Rank in Statistics
Definition
In statistics, the rank $R : \mathbb{R} \to \mathbb{N}$ is commonly used as a function to indicate the order of data when sorted in ascending order.
Explanation
$$ x_{3} < x_{1} < x_{2} \implies x_{(1)} = x_{3}, x_{(2)} = x_{1}, x_{(3)} = x_{2} $$ In statistics, the method of placing parentheses in the subscript of data is used when the order of data is required. The rank introduced in the definition can be expressed as a function of a random variable as follows, even if the data is not fixed. $$ R(X) = n \qquad , X : \Omega \to \mathbb{R} $$ This means that the random variable $X$ is the $n$th smallest in the entire sample.