Mathematical statistics

This online module provides you with the mathematical underpinning for statistical methods in general and – in particular – for other OU statistics modules. You will gain a thorough grounding in mathematical statistics, together with generic skills. You will study distribution theory, leading on to the theory of statistical inference developed under both classical and Bayesian approaches. In the classical case, you will focus on maximum likelihood estimation. You'll also explore the development of these ideas in the context of linear modelling (regression and extensions). To study this module, you should have a sound knowledge of basic statistical ideas and competence in calculus, algebra and matrices, as provided by the appropriate OU level 1 and 2 study.

Course facts
About this course:
Course code M347
Credits 30
OU Level 3
SCQF level 10
FHEQ level 6
Course work includes:
4 Tutor-marked assignments (TMAs)
14 Interactive computer-marked assignments (iCMAs)
No residential school

What you will study

Other OU statistics modules focus on hands-on practical applications of statistical techniques and interpretation of data and statistical analyses. This module complements these modules by providing the mathematical theory underlying the methods and concepts, including a treatment of both classical and Bayesian statistics. A considerable amount of mathematics is sometimes required for this development.

This module is delivered online, with integrated use of exercises, animations, audio and video segments. You will also be provided with printed versions of the main units and extra exercises.

The module is divided into four blocks of study.

Block 1: Review and distribution theory

The first block comprises a review unit and units introducing distribution theory. The review is mostly of fundamental statistical ideas of the type taught in Analysing data (M248), (see Entry section below for details); there is also a speedy reminder of important relevant methods in mathematics, including calculus and matrices. Two units in this block introduce the theory of continuous distributions. You will learn, for example, how to evaluate moments of distributions and about other properties of some important univariate distributions. The mathematical structure of multivariate distributions will be explored, with some emphasis on the multivariate normal distribution.

Block 2: Classical inference

The second block is about the classical approach to statistical inference. You will learn how to use calculus to obtain maximum likelihood estimators of parameters. You will also learn about the properties of maximum likelihood estimation and of point estimation more generally. The mathematics underlying hypothesis tests and confidence intervals will be explored. There is also a unit on asymptotic (large sample) analysis, giving an insight into how statisticians study properties of statistical procedures by approximate methods.

Block 3: Bayesian statistics

In the third block you'll consider the Bayesian approach to statistical inference. The emphasis is first on so-called conjugate analysis which constitutes the type of Bayesian analysis most amenable to straightforward mathematical development. You'll consider prior to posterior analysis first, followed by Bayesian estimation based on decision theory. Markov chain Monte Carlo (MCMC) is a technique often used for tackling Bayesian problems which are not conjugate; you'll investigate the mathematical ideas leading to the basic methods of MCMC.

Block 4: Linear modelling

The fourth and final block gives some of the mathematical development underlying linear modelling. The material covers linear regression on a single explanatory variable; multiple linear regression where there is more than one explanatory variable; and generalised linear modelling for regression situations where the normal distribution is not a suitable model for variation in the response. Both classical and Bayesian approaches to the analysis of these models are considered.

You will learn

Successful study of this module should enhance your skills in understanding some useful mathematical theory, interpreting mathematical results in a statistical context, constructing logical arguments, and finding solutions to problems.


You need no pre-requisites to study Mathematical statistics. However, we recommend that you're familiar with the following mathematical topics:

  • calculus
  • algebra
  • matrices
  • basic differentiation and integration

We recommend you also have previous basic knowledge of statistical science; we'll include some revision of the following topics:

  • normal, Poisson and binomial distributions
  • the Central Limit Theorem
  • point estimation
  • maximum likelihood estimation
  • confidence intervals
  • hypothesis testing
  • simple linear regression
  • correlation

Check you're ready to study Mathematical statistics with our self-assessed quiz.

Talk to an advisor if you're still not sure if you're ready.

Preparatory work

Mathematical statistics

If you have a disability or additional requirement

At present the mathematical equations in the module text are not accessible to a screen reader. We hope that soon they will be. Mathematics read in this way may, however, be difficult to understand.

Descriptions of core figures and animations are available, as are written transcripts of any audio component. Other alternative formats of the study materials may be available in the future.

Study materials

What's included

All the study materials (including the Study Guide), activities, assessment and study support are delivered online via the module website. You will also be provided with printed versions of the main units and extra exercises.

You will need

Calculator with basic mathematical functions (exp, log, etc), but not necessarily with statistical functions.

Computing requirements

A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

A desktop or laptop computer with either:

  • Windows 7 or higher
  • macOS 10.7 or higher

The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

To participate in our online-discussion area you will need both a microphone and speakers/headphones.

Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

Teaching and assessment

Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and who you can ask for advice and guidance. Tutorials will mainly be held online.

Contact us if you want to know more about study with The Open University before you register.


The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.

Although your scores on the TMAs and interactive computer-marked assignments (iCMAs) will not contribute directly to your final grade, and not all the TMAs and iCMAs are compulsory, you will need to complete about three-quarters of them (the total workload for all TMAs and iCMAs will be less than four standard TMAs). You will be given more information when you begin the module.

Future availability

Mathematical statistics starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2022.

This course is expected to start for the last time in October 2022.

Professional recognition

This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.

This module may also help you to apply for the professional award of Graduate Statistician conferred by The Royal Statistical Society (RSS).