A mathematical model is, plain and simple, an equation, or set of equations, that represents a relationship between two or more things. Such equations are shorthand for theories about the workings of nature and society. The theory may be supported by a substantial body of evidence -- or be just a wild guess. The language of mathematics is the same in either case.
Terms such as “predictive model,” “statistical model," or “linear model” refer to specific types of mathematical models, the names reflecting the intended use, the form, or the method of deriving a particular model. These three examples are just a few of many such possibilities.
Why use mathematical models? Couldn’t the same relationships be described using words? That's possible, yet there are certain advantages to the use of equations. These include:
- Convenience. Compared with equivalent descriptions written out in sentences, equations are brief. Mathematical symbolism has evolved specifically for the purpose of representing mathematical models; languages such as English have not.
- Clarity. Equations convey ideas succinctly and are unambiguous. They're not subject to differing interpretations based on culture, and the symbolism of mathematics is a sort of common language used widely across the globe by people who may share no other.
- Consistency. Because mathematical representations are unambiguous, the implications of any particular situation are clearly defined by a mathematical model (at least, the implications associated with the theory behind that particular model).
When a model is mentioned in a business setting, it’s most likely a model used to make predictions. Models are used to predict stock prices, product sales, and unemployment rates, among many other things. These predictions may or may not be accurate, but, for any given set of values for the known factors (also called independent variables or inputs) included in the model, there will be a well defined prediction (also called dependent variable, output, or result). Mathematical models are used for other purposes in business, as well, such as to describe the working mechanisms that drive a particular process.
Some mathematical models may be nothing more than one individual’s personal theory of how something works, but most have more substance behind them. Better models arise from a mix of carefully collected data from observation, building blocks based on statistical theory and other mathematical methods, and the experience of a knowledgeable researcher. And a model cannot be judged to be good until its performance has been tested using real data -- data that was not used to develop the model in the first place.
A model represents a theory about how some part of the world functions, but the world is mighty complex. So a model that has worked well for one purpose may not be adaptable for another. Or it may prove accurate in one time period, but not in the next. Alternative theories may come to light and prove to be even more accurate. Models don't represent reality, only our best understanding of it.