Dealing with Uncertainties: A Guide to Error Analysis

✍ Scribed by Manfred Drosg

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 243
• Category: Library

✦ Synopsis

Dealing with Uncertainties is an innovative monograph that lays special emphasis on the deductive approach to uncertainties and on the shape of uncertainty distributions. This perspective has the potential for dealing with the uncertainty of a single data point and with sets of data that have different weights. It is shown that the inductive approach that is commonly used to estimate uncertainties is in fact not suitable for these two cases. The approach that is used to understand the nature of uncertainties is novel in that it is completely decoupled from measurements. Uncertainties which are the consequence of modern science provide a measure of confidence both in scientific data and in information in everyday life. Uncorrelated uncertainties and correlated uncertainties are fully covered and the weakness of using statistical weights in regression analysis is discussed. The text is abundantly illustrated with examples and includes more than 150 problems to help the reader master the subject.

✦ Table of Contents

Preface
Preface to the First Edition
Prolog. Seven Myths in Error Analysis
Contents
Introduction
The Exactness of Science
Falsification vs. Verification
Data without Uncertainty
Uncertainties in Every-Day Life
Basics on Data
Measurement Data
The Best Estimate
Directly Measured Data
Indirectly Measured Data
Counting vs. Measuring
The Basic Counting Model
Indirect Counting
Analog vs. Digital
Analog'' Measurements Dealing with Data (Numerals) Valid Digits Truncation of Numbers Rounding Basics on Uncertainties General Characteristics of Uncertainties Shape of Uncertainty Distributions Definitions Terminology Necessary Requirements Deviations Random Uncertainties Maximum Uncertainties (Tolerances) Limits Outliers (Flyers) Uncertainty of Data Depending on One Variable Length Circular Area Multiple Uncertainty Components (Quadratic Sum) Properties of theQuadratic'' Sum
Subtraction in Quadrature
Combined Standard Uncertainty
Uncertainty Evaluations (Error Analysis)
Inductive Method
Deductive Method
Experimental Uncertainty
Counting and Measurement Uncertainties
Parameter Uncertainties
Model Uncertainties
Additional Experimental Uncertainties
Radioactive Decay, a Model for Random Events
Time Interval Distribution of Radioactive Events
Prescaling
Counting Loss (Dead Time)
Direct Correction of Dead Time
Direct Correction for Lost Counts
Loss Correction Using a Pulse Generator
Inductive Approach to Uncertainty (Example)
Properties of Data Sets and Arrays
Reproducibility within Data Sets
Linear Regression (Least-Squares Method)
Frequency and Probability Distributions
Frequency Distribution (Spectrum)
Characteristics of Distributions
Effect of Data Uncertainty on the Distribution
Probability Distributions
Binomial Distribution
Poisson Distribution
Normal (or Gaussian) Distribution
Finite Distributions
Convolution of Uncertainty Distributions
Statistical Confidence
Dealing with Probabilities
Deductive Approach to Uncertainty
Theoretical Situation
Practical Situation
Best Estimates Using Internal Uncertainties
Deductive vs. Inductive Uncertainties
The Sign of an Uncertainty
Benefits of Repeated Measurements
Regression Analysis (Least-Squares Method)
Weighted Mean
Weighted Linear Regression
General Regression Analysis
Benefits of Regression Analysis
Data Consistency within Data Sets
Criterion of Chauvenet
Discarding Data with Internal Uncertainties
Correlation
Introduction
Measure of Relation
Correlated (Systematic) Uncertainties
Sign of a Systematic'' Uncertainty Differentiation from Uncorrelated Uncertainties More Examples of Correlated Uncertainties External Scale Uncertainties? Differentiation fromSystematic Errors''
Gross Mistakes
Corrections
Correlation in Cases of Linear Regression
Weighted Linear Regression (Example)
Linear Regression without Weighting (Example)
Consistency among Data Sets
Contradictory Data Sets
Dependent (Correlated) Data Sets
Target Shooting as a Model for Uncertainties
Accuracy vs. Precision
Dealing with Internal Uncertainties
Calculations with Both Types of Uncertainties
Uncertainty of a Sum
Uncertainty of a Difference
Uncertainty of a Product
Uncertainty of a Ratio
Uncertainty of a Power (Root)
Uncertainty of More Exotic Functions
Total Uncertainty
Adding Correlated to Uncorrelated Uncertainties
Combined Uncertainty of a Single Best Estimate
Combined Uncertainty of Data Sets
Using Internal Uncertainties for Diagnosis
Quality Assurance
Analogy to Bayes' Principle
Chi-Squared Test
Presentation and Estimation of Uncertainties
Graphic Presentation, Also of Uncertainties
Uncertainty Bars (Error Bars)
Charts
Transformations
Correct Presentation of Uncertainties
Finding the Size of Internal Uncertainties
Ideal Situation in Measurements
Pragmatic Solution for Measurements
Estimating the Size of Uncertainties
Finding Upper Limits
Finding Lower Limits
Feedback of Uncertainties on Experiment Design
Optimizing Experiments
Reasons For and Against Optimization
Prevalent Design Criteria
Optimizing Background Measurements
Optimized Simple Background Measurement
Optimized Complex Background Measurement
Optimizing with Respect to Dead Time
Optimizing in View of the Mathematical Presentation
Optimizing Flat Dependences
Optimizing Linear Dependences
Optimum Angles for Cross Section Measurements
Achieving the Smallest Overall Uncertainty
The Ratio Method
Solutions
Index