
In
probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

and
statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

, the
quantile
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mo ...

function, associated with a
probability distribution
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
of a
random variable
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function, percent-point function or inverse cumulative distribution function.
Definition
With reference to a continuous and strictly monotonic distribution function, for example the
cumulative distribution function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...