Learning Outcomes:
Be able to use Python for analysis, interpretation and presentation of Physics data.
Have an understanding of the statistical methods used to analyse and interpret data including hypothesis testing and P values.
Understand the fundamental statistical distributions that describe physical measurements and their variations (e.g. binomial, Poisson, normal, Student's t).
Have an understanding of experimental measurement and uncertainties, including statistical and systematic errors, and to use appropriate precision when quoting uncertainties.
Be able to characterise data through parameters such as the mean, standard deviation, covariance, weighted mean and uncertainties on the weighted mean.
Be able to propagate errors on measurements through functions of those measurements, both analytically and numerically.
Be able to fit a function to experimental data to derive best-fit parameters including the uncertainties on the parameters and to use the best-fit covariance matrix to calculate confidence intervals.
Be able to apply a chi-squared test to assess the "goodness of fit" of a function fitted to data.
Be able to produce high-quality scientific plots with experimental data, best-fit curves and confidence intervals.
Be able to use random number generators to simulate physical processes to compare with expected distributions and lay the foundation for exploring more complex systems.
Indicative Module Content:
Python:
- Standard Python (calculations, containers, logical tests, iteration, functions)
- NumPy essentials and array-wise calculations
- SciPy.stats toolbox
- SciPy.optimize.curve_fit
- Jupyter Notebooks including markdown and LaTeX equations
- MATPLOTLIB and scientific data presentation
Experimental Measurement & Uncertainties
- accuracy, precision, statistical and non-statistical/systematic errors
- presentation of data and significant figures
- measures of a value and spread (mean and standard deviation, parent and sample)
Basic Probability Overview:
- Basic probability theory, discrete and continuous probability distributions
- Hypothesis testing and P values
- Statistical significance and confidence
- Statistical trials and corrected probability
- Investigation of probability using random numbers and simple Monte Carlo simulations
Measurement probability distributions:
- Binomial
- Poisson
- Normal
- Student's t
The weighted mean and its error
Propagation of Errors:
- propagation of errors formula and covariance terms
- examples, with and without covariance
Curve Fitting and Confidence Intervals:
- Method of least squares
- Chi-squared fitting
- Interpreting the results of the fit including errors on fit parameters and covariance terms
- Calculate uncertainty on a value calculated from fit function using best-fit parameters
- Confidence region for a fit
Chi-squared test for goodness of fit
- Chi-squared distribution, degrees of freedom, P-value
- Chi-square comparison of two distributions
Computer/Matrix Methods for Propagation of Errors