinations Table 1: Voltage and
More information1 Measurement Uncertainties
1 Measurement Uncertainties (Adapted stolen, really from work by Amin Jaziri) 1.1 Introduction No measurement can be perfectly certain. No measuring device is infinitely sensitive or infinitely precise.
More informationIntroduction to Uncertainty and Treatment of Data
Introduction to Uncertainty and Treatment of Data Introduction The purpose of this experiment is to familiarize the student with some of the instruments used in making measurements in the physics laboratory,
More informationFor a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is
Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For
More informationMEASUREMENTS AND ERRORS (OR EXPERIMENTAL UNCERTAINTIES)
MEASUREMENTS AND ERRORS (OR EXPERIMENTAL UNCERTAINTIES) Determination of Uncertainties in Measured Quantities Physics is not only a theoretical but an experimental science; it depends on measured values
More informationChapter 2: Statistical Methods. 4. Total Measurement System and Errors. 2. Characterizing statistical distribution. 3. Interpretation of Results
36 Chapter : Statistical Methods 1. Introduction. Characterizing statistical distribution 3. Interpretation of Results 4. Total Measurement System and Errors 5. Regression Analysis 37 1.Introduction The
More informationevaluate functions, expressed in function notation, given one or more elements in their domains
Describing Linear Functions A.3 Linear functions, equations, and inequalities. The student writes and represents linear functions in multiple ways, with and without technology. The student demonstrates
More informationMeasurement: The Basics
I. Introduction Measurement: The Basics Physics is first and foremost an experimental science, meaning that its accumulated body of knowledge is due to the meticulous experiments performed by teams of
More informationGravity Pre-Lab 1. Why do you need an inclined plane to measure the effects due to gravity?
Lab Exercise: Gravity (Report) Your Name & Your Lab Partner s Name Due Date Gravity Pre-Lab 1. Why do you need an inclined plane to measure the effects due to gravity? 2. What are several advantage of
More informationUncertainties & Error Analysis Tutorial
Uncertainties & Error Analysis Tutorial Physics 118/198/1 Reporting Measurements Uncertainties & Error Analysis Tutorial When we report a measured value of some parameter, X, we write it as X X best ±
More informationPhysics 403. Segev BenZvi. Propagation of Uncertainties. Department of Physics and Astronomy University of Rochester
Physics 403 Propagation of Uncertainties Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Maximum Likelihood and Minimum Least Squares Uncertainty Intervals
More informationPHYS Uncertainty Analysis
PHYS 213 1 Uncertainty Analysis Types of uncertainty We will consider two types of uncertainty that affect our measured or calculated values: random uncertainty and systematic uncertainty. Random uncertainties,
More informationCourse Outcome Summary
Course Information: Description: Instruction Level: 9-12 Total Credits: 2 Prerequisites: Textbooks: Course Algebra I Algebra is a symbolic extension of arithmetic and allows you to solve more complex problems
More informationPrinciples and Problems. Chapter 1: A Physics Toolkit
PHYSICS Principles and Problems Chapter 1: A Physics Toolkit CHAPTER 1 A Physics Toolkit BIG IDEA Physicists use scientific methods to investigate energy and matter. CHAPTER 1 Table Of Contents Section
More informationAccuracy: An accurate measurement is a measurement.. It. Is the closeness between the result of a measurement and a value of the measured.
Chemical Analysis can be of two types: Chapter 11- Measurement and Data Processing: - : Substances are classified on the basis of their or properties, such as - : The amount of the sample determined in
More informationExperiment 1 - Mass, Volume and Graphing
Experiment 1 - Mass, Volume and Graphing In chemistry, as in many other sciences, a major part of the laboratory experience involves taking measurements and then calculating quantities from the results
More informationMA.8.1 Students will apply properties of the real number system to simplify algebraic expressions and solve linear equations.
Focus Statement: Students will solve multi-step linear, quadratic, and compound equations and inequalities using the algebraic properties of the real number system. They will also graph linear and quadratic
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationTake the measurement of a person's height as an example. Assuming that her height has been determined to be 5' 8", how accurate is our result?
Error Analysis Introduction The knowledge we have of the physical world is obtained by doing experiments and making measurements. It is important to understand how to express such data and how to analyze
More informationDifferential Equations and Linear Algebra Supplementary Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University
Differential Equations and Linear Algebra Supplementary Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter 1. Linear algebraic equations 5 1.1. Gaussian elimination
More informationSwitch. R 5 V Capacitor. ower upply. Voltmete. Goals. Introduction
Switch Lab 6. Circuits ower upply Goals + + R 5 V Capacitor V To appreciate the capacitor as a charge storage device. To measure the voltage across a capacitor as it discharges through a resistor, and
More informationLab 1: Simple Pendulum 1. The Pendulum. Laboratory 1, Physics 15c Due Friday, February 16, in front of Sci Cen 301
Lab 1: Simple Pendulum 1 The Pendulum Laboratory 1, Physics 15c Due Friday, February 16, in front of Sci Cen 301 Physics 15c; REV 0 1 January 31, 2007 1 Introduction Most oscillating systems behave like
More informationSome Statistics. V. Lindberg. May 16, 2007
Some Statistics V. Lindberg May 16, 2007 1 Go here for full details An excellent reference written by physicists with sample programs available is Data Reduction and Error Analysis for the Physical Sciences,
More informationAlgebra II. A2.1.1 Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
Standard 1: Relations and Functions Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
More informationMeasurement Uncertainties
Measurement Uncertainties Introduction We all intuitively know that no experimental measurement can be "perfect''. It is possible to make this idea quantitative. It can be stated this way: the result of
More informationSwitch. R 5 V Capacitor. ower upply. Voltmete. Goals. Introduction
Switch Lab 6. Circuits ower upply Goals + + R 5 V Capacitor V To appreciate the capacitor as a charge storage device. To measure the voltage across a capacitor as it discharges through a resistor, and
More informationGraphs. 1. Graph paper 2. Ruler
Graphs Objective The purpose of this activity is to learn and develop some of the necessary techniques to graphically analyze data and extract relevant relationships between independent and dependent phenomena,
More informationERROR ANALYSIS ACTIVITY 1: STATISTICAL MEASUREMENT UNCERTAINTY AND ERROR BARS
ERROR ANALYSIS ACTIVITY 1: STATISTICAL MEASUREMENT UNCERTAINTY AND ERROR BARS LEARNING GOALS At the end of this activity you will be able 1. to explain the merits of different ways to estimate the statistical
More informationPHYS 281 General Physics Laboratory
King Abdul-Aziz University Faculty of Science Physics Department PHYS 281 General Physics Laboratory Student Name: ID Number: Introduction Advancement in science and engineering has emphasized the microscopic
More informationData Fits. We begin this discussion using a simple linear function. Later, we briefly elaborate to linearized plots and and non-linear fits.
Data Fits Introduction Most experiments involve several variables that are interdependent, such as force (F) and displacement (X) for a spring. A range of F will produce a range of X, with proportionality
More informationLab 0 Appendix C L0-1 APPENDIX C ACCURACY OF MEASUREMENTS AND TREATMENT OF EXPERIMENTAL UNCERTAINTY
Lab 0 Appendix C L0-1 APPENDIX C ACCURACY OF MEASUREMENTS AND TREATMENT OF EXPERIMENTAL UNCERTAINTY A measurement whose accuracy is unknown has no use whatever. It is therefore necessary to know how to
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationData Analysis, Standard Error, and Confidence Limits E80 Spring 2012 Notes
Data Analysis Standard Error and Confidence Limits E80 Spring 0 otes We Believe in the Truth We frequently assume (believe) when making measurements of something (like the mass of a rocket motor) that
More informationIn this unit, we will examine the movement of electrons, which we call CURRENT ELECTRICITY.
Recall: Chemistry and the Atom! What are the 3 subatomic Where are they found in the particles? atom? What electric charges do they have? How was a positive ion created? How was a negative ion created?
More informationNewton s Second Law Physics Lab V
Newton s Second Law Physics Lab V Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion. A formulated analysis of forces acting on a dynamics cart
More informationMeasurements of a Table
Measurements of a Table OBJECTIVES to practice the concepts of significant figures, the mean value, the standard deviation of the mean and the normal distribution by making multiple measurements of length
More information