Quantiles

Compact study note.

Summary

Quantile means value cutting off chosen cumulative probability. Quantiles translate CDFs into thresholds and are used for medians, percentiles, and critical values.^[1]

Prerequisites

• Cumulative Distribution Function

Notation and Assumptions

For $0<p<1$ , one $p$ -quantile is any $q_p$ satisfying $F(q_p)\ge p$ and $P(X\ge q_p)\ge 1-p$ . For continuous strictly increasing CDFs, $q_p=F^{-1}(p)$ .

Essential Result

The median is any $0.5$ -quantile.

Small Example

For $X\sim \mathrm{Uniform}(0,10)$ , $F(x)=x/10$ on $[0,10]$ , so the $0.9$ -quantile is $9$ .

Common Mistakes

• Assuming quantiles are unique for discrete distributions.
• Confusing a percentile rank $p$ with the value $q_p$ .

Connections

• Cumulative Distribution Function
• Normal Distribution
• Student's t Distribution

References