kvantili

A kvantili, or quantile, is a value that divides a probability distribution or data set into intervals with equal probability. For a random variable X with cumulative distribution function F, a p-quantile q_p is defined by F(q_p) = p. When F is continuous, q_p can be obtained as the inverse CDF, q_p = F^{-1}(p). This means that p of the probability mass lies at or below q_p.

Common quantiles include quartiles (p = 0.25, 0.5, 0.75), deciles (p = 0.1, 0.2, ..., 0.9), and percentiles (p = 0.01, 0.02, ..., 0.99). The interquartile range (IQR) is the difference between the third and first quartiles and is a measure of statistical dispersion.

Empirical quantiles are estimated from sample data. If a data set with n observations is ordered as x_(1) ≤ … ≤ x_(n), a p-quantile can be estimated using an index k = floor(p(n+1)) or similar conventions, often with interpolation between neighboring order statistics to improve accuracy.

Quantiles summarize distributional information and are robust measures of central tendency and spread in many contexts. The median is the 0.5-quantile and serves as a robust central value. Quantiles are widely used in descriptive statistics, box plots, hypothesis testing, and data visualization to compare distributions and identify outliers.

In multivariate settings, the concept of a single quantile becomes more complex, with approaches based on depth or spatial quantiles evolving to generalize the idea to higher dimensions.