fractile

Fractile is a statistical concept describing a value that partitions a probability distribution into two parts containing prescribed probabilities. Formally, the p-th fractile is a number x such that P(X ≤ x) = p for a random variable X, where p lies in (0,1). The term is often used synonymously with quantile, though some texts treat fractile as an alternative designation.

Common fractiles include the median (p = 0.5) and quartiles (p = 0.25, 0.5, 0.75). Percentiles are fractiles with p expressed as a percentile; for example, the 90th percentile corresponds to p = 0.90. Fractiles can be defined for any p, including extreme values near 0 or 1.

To compute fractiles for a continuous distribution, one uses the inverse cumulative distribution function (CDF), F^{-1}(p). For empirical data, fractile estimation relies on order statistics; various methods exist to map p to a position in a sample, such as nearest-rank or linear interpolation between adjacent data points.

Fractiles support data description, comparison of distributions, and robust analysis. They are less sensitive to outliers than moments and are used in descriptive statistics, plotting (such as box plots), normalization, and nonparametric inference. The concept is closely related to quantiles and percentiles, and the terminology varies across disciplines and languages.

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