Have an understanding of experimental measurement and uncertainties, including statistical and systematic errors, and to use appropriate precision when quoting uncertainties.

Understand the fundamental statistical distributions that apply to physical measurements.

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 a set of 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 goodness of fit and f-test to assess whether extra parameters for nested functions significantly improve the fit.

Be able to apply Kolmogorov–Smirnov test and chi-square tests to compare two distributions.

Have an understanding of and be able to apply the Permutation test and Bootstrap/Jackknife tests.

Be apply to apply the Method of Maximum Likelihood, including the Likelihood Ratio Test, for parameter estimation and significance estimation.

Be able to do all of the above in Python using appropriate libraries.

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)

Measurement probability distributions:
- Recap on basic probability theory and statistical distributions
- Binomial
- Poisson
- Normal
- Student-t
- statistical significance and confidence
- statistical trials and corrected probability

The weighted mean and its error

Propagation of Errors:
- error on x and f(x)
- 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
- Uncertainties on x & y, how to handle including combining uncertainties into one ordinate and orthogonal distance regression

Chi-squared test for goodness of fit
- Chi-squared distribution, degrees of freedom, P-value
- f-test for nested functions

Comparison of distributions of data with measured and expected distributions.?
- Chi-square comparison of two distributions
- Kolmogorov–Smirnov test

Computer/Matrix Methods for Propagation of Errors

Monte Carlo Methods:
- Permutation test
- Bootstrap/Jackknife

Method of Maximum Likelihood:
- Maximum Likelihood, log likelihood
- Parameter estimation and errors
- Likelihood ratio test

Upper-limit calculations

Python:
- essentials of programming in Python
- NumPy
- Jupyter Notebooks including markdown and LaTeX equations
- MATPLOTLIB and scientific data presentation
- Additional libraries including numdifftools, lmfit, iminuit,

Not applicable to this module.

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment

Assignments will be graded and returned to students with feedback provided.