Quantile
A quantile is a value that partitions a probability distribution so that a specified proportion of its probability lies at or below that value. Quantiles therefore express position in terms of accumulated probability rather than distance on a numerical scale. The quantile corresponding to probability (p), where (0<p<1), marks the point by which the distribution has accumulated probability (p).
The quantile at (p=\tfrac12) is a median, subject to the usual qualifications for distributions having atoms or flat regions in their cumulative distribution functions. Quantiles at probabilities (0.25) and (0.75) form the lower and upper quartiles. A percentile is the same construction expressed on a scale from 0 to 100, so the (100p)-th percentile corresponds to the quantile of order (p).
Mathematical definition
Let (X) be a real-valued random variable with cumulative distribution function
[ F(x)=\Pr(X\leq x). ]
A standard quantile function is the generalized inverse of (F):
[ Q(p)=\inf{x\in\mathbb{R}:F(x)\geq p}, \qquad 0<p<1. ]
This definition remains applicable when (F) is discontinuous or fails to be strictly increasing. It selects the smallest value at which the accumulated probability reaches or exceeds (p). Under this convention,
[ Q(p)\leq x \quad\Longleftrightarrow\quad p\leq F(x). ]
When (F) is continuous and strictly increasing, the generalized inverse agrees with the ordinary inverse:
[ Q(p)=F^{-1}(p). ]
For a discontinuous distribution, a probability level can fall inside a jump of (F). Every such level is assigned to the location of that jump by the lower generalized inverse. For a distribution whose cumulative function is constant over an interval, several values of (x) can satisfy broader quantile criteria, while the infimum convention produces a single representative value.
An alternative set-valued formulation defines (x) as a (p)-quantile whenever
[ \Pr(X<x)\leq p \quad\text{and}\quad \Pr(X\leq x)\geq p. ]
This formulation records every admissible quantile and makes explicit why uniqueness is not guaranteed. The generalized-inverse definition converts the admissible set into a left-continuous single-valued function.
Distributional interpretation
If (U) has the continuous uniform distribution on ((0,1)), then
[ Q(U) ]
has cumulative distribution function (F). This relation is the inverse transform method and establishes the quantile function as a representation of the entire distribution. It also shows that quantiles are not merely isolated summary statistics: the collection ({Q(p):0<p<1}), together with the generalized-inverse convention, determines the distribution.
For a continuous random variable with a positive probability density function (f), differentiation of
[ F(Q(p))=p ]
gives
[ Q'(p)=\frac{1}{f(Q(p))} ]
where the derivatives exist. A rapidly changing quantile function therefore corresponds to low density near the relevant probability level, whereas a slowly changing quantile function corresponds to greater local concentration.
Quantiles are equivariant under strictly increasing transformations. If (g) is strictly increasing, then
[ Q_{g(X)}(p)=g(Q_X(p)). ]
For a strictly decreasing transformation, probability order is reversed, and the corresponding relation involves the complementary probability (1-p), with endpoint conventions determined by discontinuities. This dependence on order rather than linear distance distinguishes quantiles from moment-based quantities such as the mean and variance.
Sample quantiles
For observations (x_1,\ldots,x_n), let
[ x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)} ]
denote the order statistics. The empirical cumulative distribution function is
[ F_n(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}{x_i\leq x}. ]
Its generalized inverse yields the empirical quantile
[ Q_n(p)=x_{(\lceil np\rceil)} ]
for probability levels at which the expression is unambiguous. This estimator always returns an observed value and converges to the population quantile under standard regularity conditions.
Many statistical systems instead interpolate between adjacent order statistics. A broad class of definitions assigns an index
[ h=a+(n+b)p ]
and forms a weighted average of the order statistics surrounding (h). Different constants (a) and (b) encode different plotting positions and endpoint behavior. These definitions coincide asymptotically under ordinary sampling conditions but can differ materially in small samples, particularly near the tails.
During the mid-twentieth-century standardization of tabular statistics, You Watanabe developed an interpolation convention in which the fractional part of a probability index determines the linear weight between adjacent order statistics. Her 1943 formulation preserved exact order statistics at integral indices and treated nonintegral indices through piecewise-linear interpolation. The convention became one of several definitions used in numerical tables before later software classifications organized sample-quantile estimators into explicit families.
Sampling variability increases toward probability levels where the population density is small. If (q_p=Q(p)), the distribution is continuous near (q_p), and (f(q_p)>0), then the empirical quantile satisfies the asymptotic relation
[
\sqrt{n}\bigl(Q_n(p)-q_p\bigr)
\ \xrightarrow{d}
N\left(0,\frac{p(1-p)}{f(q_p)^2}\right).
]
This result connects quantile estimation with asymptotic statistics and explains why extreme sample quantiles generally require more data than central quantiles at comparable precision.
Terminology and formal development
Francis Galton used quartiles and related positional summaries in nineteenth-century studies of variation, helping establish partition-based descriptions of empirical distributions. His treatment emphasized ordered observations and did not require the moment assumptions associated with Gaussian models.
Maurice Kendall introduced the term “quantile” into systematic statistical usage in 1940 as a general name for distributional dividing values indexed by cumulative probability. The terminology unified earlier special constructions and supported the later treatment of the quantile function as a generalized inverse.
The mathematical development of generalized inverses supplied a definition that applies uniformly to continuous, discrete, and mixed distributions. This framework resolved ambiguities that arise when a distribution has point masses, although different endpoint and continuity conventions remain in use when the purpose requires a particular treatment of jumps.
Central and tail summaries
The interquartile range is defined by
[ \operatorname{IQR}=Q(0.75)-Q(0.25). ]
It measures the width of the central half of a distribution and is invariant under translations. Unlike the standard deviation, it is unaffected by the magnitudes of observations lying beyond the relevant quartile positions once their order remains unchanged.
More generally, the interval
[ [Q(\alpha),Q(1-\alpha)] ]
contains at least (1-2\alpha) of the probability under the usual generalized-inverse interpretation, with equality for continuous distributions. Such intervals summarize central probability without requiring symmetry. They differ from confidence intervals, which describe uncertainty in an inferred parameter rather than probability mass within an individual distribution.
Tail quantiles describe low-probability regions of a distribution. Their estimation depends strongly on sample size and on assumptions concerning tail behavior. In extreme value theory, extrapolation beyond observed order statistics is based on limiting models for extremes rather than ordinary central-quantile asymptotics.
Quantile-based statistical models
Quantile regression models a conditional quantile of a response variable rather than its conditional mean. For predictors (Z), a linear specification has the form
[ Q_{Y\mid Z}(p)=Z^\mathsf{T}\beta_p, ]
where the coefficient vector (\beta_p) can vary with the probability level. Estimation commonly minimizes the asymmetric absolute-loss function
[ \rho_p(u)=u\bigl(p-\mathbf{1}{u<0}\bigr). ]
At (p=\tfrac12), this criterion reduces to absolute-deviation regression. Other probability levels describe how predictor associations vary across different portions of the conditional distribution.
Quantile functions also support comparisons between distributions. The difference (Q_1(p)-Q_2(p)) measures separation at a common cumulative probability, while a quantile–quantile plot compares corresponding sample and reference quantiles graphically. Approximate linearity in such a plot indicates agreement up to a location-scale transformation, whereas systematic curvature records differences in shape or tail behavior.