In statistics and the sciences, variables in and of themselves represent an unknown value. In the analysis of those values, particularly in empiricism and statistics, it is often important to understand the nature of those variables - not so much how they behave in the equations, but what they represent in reality.
Variables can be classified in two distinct ways by mathematicians, and several ways by the statistician.
Mathematicians will generally classify certain variables as discrete and continuous. The latter means that the variable can have any value, either in a given interval or unbounded, but any intermittent value may be treated. An example is the real number line which contains all of the fractional, irrational and transcendental values, along with the integers.
Discrete values means that the values do not exist on a continuum, but must take separate and distinct values. The most general case are the integers. However, values may take distinct fractional values, and need not be evenly spaced.
In the world of statistics, however, variables can be classified in four ways:
The first classification carries no worth in pure mathematics. The Nominal Variables, which are purely descriptive, have no numerical value. They are purely taxonomic classification, intrinsic qualities which cannot be quantified in a comparative way. Such variables include R (for "race"), where an individual may have the race-value of "Hispanic" or "Dutch" or "German" or "Chinese", etc.
The second classification carries very little worth in strict mathematics. The Ordinal Variables are merely words, nouns, that distinguish a certain set of data values from another - but still compare data values in a limited way. They are comparative and very generic. They are called "ordinal" because the data set may be ordered from one extreme to another based on a single quantifiable quality. Examples are the variables t (the number of "tall people", where a tall person is anyone above 5 feet), and s (the number of "short people", where a short person is anyone less than 5 feet). Some statisticians will have overlap variables such as m (the number of "middle-height people" who can be classed as either tall or short, because they are shorter than 5'3" but taller than 4'9"). The entire set of people fall into distinct classes, a discrete variable, based on the measurements of a quantified characteristic.
The third classification is the Interval Variables. These variables have numerical meaning and can be applied in mathematics beyond mere statistics and taxonomic classifications. Using the previous example of "height", the variable h is an interval variable. This is because the variable h can have any numerical value which represents an individuals height. Interval variables are typically continuous; but there is no restriction that they cannot be discrete, just so long as they dont qualify as purely Ordinal. There is no universal consensus on that, though. Interval variables can, however, be ordered as specific quantities can be taken and compared. Another example of Interval Values are the temperature scales of Celsius and Fahrenheit.
The fourth classification of variables are the Ratio Variables. Very similar to the Interval variables, they can carry any numerical value. They are never, however, discrete. There is very little distinction between Interval and Ratio variables, except the quality of absolutism. Interval Values may be added and subtracted and compared, but they cannot be multiplied or divided. Interval values are relative values. Ratio values, on the other hand, are absolute values. Given the example above of temperature, the interval variable can be in Celsius, but the Ratio variable would necessarily be in Kelvin.