# Quantile

One of the numerical characteristics of a probability distribution. For a real random variable with distribution function , by a quantile of order , , one means the number for which , . If is a continuous strictly-monotone function, then is the unique solution of the equation , that is, is the function of inverse to the function . If is continuous and , then the probability of the inequality , is equal to . The quantile is the median (in statistics) of the random variable . The quantiles and are called the quartiles, and , the deciles. The values of the quantiles for suitable values of enable one to form an idea about the distribution function.

For example, for the normal distribution (see Fig.)

the graph of the function can be drawn from the deciles: ; ; ; ; ; ; ; ; .

Figure: q076270a

The quartiles of the normal distribution are and .

#### Comments

#### References

[a1] | L. Breiman, "Statistics" , Houghton Mifflin (1973) pp. 231ff |

[a2] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) pp. 181; 367 |

**How to Cite This Entry:**

Quantile.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quantile&oldid=14159