Trimmed mean

The (α-) trimmed mean, also called (α-) trimmed mean or (α-) trimmed mean , is a location parameter in descriptive statistics and thus provides a measure of where the sample is located. The trimmed mean is closely related to the arithmetic mean . In contrast to this, the trimmed mean ignores a certain proportion of the largest and smallest sample elements. The trimmed mean is therefore more robust than the arithmetic mean, i.e. it changes less when the sample is modified.

definition

It denotes the rounding function , which assigns the next smaller or the same whole number to each number. So it is and . ${\displaystyle \lfloor \cdot \rfloor }$ ${\displaystyle \lfloor 1{,}8\rfloor =1}$ ${\displaystyle \lfloor 3{,}1\rfloor =3}$

A sample is given

${\displaystyle x=(x_{1};x_{2};\dotsc ;x_{n})}$

with elements. Be ${\displaystyle n}$

${\displaystyle x_{\text{sort}}=(x_{(1)};x_{(2)};\dotsc ;x_{(n)})}$

the sample sorted according to size and be

${\displaystyle \alpha \in (0,{\tfrac {1}{2}})}$

a real number. Set

${\displaystyle k=\lfloor \alpha \cdot n\rfloor }$.

Then is called

${\displaystyle x_{t;\alpha }={\frac {1}{n-2k}}\sum _{i=k+1}^{n-k}x_{(i)}}$

the -trimmed mean. It corresponds to the arithmetic mean in which a proportion of the sample elements, i.e. not included in the calculation: the proportion of the largest sample elements and the proportion of the smallest sample elements. Values between and are common . ${\displaystyle \alpha }$ ${\displaystyle 2\alpha }$ ${\displaystyle 200\alpha \;\%}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle 0{,}1}$ ${\displaystyle 0{,}2}$

example

Look at the sample

${\displaystyle x=(5;30;29;15;25;5;13;28;24;29)}$.

It consists of 10 elements, so is . Sorting by size gives ${\displaystyle n=10}$

${\displaystyle x_{\text{sort}}=(5;5;13;15;24;25;28;29;29;30)}$.

If you choose , the largest 10% and the smallest 10% of the sample are not included. It is ${\displaystyle \alpha =0{,}1}$

${\displaystyle k=\lfloor 0{,}1\cdot 10\rfloor =1}$,

because a proportion of 0.1 for 10 sample elements corresponds to exactly one element. So the 0.1 trimmed mean is

${\displaystyle {\begin{aligned}x_{t;0{,}1}&={\frac {1}{10-2\cdot 1}}\sum _{i=1+1}^{10-1}x_{(i)}\\&={\frac {1}{8}}(5+13+15+24+25+28+29+29)\\&={\frac {168}{8}}=21\end{aligned}}}$

In particular, the largest value of the sample could be replaced by any value without affecting the 0.1-trimmed mean, since the largest value is always not included in the calculation. In general, outliers (up or down) only affect the trimmed mean if their proportion of the sample is greater than . ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

Individual evidence

1. ↑ 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. 30 , doi : 10.1007 / 978-3-658-03077-3 .
2. ↑ Ulrich Krengel : Introduction to Probability Theory and Statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , p. 170-171 , doi : 10.1007 / 978-3-663-09885-0 .