 a statistical model is a set of assumptions concerning the generation of the observed data, and similar data from a larger population. the model represents, often in considerably idealized form, the data-generating process. a statistical model is used to describe a set of probability distributions, some of which are assumed to reasonably approximate the distribution from which a particular data set is sampled. the model is formally specified by relationships among one or more random variables and other (non-random) variables; the relationships are usually given as mathematical equations. herman adÃ¨r quotes kenneth bollen as saying, "a model is a formal representation of a theory".adÃ¨r all statistical tests can be described in the form of statistical models. for example, the student's t-test for comparing the means of two sets can be formulated as determining if an estimated parameter in the model is different from 0. similarly, all statistical estimates, such as confidence intervals, are derived from statistical models. ==formal definition== in mathematical terms, a statistical model is usually thought of as a pair (s, \mathcal{p}), where s is the set of possible observations, i.e. the sample space, and \mathcal{p} is a set of probability distributions on s.mccullagh the intuition behind this definition is as follows. it is assumed that there is a "true" probability distribution that generates the observed data. we choose \mathcal{p} to represent a set (of distributions) which contains a distribution that adequately approximates the true distribution. note that we do not require that \mathcal{p} contains the true distribution, and in practice that is rarely the case. indeed, as burnham & anderson state, "a model is a simplification or approximation of reality and hence will not reflect all of reality".burnham & anderson, Â§1.2.5 the set \mathcal{p} is usually parameterized: \mathcal{p}=\{p_{\theta} : \theta \in \theta\}. the set \theta defines the parameters of the model. ==an example== height and age are each probabilistically distributed over humans. they are stochastically related: when you know that a person is of age 10, this influences the chance of the person being 6 feet tall. you could formalize that relationship in a linear regression model with the following form: heighti&nbsp;= b_0&nbsp;+ b_1agei&nbsp;+ Îµi, where b_0 is the intercept, b_1 is a parameter that age is multiplied by to get a prediction of height, Îµ is the error term, and i identifies the person. this means that height is predicted by age, with some error. a model must fit all the data points. thus, the straight line (heighti&nbsp;= b_0&nbsp;+ b_1agei) 
Scientific_modeling Statistical_models Statistical_theory