View source for Quantile

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{{MSC|60-01|62-01}}

One of the numerical characteristics of a [[Probability distribution|probability distribution]]. For a real random variable $X$ with distribution function $F$, by a quantile of order $p$, $0<p<1$, one means the number $K_p$ for which $F(K_p)\leq p$, $F(K_p+0)\geq p$. If $F$ is a continuous strictly-monotone function, then $K_p$ is the unique solution of the equation $F(x)=p$, that is, $K_p$ is the function of $p$ inverse to the function $F$. If $F$ is continuous and $p'>p$, then the probability of the inequality $K_p<X<K_{p'}$, is equal to $p'-p$. The quantile $K_{1/2}$ is the [[Median (in statistics)|median (in statistics)]] of the random variable $X$. The quantiles $K_{1/4}$ and $K_{3/4}$ are called the quartiles, and $K_{0.1},\ldots,K_{0.9}$, the deciles. The values of the quantiles for suitable values of $p$ enable one to form an idea about the distribution function.

For example, for the [[Normal distribution|normal distribution]] (see Fig.)

$$\Phi(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^xe^{-t^2/2}dt$$

the graph of the function $\Phi$ can be drawn from the deciles: $K_{0.1}=-1.28$; $K_{0.2}=-0.84$; $K_{0.3}=-0.52$; $K_{0.4}=-0.25$; $K_{0.5}=0$; $K_{0.6}=0.25$; $K_{0.7}=0.52$; $K_{0.8}=0.84$; $K_{0.9}=1.28$.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/q076270a.gif" />

Figure: q076270a

The quartiles of the normal distribution $\Phi$ are $K_{1/4}=-0.67$ and $K_{3/4}=0.67$.



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====References====
{|
|valign="top"|{{Ref|B}}|| L. Breiman, "Statistics" , Houghton Mifflin (1973) pp. 231ff {{MR|0359089}} {{ZBL|0289.62001}}
|-
|valign="top"|{{Ref|C}}|| H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) pp. 181; 367 {{MR|0016588}} {{ZBL|0063.01014}}
|}