Sample means

The sample mean , also known as the sample mean , arithmetic mean or arithmetic mean , is a special estimation function in mathematical statistics . It plays an important role in estimating the expected value of unknown probability distributions and also occurs in the construction of confidence intervals and statistical tests .

Its empirical counterpart is the empirical mean . It corresponds to a realization of the sample mean.

definition

Let be independent and identically distributed random variables . Then the sample mean is defined as ${\ displaystyle X_ {1}, X_ {2}, \ dots, X_ {n}}$

{\ displaystyle {\ overline {X}} = {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} X_ {i}}

.

Sometimes the number of random variables is also noted as an index, especially when considering limit values. The sample mean is then noted as. ${\ displaystyle {\ overline {X}} _ {n}}$

properties

The sample mean is the first sample moment and thus the expected value of the empirical distribution . It follows directly from this that the sample mean is the moment estimator for the expected value (for a derivation see moment method # Estimation of the expected value ).

The estimator thus obtained is expected faithful to the unknown expectation and thus has a distortion of zero. This follows directly from the linearity of the expected value , for it is ${\ displaystyle \ mu}$

{\ displaystyle \ operatorname {E} ({\ overline {X}}) = \ operatorname {E} \ left ({\ frac {1} {n}} \ sum _ {i = 1} ^ {n} X_ { i} \ right) = {\ frac {1} {n}} \ operatorname {E} \ left (\ sum _ {i = 1} ^ {n} X_ {i} \ right) = {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} \ operatorname {E} (X_ {i}) = {\ frac {1} {n}} \ cdot n \ cdot \ mu = \ mu}

,

what exactly corresponds to the expected value of the underlying probability measure. Furthermore, due to the central limit theorem , the sample mean is always asymptotically normally distributed and, according to the strong law of large numbers, also strongly consistent .

Next applies to independent ${\ displaystyle X_ {i}}$

{\ displaystyle \ operatorname {Var} ({\ overline {X}}) = \ operatorname {Var} \ left ({\ frac {1} {n}} \ sum _ {i} X_ {i} \ right) = {\ frac {1} {n ^ {2}}} \ operatorname {Var} \ left (\ sum _ {i} X_ {i} \ right) = {\ frac {1} {n ^ {2}}} \ sum _ {i} \ operatorname {Var} (X_ {i}) = {\ frac {1} {n ^ {2}}} \ cdot n \ cdot \ sigma ^ {2} = {\ frac {\ sigma ^ {2}} {n}}}

Web links

Eric Weisstein: Sample Mean on MathWorld (English)

Individual evidence

↑ Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 99 , doi : 10.1007 / 978-3-642-45387-8 .
↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 5 , doi : 10.1007 / 978-3-642-17261-8 .
↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 26 , doi : 10.1007 / 978-3-642-17261-8 .
↑ Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 246 , doi : 10.1007 / 978-3-642-45387-8 .

[Kusolitsch99-1] Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 99 , doi : 10.1007 / 978-3-642-45387-8 .

[Czado5-2] Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 5 , doi : 10.1007 / 978-3-642-17261-8 .

[Czado26-3] Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 26 , doi : 10.1007 / 978-3-642-17261-8 .

[Kusolitsch246-4] Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 246 , doi : 10.1007 / 978-3-642-45387-8 .