Namespaces
Variants
Actions

Difference between revisions of "Stratified sample"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Undo revision 48869 by Ulf Rehmann (talk))
Tag: Undo
m (tex encoded by computer)
Line 1: Line 1:
A [[Sample|sample]] which is broken up into several samples of smaller sizes by certain distinguishing marks (characteristics). Let each element of some sample of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904401.png" /> possess one and only one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904402.png" /> possible marks. Then the original sample can be broken into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904403.png" /> samples of sizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904404.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904405.png" />:
+
<!--
 +
s0904401.png
 +
$#A+1 = 17 n = 0
 +
$#C+1 = 17 : ~/encyclopedia/old_files/data/S090/S.0900440 Stratified sample
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904406.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904407.png" />-th sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904408.png" /> contains only those elements of the original sample which have the mark <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s0904409.png" />. As a result of this decomposition, the original sample becomes stratified into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044010.png" /> strata <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044012.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044013.png" />-th stratum contains information about the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044014.png" />-th mark. This notion gives rise, for example, to realizations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044015.png" />-component of a two-dimensional random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044016.png" /> whose second component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090440/s09044017.png" /> has a discrete distribution.
+
A [[Sample|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.G. Cochran,  "Sampling techniques" , Wiley  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.G. Cochran,  "Sampling techniques" , Wiley  (1977)</TD></TR></table>

Revision as of 14:55, 7 June 2020


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=49450
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article