Many universities use diagnostic tests to assess incoming students' preparedness for mathematics courses. Diagnostic test results can help students to identify topics where they need more practice and give lecturers a summary of strengths and weaknesses in their class. We demonstrate a process that can be used to make improvements to a mathematics diagnostic test, so that it provides better information about students' abilities. To identify questions that could be improved, our process uses item response theory to analyse student data. We provide an accessible introduction to the key ideas of item response theory, and we describe how we applied the process of improvement of diagnostic tests at two universities. Our method provides a model for others wishing to make improvements to mathematics diagnostic tests.