Variable (Statistics)

In statistics, a variable is a characteristic of a subject that varies in a non-random way.

Overview and Related Definitions

At the top level of statistics, we investigate a population: a set of units that we are interested in studying.

Populations are almost always impossible to study due to their massive size and other constraints. Therefore, we take a sample: a subset of the population.

A subject is a unit that we study in a population or a sample, and a variable is a particular characteristic of the subject that we are interested in studying.

Types of Variables

There are two types of variables, each with two sub-categories that have useful properties:

• Quantitative/Numerical variables are measured with numbers. Their sum has meaning.
  • Continuous numerical variables can theoretically take on any number within an interval, whereas
  • Discrete numerical variables have natural gaps
• Qualitative/Categorical variables are measured as labels. Their sum does not have meaning.
  • Ordinal categorical variables have a natural ordering, whereas
  • Nominal categorical variables do not

It is possible for a categorical variable to be denoted with numbers; a common example would be an ID number. The biggest difference between categorical variables denoted as numbers and numerical variables is the fact that the sum/mean of categorical variables does not have meaning, whereas that of numerical variables do.

Notation

A capitalized character (usually $X,Y,Z$) is used to denote all possible values of a variable. This is called a major. When we say "variable", we usually mean this.

A lower case character corresponding to the major is used to denote a specific value of that major (such as $x,y,z$. This is called a statistic.

Random Variable

A random variable is a numerical variable whose outcome is the result of a random process (i.e. we don't know what will happen for certain). Notably, the numerical interpretation of the outcome of an experiment is a random variable. They come in two types, deserving their own pages. See Continuous Random Variable and Discrete Random Variable.