Stratified sample

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)

References

[a1]	W.G. Cochran, "Sampling techniques" , Wiley (1977)