## Measurements and Their Uncertainties: A Practical Guide to Modern Error AnalysisThis hands-on guide is primarily intended to be used in undergraduate laboratories in the physical sciences and engineering. It assumes no prior knowledge of statistics. It introduces the necessary concepts where needed, with key points illustrated with worked examples and graphic illustrations. In contrast to traditional mathematical treatments it uses a combination of spreadsheet and calculus-based approaches, suitable as a quick and easy on-the-spot reference. The emphasis throughout is on practical strategies to be adopted in the laboratory. Error analysis is introduced at a level accessible to school leavers, and carried through to research level. Error calculation and propagation is presented though a series of rules-of-thumb, look-up tables and approaches amenable to computer analysis. The general approach uses the chi-square statistic extensively. Particular attention is given to hypothesis testing and extraction of parameters and their uncertainties by fitting mathematical models to experimental data. Routines implemented by most contemporary data analysis packages are analysed and explained. The book finishes with a discussion of advanced fitting strategies and an introduction to Bayesian analysis. |

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### Contents

1 Errors in the physical sciences | 1 |

2 Random errors in measurements | 9 |

3 Uncertainties as probabilities | 23 |

4 Error propagation | 37 |

5 Data visualisation and reduction | 53 |

6 Leastsquares fitting of complex functions | 67 |

7 Computer minimisation and the error matrix | 85 |

8 Hypothesis testinghow good are our models? | 101 |

9 Topics for further study | 121 |

131 | |

133 | |

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### Common terms and phrases

accepted value appropriate approximation best estimate best-fit parameters best-fit straight line calculate central limit theorem Chapter summary confidence limits constant contour correlation covariance curvature matrix curve data points data set defined degrees of freedom digit discussed error analysis error bars error surface evaluated example experiment experimental data fit parameters Gaussian distribution gradient graph Hessian matrix histogram independent variable intercept iterative least-squares fit linear look-up tables method minimisation minimum multi-variable functions nonlinear function normalised residuals null hypothesis number of counts number of data number of degrees number of measurements obtaining a value occurrences parent distribution peak Poisson distribution precision probability distribution function probability of obtaining procedure propagation quantity radioactive decay range repeat measurements sample distribution Section shown in Fig shows significant figures single-variable function spreadsheet squares standard deviation standard error systematic errors theoretical model tion uncertainty voltage weighted width zero