Perzentil

Das $\alpha$ Perzentil

Der Wert, für den gilt, dass $\alpha$ Prozent aller Datenpunkte kleiner sind und $1-\alpha$ Prozent der Werte größer sind.

Definition

Let $F$ be a CDF. Then $x_p$ is called $p$-Quantile if $F\left(x_p-\right)\le p\le F\left(x_p\right)$

• $F$ has exactly one $p$-quantile when there is at most one $t$ with $F(t)=p$
• Similar definition when working with real probabilities $P((-\infty ,x_p])\ge p$ and $P([x_p,\infty ))\ge 1-p$

$x_{0.25},x_{0.5},x_{0.75}$ are called quartiles.

We can use quantiles to calculate

• IQR
• Median

We use the Quantile Function to quickly calculate quantiles of a given CDF.

Quantiles of a data sequence can be best visualized using a Boxplot.

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Empirical Quantiles

$$\begin{array}{rl}& Q_{F_n(\cdot ;x)}^{-}(p)=\min \left\{t\in \mathbb{R}\mid \sum _{i=1}^n{1}_{(-\infty ,t]}\left(x_{(i)}\right)\ge np\right\}=x_{(\lceil np\rceil )}\\ & Q_{F_n(\cdot ;x)}^{+}(p)=\max \left\{t\in \mathbb{R}\mid \sum _{i=1}^n{1}_{(-\infty ,t)}\left(x_{(i)}\right)\le np\right\}=x_{(\lfloor np\rfloor +1)}\end{array}$$

Smallest p-Quantile

$Q_F^{-}(p):=\min \{t\in \mathbb{R}\mid F(t)\ge p\}$ > Proof

• $F(t)\ge p$ exists since $F$ is bounded by $0$ and $1$ from below and above
• Hence $t_p=\inf (F(t)\ge p)$
• Due to right continuity we even have $t_p=\min (F(t)\ge p)$

Largest p-Quantile

$Q_F^{+}(p):=\max \{t\in \mathbb{R}\mid F(t-)\le p\}$

Other Definition

$$\begin{array}{c}{x}_p:=\{\begin{array}{ll}\frac{x_{N\cdot p}+x_{N\cdot p+1}}{2}& \text{ falls }N\cdot p\text{ ganzzahlig }\\ x_{\lfloor N\cdot p+1\rfloor }& \text{ falls }N\cdot p\text{ nicht ganzzahlig }\end{array}\end{array}$$