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Difference between revisions of "Stratified sample"

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$$
 
$$
  
where the  $  i $-
+
where the  $  i $-th sample  $  X _ {i1} \dots X _ {in _ {i}  } $
th sample  $  X _ {i1} \dots X _ {in _ {i}  } $
 
 
contains only those elements of the original sample which have the mark  $  i $.  
 
contains only those elements of the original sample which have the mark  $  i $.  
 
As a result of this decomposition, the original sample becomes stratified into  $  k $
 
As a result of this decomposition, the original sample becomes stratified into  $  k $
 
strata  $  X _ {i1} \dots X _ {in _ {i}  } $,  
 
strata  $  X _ {i1} \dots X _ {in _ {i}  } $,  
 
$  i = 1 \dots k $,  
 
$  i = 1 \dots k $,  
where the  $  i $-
+
where the  $  i $-th stratum contains information about the  $  i $-th mark. This notion gives rise, for example, to realizations of the  $  X $-component of a two-dimensional random variable  $  ( X, Y) $
th stratum contains information about the  $  i $-
 
th mark. This notion gives rise, for example, to realizations of the  $  X $-
 
component of a two-dimensional random variable  $  ( X, Y) $
 
 
whose second component  $  Y $
 
whose second component  $  Y $
 
has a discrete distribution.
 
has a discrete distribution.

Latest revision as of 14:55, 1 March 2022


A sample which is broken up into several samples of smaller sizes by certain distinguishing marks (characteristics). Let each element of some sample of size $ N \geq 2 $ possess one and only one of $ k \geq 2 $ possible marks. Then the original sample can be broken into $ k $ samples of sizes $ n _ {1} \dots n _ {k} $, respectively $ ( n _ {1} + \dots + n _ {k} = N) $:

$$ \begin{array}{c} X _ {11} \dots X _ {1n _ {1} } , \\ X _ {21} \dots X _ {2n _ {2} } , \\ {} \dots \dots \dots \\ X _ {k1} \dots X _ {kn _ {k} } , \\ \end{array} $$

where the $ i $-th sample $ X _ {i1} \dots X _ {in _ {i} } $ contains only those elements of the original sample which have the mark $ i $. As a result of this decomposition, the original sample becomes stratified into $ k $ strata $ X _ {i1} \dots X _ {in _ {i} } $, $ i = 1 \dots k $, where the $ i $-th stratum contains information about the $ i $-th mark. This notion gives rise, for example, to realizations of the $ X $-component of a two-dimensional random variable $ ( X, Y) $ whose second component $ Y $ has a discrete distribution.

References

[1] S.S. Wilks, "Mathematical statistics" , Wiley (1962)

Comments

References

[a1] W.G. Cochran, "Sampling techniques" , Wiley (1977)
How to Cite This Entry:
Stratified sample. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stratified_sample&oldid=49608
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article