Difference between revisions of "Quantile"
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One of the numerical characteristics of a [[Probability distribution|probability distribution]]. For a real random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762701.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762702.png" />, by a quantile of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762705.png" />, one means the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762706.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762708.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762709.png" /> is a continuous strictly-monotone function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627010.png" /> is the unique solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627011.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627012.png" /> is the function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627013.png" /> inverse to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627015.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627016.png" />, then the probability of the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627017.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627018.png" />. The quantile <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627019.png" /> is the [[Median (in statistics)|median (in statistics)]] of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627020.png" />. The quantiles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627022.png" /> are called the quartiles, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627023.png" />, the deciles. The values of the quantiles for suitable values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627024.png" /> enable one to form an idea about the distribution function. | One of the numerical characteristics of a [[Probability distribution|probability distribution]]. For a real random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762701.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762702.png" />, by a quantile of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762705.png" />, one means the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762706.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762708.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q0762709.png" /> is a continuous strictly-monotone function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627010.png" /> is the unique solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627011.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627012.png" /> is the function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627013.png" /> inverse to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627015.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627016.png" />, then the probability of the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627017.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627018.png" />. The quantile <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627019.png" /> is the [[Median (in statistics)|median (in statistics)]] of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627020.png" />. The quantiles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627022.png" /> are called the quartiles, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627023.png" />, the deciles. The values of the quantiles for suitable values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076270/q07627024.png" /> enable one to form an idea about the distribution function. | ||
Revision as of 08:22, 8 April 2012
2020 Mathematics Subject Classification: Primary: 60-01 Secondary: 62-01 [MSN][ZBL]
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 MR0359089 Zbl 0289.62001 |
| [a2] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) pp. 181; 367 MR0016588 Zbl 0063.01014 |
How to Cite This Entry:
Quantile. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantile&oldid=24270
Quantile. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantile&oldid=24270
This article was adapted from an original article by V.V. Senatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article