Cover

Half-title

Title

Copyright

Dedication

Contents

Preface

Part I: Data and error analysis

1 Introduction

2 The presentation of physical quantities with their inaccuracies

2.1 How to report a series of measurements

List all data, a histogram or percentiles

List properties of the data set

Example: 30 observations

2.2 How to represent numbers

Decimal separator: comma or period?

Significant figures

Examples, for the English language

2.3 How to express inaccuracies

Absolute and relative errors

Examples

Using probability distributions

Examples

2.4 Reporting units

SI units

Non-SI units

Typographical conventions

Examples

2.5 Graphical presentation of experimental data

Exercises

3 Errors: classification and propagation

3.1 Classification of errors

Systematic errors

Random errors or uncertainties

Know where the errors are

3.2 Error propagation

Propagation through functions

Example

Combination of independent terms

Example 1

Example 2

Combination of dependent terms: covariances

Systematic errors due to random deviations

Monte Carlo methods

Exercises

4 Probability distributions

4.1 Introduction

4.2 Properties of probability distributions

Normalization

Expectation, mean and variance

Moments and central moments

Cumulative distribution function

Characteristic function

A word on nomenclature

Numerical values of distribution functions

4.3 The binomial distribution

Definition and properties

Variance proportional to number

Examples

From binomial to multinomial

4.4 The Poisson distribution

4.5 The normal distribution

The Gauss function

Relation of cdf to error function

Probability scales

Significant deviations

4.6 The central limit theorem

4.7 Other distributions

Log-normal distribution

The Lorentz distribution: undefined variance

Lifetime and exponential distributions

The hazard function

The exponential distribution

Population statistics

Chi-squared distribution

Student’s t-distribution

F-distribution

Example

Exercises

5 Processing of experimental data

5.1 The distribution function of a data series

5.2 The average and the mean squared deviation of a data series

5.3 Estimates for mean and variance

5.4 Accuracy of mean and Student's t-distribution

5.5 Accuracy of variance

5.6 Handling data with unequal weights

Accuracy of the estimated mean

5.7 Robust estimates

Elimination of outliers

Rank-based estimates

Sign-based confidence intervals

The bootstrap method

Exercises

6 Graphical handling of data with errors

6.1 Introduction

6.2 Linearization of functions

Example: urease kinetics

6.3 Graphical estimates of the accuracy of parameters

6.4 Using calibration

Exercises

7 Fitting functions to data

7.1 Introduction

7.2 Linear regression

Uncertainties in x

The best parameter estimates

Uncertainties in the parameters

Covariances between parameters

Should you use S0 or…

Correlation coefficient between x and y values of a data series

7.3 General least-squares fit

Example: nonlinear fit

7.4 The chi-squared test

Example 1

Example 2

7.5 Accuracy of the parameters

Covariances of the parameters

Relation between chi2 and 1-D parameter distribution

Relation between chi2 and 2-D parameter distribution

Example

7.6 F-test on significance of the fit

Example

Exercises

8 Back to Bayes: knowledge as a probability distribution

8.1 Direct and inverse probabilities

8.2 Enter Bayes

8.3 Choosing the prior

8.4 Three examples of Bayesian inference

Updating knowledge: Avogadro’s number

Inference from a series of normally distributed samples

Infer a rate constant from a few events

8.5 Conclusion

References

Answers to exercises

Part II: Appendices

A1 Combining uncertainties

Why do squared uncertainties add up in sums?

A2 Systematic deviations due to random errors

A special case: sampling exponential functions

A3 Characteristic function

A4 From binomial to normal distributions

A4.1 The binomial distribution

A4.2 The multinomial distribution

A4.3 The Poisson distribution

From binomial to Poisson

Properties of the Poisson distribution

A4.4 The normal distribution

From Poisson to normal

A5 Central limit theorem

A6 Estimation of the variance

Why is the best estimate for the variance larger than the mean squared deviation of the average?

Uncorrelated data points

Correlated data points

A7 Standard deviation of the mean

Why is the variance of the mean of n independent data equal to the variance of x itself divided by n?

How is this result influenced when the data are correlated?

Example

How accurate is the estimated standard deviation?

A8 Weight factors when variances are not equal

What is the “best” determination of the mean of a number of data xi with the same expectations μ but with unequal standard deviations σi?

How large is the variance in ?

A9 Least-squares fitting

A9.1 How do you find the best parameters a and b in y approx ax + b?

A9.2 General linear regression

A9.3 SSQ as a function of the parameters

A9.4 Covariances of the parameters

Why is the s.d. of a parameter given by the projection of the ellipsoid…

Nonlinear least-squares fit

Part III: Python codes

Part IV: Scientific data

Chi-squared distribution

Probability distribution sum of squares

Cumulative chi2-distribution

Chi-squared distribution

Values of chi2 for 1%, 10%, 50%, 90%, and 99%

F-distribution

F-distribution

Use in ANOVA (analysis of variance) in regression

F-distribution, percentage points 95% and 99%

Least-squares fitting

General least-squares fitting

Linear in the parameters

Correlation coefficient r of x and y

Normal distribution

One-dimensional Gauss function

Multivariate Gauss functions

Probability that…

Probability that…

Physical constants

Relative standard deviations

Accuracies of derived quantities

Probability distribution

Continuous one-dimensional probability functions

Continuous two-dimensional probability functions

Student's t-distribution

Student’s t-distribution

Application: accuracy of the mean

Properties and moments

Values of t at 75%, 90%, 95%, 99%, and 99.5%

Units

Definitions SI basic units

Derived SI units

Atomic units (a.u.)

Molecular units

Index