Kvantil

Kvantil, or quantile, is a value that partitions a probability distribution into intervals carrying specified probabilities. For a random variable X with distribution function F, the p-quantile q_p is a value for which F(q_p) is at least p and F(q_p−) is at most p. In continuous distributions, the p-quantile is unique and can be written as the inverse CDF, q_p = F^{-1}(p). Quantiles provide a concise summary of where most of the distribution’s mass lies.

Common quantiles include quartiles (p = 0.25, 0.5, 0.75), deciles (p = k/10), and percentiles (p = k/100). The p = 0.5 quantile is the median, while p = 0.25 and p = 0.75 are the lower and upper quartiles, respectively. Quantiles are widely used to describe distributions, compare data sets, and summarize robustly where observations tend to lie.

In practice, quantiles are estimated from samples. If x_(1) ≤ x_(2) ≤ … ≤ x_(n) are the order statistics, various methods exist to define the empirical p-quantile, often involving interpolation between order statistics. The choice of method affects edge cases, especially for small samples or extreme p values. For large samples, the empirical quantiles converge to the true quantiles, with asymptotic variance that depends on the density at the quantile.

Quantiles have broad applications in statistics and data analysis, including descriptive summaries, box plots, and risk assessment. They relate to percentile notions and are foundational in understanding the spread and central tendency of a distribution beyond the mean.