The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that satisfy the basic principles stated for these different approaches. Within a given approach, statistical theory gives ways of comparing statistical procedures; it can find a best possible procedure within a given context for given statistical problems, or can provide guidance on the choice between alternative procedures. Apart from philosophical considerations about how to make statistical inferences and decisions, much of statistical theory consists of mathematical statistics, and is closely linked to probability theory, to utility theory, and to optimization. Scope Statistical theory provides an underlying rationale and provides a consistent basis for the choice of methodology used in applied statistics. Modelling Statistical models describe the sources of data and can have different types of formulation corresponding to these sources and to the problem being studied. Such problems can be of various kinds: Sampling from a finite population Measuring observational error and refining procedures Studying statistical relations Statistical models, once specified, can be tested to see whether they provide useful inferences for new data sets. Data collection Statistical theory provides a guide to comparing methods of data collection, where the problem is to generate informative data using optimization and randomization while measuring and controlling for observational error. Optimization of data collection reduces the cost of data while satisfying statistical goals, while randomization allows reliable inferences. Statistical theory provides a basis for good data collection and the structuring of investigations in the topics of: Design of experiments to estimate treatment effects, to test hypotheses, and to optimize responses. Survey sampling to describe populations Summarising data The task of summarising statistical data in conventional forms (also known as descriptive statistics) is considered in theoretical statistics as a problem of defining what aspects of statistical samples need to be described and how well they can be described from a typically limited sample of data. Thus the problems theoretical statistics considers include: Choosing summary statistics to describe a sample Summarising probability distributions of sample data while making limited assumptions about the form of distribution that may be met Summarising the relationships between different quantities measured on the same items with a sample Interpreting data Besides the philosophy underlying statistical inference, statistical