Forest Error Analysis for the Physical Sciences

Class Admin

Homework

Homework is due at the beginning of class on the assigned day. If you have a documented excuse for your absence, then you will have 24 hours to hand in the homework after being released by your doctor.

Class Policies

http://wiki.iac.isu.edu/index.php/Forest_Class_Policies

Instructional Objectives

Course Catalog Description

Error Analysis for the Physics Sciences 3 credits. Lecture course with computation requirements. Topics include: Error propagation, Probability Distributions, Least Squares fit, multiple regression, goodness of fit, covariance and correlations.

Prequisites:Math 360.

Course Description

The application of statistical inference and hypothesis testing will be the main focus of this course for students who are senior level undergraduates or beginning graduate students. The course begins by introducing the basic skills of error analysis and then proceeds to describe fundamental methods comparing measurements and models. A freely available data analysis package known as ROOT will be used. Some programming skills will be needed using C/C++ but a limited amount of experience is assumed.

Objectives and Outcomes

Forest_ErrorAnalysis_ObjectivesnOutcomes

Suggested Text

Data Reduction and Error Analysis for the Physical Sciences by Philip Bevington ISBN: 0079112439

Homework

TF_ErrAna_Homework

Class Labs

TF_ErrAna_InClassLab

Systematic and Random Uncertainties

Although the name of the class is "Error Analysis" for historical purposes, a more accurate description would be "Uncertainty Analysis". "Error" usually means a mistake is made while "Uncertainty" is a measure of how confident you are in a measurement.

Accuracy -vs- Precision

Accuracy

How close does an experiment come to the correct result

Precision

a measure of how exactly the result is determine. No reference is made to what the result means.

Systematic Error

What is a systematic error?

A class of errors which result in reproducible mistakes due to equipment bias or a bias related to its use by the observer.

Example:

1.) A ruler

a.) A ruler could be shorter or longer because of temperature fluctuations

b.) An observer could be viewing the markings at a glancing angle.

In this case a systematic error is more of a mistake than an uncertainty.

In some cases you can correct for the systematic error. In the above Ruler example you can measure how the ruler's length changes with temperature. You can then correct this systematic error by measuring the temperature of the ruler during the distance measurement.

Correction Example:

A ruler is calibrated at 25 C an has an expansion coefficient of (0.0005 [math]\pm[/math] 0.0001 m/C.

You measure the length of a wire at 20 C and find that on average it is [math]1.982 \pm 0.0001[/math] m long.

This means that the 1 m ruler is really (1-(20-25 C)(0.0005 m/C)) = 0.99775

So the correction becomes

1.982 *( 0.99775) =1.977 m

Note

The numbers above without decimal points are integers. Integers have infinite precision. We will discuss the propagation of the errors above in a different chapter.

Error from bad technique:

After repeating the experiment several times the observer discovers that he had a tendency to read the meter stick at an angle and not from directly above. After investigating this misread with repeated measurements the observer estimates that on average he will misread the meter stick by 2 mm. This is now a systematic error that is estimated using random statistics.

Reporting Uncertainties

Notation

X [math]\pm[/math] Y = X(Y)

Significant Figures and Round off

Significant figures

Most Significant digit

The leftmost non-zero digit is the most significant digit of a reported value

Least Significant digit

The least significant digit is identified using the following criteria

1.) If there is no decimal point, then the rightmost digit is the least significant digit.

2.)If there is a decimal point, then the rightmost digit is the least significant digit, even if it is a zero.

In other words, zero counts as a least significant digit only if it is after the decimal point. So when you report a measurement with a zero in such a position you had better mean it.

The number of significant digits in a measurement are the number of digits which appear between the least and most significant digits.

examples:

Measurement	most Sig. digit	least Sig.	Num. Sig. Dig.	Scientific Notation
5	5	5	1*	[math]5[/math]
5.0	5	0	2	[math]5.0 [/math]
50	5	0	2*	[math]5 \times 10^{1}[/math]
50.1	5	1	3	[math]5.01 \times 10^{1}[/math]
0.005001	5	1	4	[math]5.001\times 10^{-3}[/math]

• Note

The values of "5" and "50" above are ambiguous unless we use scientific notation in which case we know if the zero is significant or not.

Round Off

Measurements that are reported which are based on the calculation of more than one measured quantity must have the same number of significant digits as the quantity with the smallest number of significant digits.

To accomplish this you will need to round of the final measured value that is reported.

To round off a number you:

1.) Increment the least significant digit by one if the digit below it (in significance) is greater than 5.

2.) Do nothing if the digit below it (in significance) is less than 5.

Then truncate the remaining digits below the least significant digit.

What happens if the next significant digit below the least significant digit is exactly 5?

To avoid a systematic error involving round off you would ideally randomly decide to just truncate or increment. If your calculation is not on a computer with a random number generator, or you don't have one handy, then the typical technique is to increment the least significant digit if it is odd (or even) and truncate it if it is even (or odd).

Examples

The table below has three entries; the final value calculated from several measured quantities, the number of significant digits for the measurement with the smallest number of significant digits, and the rounded off value properly reported using scientific notation.

Value	Sig. digits	Rounded off value
12.34	3	[math]1.23 \times 10^{1}[/math]
12.36	3	[math]1.24 \times 10^{1}[/math]
12.35	3	[math]1.24 \times 10^{1}[/math]
12.35	2	[math]1.2 \times 10^{1}[/math]

Statistics abuse

http://www.worldcat.org/oclc/28507867

http://www.worldcat.org/oclc/53814054

Forest_StatBadExamples

Statistical Distributions

Forest_ErrAna_StatDist

Propagation of Uncertainties

TF_ErrorAna_PropOfErr

Statistical inference

TF_ErrorAna_StatInference

Final Exam

The Final exam will be to write a report describing the analysis of the data in TF_ErrAna_InClassLab#Lab_16

Grading Scheme:

Grid Search method results

10% Parameter values

20% Parameter errors

30% Probability fit is correct

40% Grammatically correct written explanation of the data analysis with publication quality plots

Report and source code due in my office by Thursday May 3, 2:30 pm (MST)

Report length is between 3 and 15 pages all inclusive.

WIP

TF_ErrorAnalysis_WIP

http://root.cern.ch/root/htmldoc/MATH_Index.html