Interquartile range (descriptive statistics)

The interquartile range , also briefly interquartile called and IQA or IQR (after the English name interquartile range for short), is a measure of dispersion in the descriptive statistics . If you sort a sample according to size, the interquartile range indicates how wide the interval is in which the middle 50% of the sample elements lie.

definition

A sample is given

${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$

with elements sorted by size. So it applies ${\displaystyle n}$

${\displaystyle x_{1}\leq x_{2}\leq \dots \leq x_{n}}$.

Furthermore, let the lower quartile and the upper quartile. These are defined as ${\displaystyle x_{0{,}25}}$ ${\displaystyle x_{0{,}75}}$

${\displaystyle x_{0{,}25}={\begin{cases}{\tfrac {1}{2}}(x_{n\cdot 0{,}25}+x_{n\cdot 0{,}25+1}),&{\text{wenn }}n\cdot 0{,}25{\text{ ganzzahlig,}}\\x_{\lfloor n\cdot 0{,}25+1\rfloor },&{\text{wenn }}n\cdot 0{,}25{\text{ nicht ganzzahlig.}}\end{cases}}}$and . ${\displaystyle x_{0{,}75}={\begin{cases}{\tfrac {1}{2}}(x_{n\cdot 0{,}75}+x_{n\cdot 0{,}75+1}),&{\text{wenn }}n\cdot 0{,}75{\text{ ganzzahlig,}}\\x_{\lfloor n\cdot 0{,}75+1\rfloor },&{\text{wenn }}n\cdot 0{,}75{\text{ nicht ganzzahlig.}}\end{cases}}}$

Here is the rounding function . It rounds each number down to the nearest whole number. For example, and . ${\displaystyle \lfloor x\rfloor }$ ${\displaystyle x}$ ${\displaystyle \lfloor 1{,}2\rfloor =1}$ ${\displaystyle \lfloor 3{,}99\rfloor =3}$

The interquartile range is then defined as

${\displaystyle \operatorname {IQA} =x_{0{,}75}-x_{0{,}25}}$

and is therefore exactly the difference between the upper and lower quartile.

example

Look at the sample

${\displaystyle {\tilde {x}}=(25;28;4;28;19;3;9;17;29;29)}$

with elements. If you sort the elements according to size, you get ${\displaystyle n=10}$

${\displaystyle x=(3;4;9;17;19;25;28;28;29;29)}$.

To determine the lower quantile, one calculates what is not an integer. Therefore, according to the definition given above ${\displaystyle n\cdot 0{,}25=2{,}5}$

${\displaystyle x_{0{,}25}=x_{\lfloor n\cdot 0{,}25+1\rfloor }=x_{\lfloor 2{,}5+1\rfloor }=x_{\lfloor 3{,}5\rfloor }=x_{3}=9}$.

Analogously follows

${\displaystyle x_{0{,}75}=x_{\lfloor n\cdot 0{,}75+1\rfloor }=x_{\lfloor 7{,}5+1\rfloor }=x_{\lfloor 8{,}5\rfloor }=x_{8}=28}$.

This gives for the interquartile range

${\displaystyle \operatorname {IQA} =x_{0{,}75}-x_{0{,}25}=28-9=19}$.

Constructive terms

Based on the interquartile range, the mean interquartile range is defined, which is abbreviated to MQA or QD (after the English term quartile deviation ). It is defined as

${\displaystyle \operatorname {MQA} ={\frac {1}{2}}\operatorname {IQA} ={\frac {1}{2}}\left({x_{0{,}75}-x_{0{,}25}}\right)}$.

In the example above, the mean interquartile range would be

${\displaystyle \operatorname {MQA} ={\frac {1}{2}}\cdot 19=9{,}5}$.

Individual evidence

1. ↑ ^{a b c d e} Thomas Cleff: Descriptive Statistics and Exploratory Data Analysis . A computer-aided introduction with Excel, SPSS and STATA. 3rd, revised and expanded edition. Springer Gabler, Wiesbaden 2015, ISBN 978-3-8349-4747-5 , p. 54 , doi : 10.1007 / 978-3-8349-4748-2 .
2. ↑ Norbert Henze: Stochastics for beginners . An introduction to the fascinating world of chance. 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 32 , doi : 10.1007 / 978-3-658-03077-3 .
3. ^ Eric W. Weisstein : Interquartile Range . In: MathWorld (English).
4. ^ Eric W. Weisstein : Quartile Deviation . In: MathWorld (English).